\input texinfo

@c set spell spelllang=en_us
@c 

@setfilename apl.info
@documentencoding UTF-8
@settitle GNU APL

@c ============================================================================
@c ====== Part 2: Summary Description and Copyright ===========================
@c ============================================================================

@copying
This is the reference manual for GNU APL, an implementation of ISO
standard 13751 "Programming Language APL, Extended".

Copyright @copyright{} 2013 Free Software Foundation, Inc.
@end copying

@dircategory GNU programming tools
@direntry
* APL: (apl).                  the programming language APL - ISO 13751.

@end direntry

@c ============================================================================
@c ====== Part 3: Titlepage, Contents, Copyright ==============================
@c ============================================================================

@titlepage
@title GNU APL Reference Manual

@c The following two commands start the copyright page.
@page
@vskip 0pt plus 1filll
@insertcopying
@end titlepage

@c Output the table of contents at the beginning.
@contents

@c ============================================================================
@c ====== Part 4: ‘Top’ Node and Master Menu ==================================
@c ============================================================================

@node Top
@top GNU APL Reference Manual
@ifnottex

@today{}.
This manual briefly documents GNU APL, an interpreter for APL as described
in ISO standard 13751, aka. "Programming Language APL, Extended".

This manual for GNU APL can be distributed under the terms of the
GNU Free Documentation License, *Note Chapter 8.

This manual does NOT describe APL itself; there exist a number of good books
and texts about APL. We would like to refer the reader to the following:

    ISO Standard for APL: @*
    www.math.uwaterloo.ca/~ljdickey/apl-rep/docs/is13751.pdf

Note that the file @b{is13751.pdf} above is compressed with @b{gzip},
therefore it must be expanded with e.g. @b{gunzip} before it can be read.

    More pointers to APL resources on the Web: @*
    ftp://rtfm.mit.edu/pub/usenet-by-group/comp.lang.apl/APL_language_FAQ

@end ifnottex


@c ============================================================================
@c ====== Part 5: The Body of the Document ====================================
@c ============================================================================

@c ============================================================================
@node Installing and Starting GNU APL
@chapter Installing and Starting GNU APL

@node Installing APL
@section Installing APL

APL is built and installed like this:

@itemize
@item download the tar file containing GNU APL, from
 @uref{ftp://ftp.gnu.org/gnu/apl} or from one of its mirrors
 (see @uref{http://www.gnu.org/prep/ftp.html}), and
 un-tar the file.

@item read (and follow) the instructions given in file README in the
 unpacked directory.
@end itemize

For the experienced, but impatient reader: it is the well-known sequence

@example
$ ./configure [configure-option]...
$ make
$ sudo make install
@end example

On success, @b{make} produces an executable filee named @b{apl} in
sub-directory @b{src} while @b{make install} installs that file (in
@b{/usr/local/bin}) as well as some other files in other locations
(that can be controlled via @b{configure-options}).

File README contains further instructions about the installation GNU APL.

@node Starting APL
@section Starting APL

Last things first: before explaining how to start APL, it is important to
 remember how to stop (i.e. exit) it. Neither @b{^C} nor @b{^D} will stop
APL - these serve other purposes. Instead, you leave APL with the command
 @b{)OFF} or @b{)off} (on a separate line) like this:

@example
      )OFF
@end example


If APL is computing a function (and possibly caught in an endless loop),
then you may have to press @b{^C} (this is called @i{ATTENTION} in APL) to
return to APL's command mode so that the @b{)OFF} command can be entered.
In some circumstances it may be necessary to press @b{^C} twice within a
short time interval (this is called @i{INTERRUPT} in APL).

Using the )OFF command will not automatically save your work, however, a
history of your GNU APL commands will have been preserved. Having that said,
APL is started like every other program - by entering its name and optional
command line parameters, for example:

@example
$ apl
@end example

or:

@example
$ apl -id 1001
@end example

GNU APL is script-able; a text file whose first line looks like this (assuming
the APL interpreter binary is called 'apl' and is located in the current
directory):

@example
#! ./apl 
@end example

or (if the APL interpreter binary is not in the current directory but in
/mypath/apl):

@example
#! /mypath/apl
@end example

The path to the 'apl' binary can be missing (like in the first example),
relative, or absolute (second example). The exact details of how the first
line of a GNU APL script shall look vary slightly among operating systems.
Please consult the @b{info} (or @b{man}) pages for @b{path_resolution}
and/or @b{execve}
on your operating system. After the name of the binary, optional GNU APL
command line options, usually @b{--script}, can be provided.

The text file must, of course, have execute permission, and should be ASCII
or, more likely, UTF-8 encoded. The APL characters in the script shall be
those defined in the Unicode character set (most of them in the U+2200 -
U+23FF range).

@node Command Line Options
@section Command Line Options

GNU APL understands the following command line options:

@itemize @bullet

@item
@b{-C new_root}

Perform chroot("new_root") followed by chdir("/"). This restricts the access
of the process running GNU APL to files in or below directory new_root, and
it also changes the current directory (which could have resided above
new_root before the chroot() was executed).

The -C option is intended to be a security feature for GNU APL interpreters
that are facing hostile environments such as the public Internet.

NOTES:
@itemize @bullet

@item
For -C to work, new_root needs to contain a minimal set of binaries, in
particular a shell, and possibly libraries needed by the shell. Consult
'info chroot invocation' for issues to consider and common pitfalls.

@item
GNU APL processes the -C option (i.e. it calls chroot("new_root")) before
all other command line options. As a consequence, file names in other command
line options of GNU APL are also affected by -C. That means that the file
names in command line options are being interpreted relative to new_root
and @b{not} relative to the current directory of the process that had
started GNU APL.

@item
GNU APL may automatically start other processes such as APserver and friends
after processing the -C option. These processes (and the libraries that they
depend on) should reside in the proper directory below new_root. For example,
with the default configuration of GNU APL, GNU APL installs itself in
/usr/local/bin and also expects APserver in the same directory. After -C
new_root, however, /usr/local/bin is no longer accessible and one would need
to copy /usr/local/bin/APserver to new_root/usr/local/bin/APserver

@item
Depending on your platform, the process using @b{-C} may need root privileges.

@end itemize

@item @code{--cfg}

Show ./configure options that were used to configure GNU APL, and exit.

@item @b{--[no]Color}

Start with ]COLOR ON [OFF].

@item
@b{--CPU_limit_secs sec}

Set CPU time limit to @i{sec} seconds.

@item
@code{-d}

Run the APL interpreter (or APL script) in the background (i.e. as a daemon).
For this to work you need to provide some input to the background process,
e.g. via the -f option.

@item
@code{--emacs}

Run in (old) Emacs mode.

@item
@code{--emacs_arg arg}

Run in (new) Emacs mode with argument arg.

@item
@code{--eval line}

Evaluate one APL @i{line} and exit (-s without -f is implied). This option
can be given several times; in that case several lines will be executed in
command line order before GNU apl exits. Keep in mind that command line
options are normally processed by the shell before being passed to apl.
Therefore quoting the argument of --eval may often be necessary and it is
common wisdom to always quote the --eval argument.

@item
@code{-f file}

Read input from @i{file} rather than from the keyboard. When the end of
the file is reached, input is switched back to the keyboard. 
If you want to terminate the APL interpreter after executing the file,
then use )OFF as last line in the file or include the --OFF option.

@item
@code{--gpl}

Show GNU APL license (GPL) and exit.

@item
@code{-L wsname}

)LOAD wsname on start-up.

@item
@code{--LX expr}

Execute @b{expr} first. The workspace behaves as if ⎕LX (latent
expression) were set to @b{expr} in the workspace. This can be used,
for example, to start the same workspace with different start-up values.

@item
@code{-h}, @code{--help}

Print all command line options with a brief hint about what they do.

@item
@code{--id proc}

Use processor ID proc for this interpreter. If no ID is provided, then the
first unused ID > 1000 is taken by this interpreter and the ID becomes used
as long as the interpreter runs. Processor IDs are used by shared variables
to identify share partners.

@item
@code{-l num}

Turn logging facility @b{num} ON (provided that dynamic logging was
./configure'd). The logging facility 37 (start-up messages) is of particular
importance for troubleshooting and it works even if dynamic logging
was not ./configure'd.

@item
@code{--mem [memory-limit]}

Tell the interpreter not to use more than memory-limit bytes of RAM. By
using this option, the user is fully responsible for ensuring that the
specified amount of memory will always be available. The following rules
should be observed.

The interpreter will exit at start-up if @code{-mem} is used and:

@itemize
@item the platform on which the interpreter runs has no /proc/meminfo, or
@item the platform has no /proc/sys/vm/overcommit_memory, or
@item /proc/sys/vm/overcommit_memory is not 2 (aka. 'never overcommit')
@end itemize

On GNU/Linux systems these conditions are normally satisfied, but the root
user has to set /proc/sys/vm/overcommit_memory to 2 which differs from the
default value 0 (aka. overcommit allowed).

If no memory-limit is given, then a memory-limit of 50% is used.

If a memory-limit is provided then it must have a unit of %, kB, MB, or GB.
If the unit is % then the limit is computed as that percentage (between 5%
and 95%) of parameter 'MemFree:' in /proc/meminfo. Otherwise the limit
is the given amount in kB, MB, or GB.

For example:

@itemize
@item @code{--mem}      (50% of MemFree: in /proc/meminfo are guaranteed)
@item @code{--mem 80%}  (80% of MemFree: in /proc/meminfo are guaranteed)
@item @code{--mem 5G}   (5 GB are guaranteed)
@end itemize

WARNING: The memory-limit is checked against parameter 'MemFree' in
/proc/meminfo when GNU APL starts, but this does not protect against
other processes consuming the available free memory sometime later.

If that happens (and according to the rules above the user has the
responsibility to prevent it), then ⎕WA becomes unreliable and the
interpreter may crash badly (i.e. without a WS FULL error, and without
any chance to )SAVE the workspace) when the available memory is exhausted.

@item
@code{--echoCIN}

Copy the input line (after editing) to stdout. For creating session logs.

@item
@code{--noCIN}

Do not echo stdin to stdout. Almost a must for scripting (unless you
intend to troubleshoot a script).

@item
@code{--to_COUT}

normally GNU APL writes its output to stderr (i.e. file descriptor 2) so that,
when GNU APL is started in a script, the output of the script appears on
stdout (i.e. file descriptor 1) while the output of GNU APL appears on stderr.

This option redirects the stderr output of GNU APL to stdout. The same effect
can be achieved with the option "OUTPUT-TO-COUT Yes" in a preferences file.

@item
@code{--tcp_port PORT}

This option starts GNU APL as a server that listens on TCP port PORT.
Every TCP connection accepted by the server forks a new GNU APL instance
which has its stdin, stdout, and stderr redirected to the TCP connection.

WARNING: This option is dangerous if PORT is directly exposed to the Internet!

@item
@code{--noCONT}

Do not load a SETUP or CONTINUE workspace on start-up.

@item
@code{--OFF}

This option causes GNU APL to perform an automatic )OFF command after the
last line of the last input file (as per -f option) was executed.

@item
@code{--PW COLS}

set the initial value of ⎕PW to COLS (min. 30, max. 10000)

@item
@code{--[no]SV}

do [not] start APserver (a shared variable server) on start-up. This disables
communication with other workspaces or auxiliary processors through
shared variables.

@item
@code{-p N}

use profile number N in preferences files. A preference file may contain
several sets of settings for different purposes; the profile number selects
one of these sets.

@item
@code{--par pproc}

Use processor parent ID pproc (default: no parent ID).

@item
@code{--rawCIN}

Do not emit ESC sequences. Normally ESC sequences are emitted for colored
output and during line editing. In scripts, however, ESC sequences usually
are not wanted and can be turned off with this option.

@item
@code{-s}, @code{--script}

This option is an abbreviation for:
 @code{--silent} @code{--noCIN} @code{--noCONT} @code{-f -}
(which is a typical combination of options for APL scripts).

@item
@code{-q}, @code{--silent}

Suppress printing of the GNU APL welcome message. Useful for scripts.

@item
@code{--safe}

disable shared variables and native functions

@item
@code{--show_bin_dir}

Display the binary directory where, according to ./configure, the programs
apl, APserver, AP100, and AP210 are installed. Then exit.

@item
@code{--show_doc_dir}

Display the directory where, according to ./configure, documentation files
for GNU APL are installed. Then exit.

@item
@code{--show_etc_dir}

Display the system configuration directory where, according to ./configure,
the preferences file for GNU APL is installed. Then exit.

@item
@code{--show_lib_dir}

Display the library directory where, according to ./configure, shared library
files and the workspaces shipped with GNU APL are installed. Then exit.

@item
@code{--show_src_dir}

display the source directory where, according to ./configure, GNU APL
was compiled. Then exit. This can be used, for example, by native functions
that are built outside the GNU APL source tree to find GNU APL header files
that are needed to compile the native function.

@item
@code{--show_all_dirs}

Display all the directories above. Then exit.

@item
@code{-T testcases ...}
run testcases. Testcases are text files that contain both input to the APL
interpreter and the expected response from the interpreter. The output from
the interpreter is compared with the expected output in the testcase file(s)
and differences are marked. In addition a summary file is created that tells
whether or not each of the testcases was successful.

@item
@code{--TM mode}

Test mode. This option specifies how the interpreter shall behave
when running a number of testcases (as specified with the @code{-T} option).

@code{--TM 0} (default) run all testcases and exit after the last testcase.

@code{--TM 1} like @code{--TM 0} if no error was detected.
However, if one of the testcases has failed, then the interpreter
does not exit so that the user can investigate the state of APL
(SI, variable values, etc.).

@code{--TM 2} like @code{--TM 1}, but stay in the interpreter even if
all testcases have passed. This can be useful for quickly bringing the
interpreter into a specific state and continue manual troubleshooting
from that state.

@code{--TM 3} like @code{--TM 1}, but stop testcase execution after the
first failed testcase (i.e. do not exit).

@code{--TM 4} like @code{--TM 3}, but exit after the first failed
testcase. This is useful for automatic regression tests, where no errors
are expected.

@item
@code{--TR}

Executes test case files in random order.

@item
@code{--TS}

Normally, when the interpreter is run with the @code{T} option,
an existing summary.log file is overwritten without notice.
This option causes new test results to be appended to a possibly
existing summary.log instead of overwriting it.

@item
@code{-v}, @code{--version},

Show version information and exit.

@item
@code{-u UID}

Run as user with UID 0. This option can only be used by the root user (who
then wants to run as a different user).

@item
@code{-w milli}

wait milli milliseconds at start-up. Useful to give other programs that are
started together with this interpreter time to initialize themselves.

@item @code{|  +APPOPT|}
@item @code{| ++APPOPT ARG1|}
@item @code{|+++APPOPT ARG1 ARG2|}
@item ...

   Those command line options above that start with + are understood by the
   GNU APL binary and their arguments must follow the description given for
   them. In addition the interpreter also accepts command line options that
   start with +. However, these options are not checked by the interpreter
   in any way, but are merely copied to @b{⎕ARG} (see below). In these options,
   APPOPT, ARG1, ARG2, ... are arbitrary strings that should not contain any
   whitespace characters. The purpose of these options is to control aspects
   of the APL application from the command line.

@item
@code{-}

End of command line options for the interpreter. GNU APL provides the system
variable ⎕ARG that returns all command line options with which the GNU APL
interpreter was invoked (similar to variable argv in main(int argc,
char * argv[]) in C/C++). Option @code{-} can be used to separate command line
options for the APL interpreter from command line options understood by APL
applications.

Thus,

@code{(⎕ARG ⍳ ⊂'--') ↑ ⎕ARG} returns the options for the APL interpreter, while

@code{(⎕ARG ⍳ ⊂'--') ↓ ⎕ARG} returns the options for the APL application.

All command line options after @code{-} are ignored by the interpreter
(except for including them in ⎕ARG).

@end itemize

@node Configuration File for GNU APL
@section Configuration File for GNU APL

The default values for some of the command line options discussed in the
previous section can be set in a configuration file for GNU APL. The name
of the configuration file is 'preferences' and it should live in one of
the following directories:

@itemize

@item
in the sub-directory gnu-apl.d of the system configuration directory, or

@item
in the sub-directory .config/gnu-apl of the user's home directory (as per
$HOME).

@end itemize

The system configuration directory is usually /etc or /usr/local/etc and is
configurable via @option{./configure --sysconfdir}. An empty (i.e. most
settings commented out) preferences file is also installed in the
system configuration directory when GNU APL is installed. You can edit
it, use it as a template or read it to see which options can be controlled.

If file 'preferences' exists in both directories, the settings in
$HOME/.config/gnu-apl/preferences override settings in, for example,
/etc/gnu-apl.d/preferences.

Command line options in turn override settings in 'preferences' files.

@node File Names and Paths
@section File Names and Paths

The GNU APL interpreter is a binary file named @b{apl}. It is usually
installed in directory @b{/usr/bin/} or in @b{/usr/local/bin/}. The location
where @b{apl} is installed can be changed via @b{./configure} options
(see file @b{INSTALL}).

GNU APL understands 4 file types:

@enumerate
@item @i{APL workspaces} that can be manipulated with the @b{)LOAD}, @b{)SAVE},
      @b{)COPY}, and @b{)DROP} commands. APL workspaces are XML files and shall
      have a file extension of @b{.xml} to be accepted by GNU APL. APL
      workspaces can only be exchanged between machines that all run GNU APL.

@item @i{APL exchange files} can be manipulated with the @b{)IN} and @b{)OUT}
      commands. APL exchange files are text files in @b{⎕TF} format defined
      by IBM (basically APL expressions that create variables or functions)
      and shall have a file extension of @b{.atf} to be accepted by GNU APL.
      APL exchange files can be exchanged between machines running APL
      interpreters from different vendors. The @b{⎕TF} format can be easily
      emulated on machines that do not not provide compatible @b{)IN} and
      @b{)OUT} commands.

@item @i{APL scripts} consist of APL commands and APL expressions (including
      function definition via @b{∇}) like they would be entered by the user.
      APL scripts should have a file extension of @b{.apl} but other extensions
      are also accepted by GNU APL. APL scripts are, for example, the files
      expected for the -f command line option. A workspace can be written in this
      format with the i@b{)DUMP} command and can become an active workspace
      using the @b{)LOAD} command.

@item @i{APL testcase files} are similar to APL scripts, but in addition to
      the APL commands and expressions they also contain the expected output
      from the commands.
      APL testcase files normally have a file extension of @b{.tc} for normal
      (functional) testcases and @b{.pt} for performance testcases.
      APL testcase files are, for example, the files expected for the
      @b{-T} command line option.
@end enumerate

The following APL commands are related to file names:

@itemize
@item )LOAD [lib] name[.xml|.apl]
@item )SAVE [lib] [name[.xml]]
@item )COPY [lib] [name[.xml|.apl]]
@item )PCOPY [lib] [name[.xml|.apl]]
@item )DROP [lib] [name[.xml|.apl]]
@item )IN   [lib] name[.atf]
@item )PIN   [lib] name[.atf]
@item )OUT  [lib] name[.atf]
@item )DUMP [lib] [name[.apl]]
@item )LIB  [lib]
@item )LIBS [new-lib-root]
@item )WSID       [name]
@end itemize

The rules for how file names are constructed from the argument(s) of an APL
command above are:

@enumerate
@item command arguments shown in brackets are optional.
@item @b{lib} is a number from 0 to 9. If lib is not present then 0 is taken
      as default.
@item if the @b{name} is optional and missing then the workspace ID (the name
      set with the @b{)WSID} command) is used.
@item if the file extension (i.e. @b{.apl}, @b{.xml} or @b{.atf}) is missing
      then it is appended automatically to name.
@end enumerate

If the @b{name} starts with '/' then it is taken as an absolute path to
the file (an absolute file name) and no further computation is done
with the name.

Otherwise @b{name} is a relative path which is relative to some directory
library-root and a sub-directory of library-root that is determined by
the @b{lib} number. The library numbers 0-9 correspond to the following
directories:

    0: library-root/workspaces/

    1: library-root/wslib1/

    2: library-root/wslib2/

    ...

    9: library-root/wslib9/

The command @b{)LIBS} without arguments shows the mapping between library
numbers and paths. The command )LIBS with an argument sets a new lib-root.

The command )LIB [lib] shows the files in library (i.e. directory) lib.

The directory library-root is computed as follows when the interpreter starts:

If environment variable @b{APL_LIB_ROOT} is defined, then its value is used
as the library-root. Otherwise the path from the current directory (".") up to
the root directory ("/") is searched until a directory containing two
sub-directories @b{workspaces} and @b{wslib1} is found. Normally @b{workspaces}
and @b{wslib1} are directories, but for the computation of library-root files
suffice.

If such a directory is found, then it is used as library-root; otherwise the
current directory (i.e. ".") is used and converted to an absolute path.

For example, if library-root is "." then the command

@example
)LOAD 2 test
@end example

will try to load the workspace file

./wslib2/test.xml

Using a library root implies that all 10 library directories are contained
in the same directory. This is good enough for single-user environments but
is often not adequate for multi-user environments where some directories
are not writable by users and different users have different home directories.

For that reason the above library root scheme can be overridden by the
GNU APL configuration files (named @b{preferences}). In these files you can
un-comment any of the @b{LIBREF-0} to @b{LIBREF-9} settings (which correspond
to library numbers 0 to 9) and provide your own paths. The library numbers
that are @b{NOT} overridden in a @b{preferences} file still follow the
library root scheme.

@c ============================================================================
@node Non-standard GNU APL Features
@chapter Non-standard GNU APL Features

There are a few hopefully useful features in GNU APL:

@c ----------------------------------------------------------------------------
@node APL Scripting
@section APL Scripting

@node Introduction and Terminology
@subsection Introduction and Terminology

A @i{script} is a text file that contains the source code of an interpreted
programming language. An @i{APL script} is a text file that contains @i{APL
commands} (like @b{)LOAD} or @b{)OFF}) and @i{APL expressions} like
@b{1 2 3+ 4 5 6}. In GNU APL a script can also contains instructions for
the well-known ∇-editor in order to create defines functions.

On most platforms are the lines of a script separated by the single ASCII
character linefeed (@b{LF}, ASCII code 10); on Microsoft operating systems
the lines are separated by two characters carriage return (@b{CR}, ASCII
code 13) and (@b{LF}. ASCII code 10). GNU APL ignores @b{CR} in most contexts
so that the same APL script can be executed on both kinds of platforms.

The first line of a script is called the @i{shebang line}. The shebang line
determines how and with which arguments the script shall be executed. When
the operating system (or, more precisely: a shell of the operating system)
executes a script then it parses the shebang line and calls a function
@b{execve} whose arguments are the result of that parsing. Much of how
GNU APL handles APL scripts can be learned by consulting the 
@b{man} page for function @b{execve}. To quote that @b{man} page:

@example@
    ...

Interpreter scripts
       An  interpreter script is a text file that has execute permission enabled and whose
       first line is of the form:

           #!interpreter [optional-arg]

       The interpreter must be a valid pathname for an executable file.

       If the pathname argument of execve() specifies an interpreter script,
       then interpreter will be invoked with the following arguments:

           interpreter [optional-arg] pathname arg...

       where pathname is the absolute pathname of the file specified as the
       first argument of execve(), and arg... is the series of words pointed
       to by the argv argument of execve(), starting at argv[1]. Note that
       there is no way to get the argv[0] that was passed to the execve()
       call.

    ...
@end example

It is important to note the @i{singular} in @b{[optional-arg]} above.
Traditionally the interpreter would have been @b{/usr/local/bin/apl}
(if GNU APL was installed in directory @b{/usr/local/bin/} or else
@b{/usr/bin/apl} (if GNU APL was installed in directory @b{/usr/bin/}).
These days, however, it has become common practice not to call the
interpreter directly, but to call the program @b{/usr/bin/env} as the
interpreter and passing the real interpreter (i.e. /usr/local/bin/apl or
 apl) to @b{/bin/env}. For example:

@example
Traditional shebang line (full path to the interpreter):

    #!/usr/local/bin/apl --script

Traditional shebang line (interpreter name only):

    #!apl --script

Contemporary shebang line:

    #!/bin/env -S apl --script
@end example

For a single argument the shebang lines above have essentially the same
effect. The long form (full path to the interpreter) is needed only if the
interpreter interpreter can not be found in one of the directories listed
in the environment variable @b{$PATH}, for example if a user installs GNU APL
in a non-standard directory (i.e. other than in @b{/bin}, @b{/usr/bin}, or
@b{/usr/local/bin}).

A minor disadvantage of using @b{env} as the interpreter is that
@b{execve()} is called twice: first by the shell to run @b{env} and then
by @b{env} to run @b{apl}.

A potential advantage of using @b{/bin/env} as the interpreter is the @b{-S}
option of @b{/bin/env} which works around the single argument limitation
of @b{execve}. However, this advantage only to programs other than
GNU APL because GNU APL comes with a built-in capability that is
essentially doing the same as the @b{-S} option of @b{/bin/env}. Another,
more important, aspect is portability. On a GNU/Linux system the standard
shells (@b{bash}, @b{dash}) seem to pass multiple arguments in the shebang
line to the interpreter in a way that GNU APL understands. Whether all
systems behave in the same way is, however, not known. Therefore using
@b{/bin/env} (with multiple arguments quoted) might be a more portable
approach than starting GNU APL directly.

For multiple arguments the situation is more tricky and the
two methods differ slightly. The @b{/bin/env}
method requires that multiple arguments are being quoted in the shebang
line (so that they beconme a single argument which @b{env -S} splits into
several arguments passed to @b{execve}). the direct @b{apl} method does
the same,  but without the need to quote multiple arguments in the
shebang line. For example:

@example
    #!apl --script --id 1010

    #!/bin/env -S apl "--script --id 1010"
@end example

A nasty pitfall of @b{/bin/env} is not to quote multiple arguments:

@example
$ cat SCRIPT2.apl
#!/usr/bin/env -S apl --id 1010

$ ./SCRIPT2.apl
unknown option './SCRIPT2.apl'
...

$ cat SCRIPT3.apl
#!/usr/bin/env apl --id 1010

$ b./SCRIPT2.apl
/usr/bin/env: ‘apl --id 1010’: No such file or directory
/usr/bin/env: use -[v]S to pass options in shebang lines
@end example

@c ............................................................................
@node Redirect the script file to the stdin of the GNU APL interpreter
@subsection Redirect the script file to the stdin of the GNU APL interpreter

Another method to execute an APL script file, say @b{SCRIPT.apl}, is to
use the input redirection of the shall:

@example
$ apl < SCRIPT.apl   # input redirection of SCRIPT.apl to stdin of apl
@end example

Alternatively, the GNU APL command line option @b{-f} or @b{-s} can be used:

@example
$ apl -f SCRIPT.apl
$ apl -s SCRIPT.apl
@end example

Both ways of redirecting the file are almost identical; in the first case
redirection was performed by the shell running apl, while in the second case
the redirecting of the file was performed by apl itself.

@c ............................................................................
@node Make the script file executable
@subsection Make the script file executable

Alternatively the script can be made executable, specifying apl as its
interpreter (see also @code{'man 2 execve'}). For this to work, the
first line of the script must have a special format:

@example
    #!/usr/bin/env apl --script
@end example

@code{--script} prevents: printing of a welcome banner, 
echoing of stdin to stdout, automatic loading of a )CONTINUE workspace,
and prevents starting a thread for shared variable communication.

If the path to the interpreter is relative (which might be useful if you
do not know in advance where the APL interpreter will be installed) then
it must be in @b{$PATH} of the shell that runs the script.

@c ............................................................................
@node Starting GNU APL Scripts: a Conclusion
@subsection Starting GNU APL Scripts: a Conclusion

The different methods for starting a GNU APL described above have emerged
over time. They can be seen as different front-ends to the GNU APL
interpreter that slightly differ in how command line options are being passed
from the shell to GNU APL. However, sooner or later each of the methods wiil
call @b{execve(pathname, argv, envp)} where:

@itemize
@item @b{pathname} is the name or absolute path to the GNU APL interpreter,
@item @b{argv} is the command line arguments for the GNU APL interpreter, and
@item @b{envp} is the environment variables for the GNU APL interpreter.
@end itemize

The different front-ends differ slightly as to how @b{argv} and @b{envp} are
being constructed; @b{/bin/env} allows the user to modify the environment 
before the GNU APL interpreter is called, input redirection allows the user
to modifu the apl script file before the GNU APL interpreter is called, and
@b{-f} allows the user to select the script file to be executed. One method
may be simpler to use in one situation and another method may be simpler to 
use in another situation and the final verdict as to which method is the
best is left to the user.

@c ............................................................................
@node How command line arguments are handled
@subsection How command line arguments are handled

As we have seen, GNU APL can be started directly (by entering the name
of the GNU APL binary in the command line of a shell, or indirectly by
entering the name of a script that provides the name of the GNU APL binary
in the first line of the script. In the latter case, the shell uses function
execve() to start APL, and one can provide additional arguments that are
passed on to APL. We briefly discuss both cases:

0. General Remark

There is a convention in (not only) GNU APL, that the command line options
(and possibly an optional option argument) that occur left of @code{--} are
options directed to (and understood by) the APL interpreter while the options
right of @code{--} are passed to the APL application (and their exact format is
then defined by the APL application). All command line options that were used
to start APL can later be retrieved with the system variable @b{⎕ARG} in APL.

Example 1: APL started directly

We start APL with some command line options and display them with ⎕ARG.
The command line options for the interpreter are --silent, -l, and 37,
while the options app1 and app2 are for the APL application:

@example
$ apl --silent -l 37 -- app1 app2
      ⊃⎕ARG
apl     
--silent
-l      
37      
--      
app1    
app2    
@end example

Example 2: APL started indirectly

A script is a text file which has execute permissions. The first line
of a script, commonly known as the @b{shebang} of the script, has a
special format which specifies how the subsequent lines of the script file
shall be processed, see @b{man execve} for details.

A GNU APL script is then a script whose shebang indicates the apl binary as
the interpreter for the subsequent lines where those subsequent lines are
either APL commands or APL statements. With some limitations, executing such
an APL script yields the same result as entering the subsequent lines in
immediate execution mode.

When APL is started directly, there is only one (possibly empty) set of
command line options, which are processed by the shell and then passed
to the apl binary.

Starting APL indirectly refers to executing an APL script. In this case
there are two sets of command line options:

@itemize
@item those passed to the script as command line arguments, and
@item those in the shebang line of the script
@end itemize

When the APL script is started, possibly with command line arguments, the
command line arguments entered by the user are combined with the arguments
in the shebang line of the script in a way that is described in 'man 2 execve'.

Assume the script file SCRIPT.apl contains this (the file is shipped
with GNU APL):

@example
# !/usr/bin/env -S apl "--script --id 1010"

      ⊃⎕ARG     ⍝ show command line options
      )OFF      ⍝ leave the interpreter
@end example

Assume further that the script is started like this:

@example
$ ./SCRIPT.apl -- sarg1 sarg2
@end example

Then the following output, which shows the order of command line options
as seen by ⎕ARG, is produced:

@example
--script    
--id        
1010        
./SCRIPT.apl
--          
sarg1       
sarg2       
@end example

In the example above please note the following:

@itemize
@item @b{--} separates the arguments passed to the interpreter (left of @b{--})
      from the arguments passed to the APL application,
@item The first line of the text file @b{SCRIPT.apl} is commonly referred to
      as the @i{shebang line}. If the shebang contains more than one argument
      (like @b{--script}, @b{--id}, and @b{1010} above), then the arguments
      need to be quoted, and
         then the arguments must be quoted, and
@item in contrast, multiple arguments in the command line (like @b{sarg1} and
      @b{sarg2} above) must not be quoted (or else they will appear as a single
      argument).
@end itemize

⎕ARG shows the arguments in the shebang line and in the command line in
the following order:

@enumerate
@item Interpreter name (the first name in the shebang, here: @b{apl})
@item Optional Interpreter argument(s) (here: @b{--script}, @b{--id}, and
      @b{1010})
@item Script filename (here: @b{./SCRIPT.apl})
@item Optional Script argument(s) (here: @b{--}, @b{sarg1}, and @b{sarg2})
@end enumerate

CAUTION: as discussed in 'man 2 execve' the interpretation of the optional
interpreter arguments (2. above) is system specific and possibly not portable.
For that reason the optional interpreter argument should not be omitted (you
can use -- to provide at least one argument) and also not more than one
argument. On GNU/Linux systems the following shebang lines of the script are
understood:

@example
#!/usr/bin/env -S apl "--script --id 1010"
#! /usr/local/bin/apl
#! /usr/local/bin/apl --
#! /usr/local/bin/apl -s
#! /usr/local/bin/apl --script
#! /usr/local/bin/apl -s --
#! /usr/local/bin/apl --script --
@end example

The recommended form is: @code{|/usr/local/bin/apl --script|}.

Note that @code{-} and any options that follow it in the shebang line
MAY OR MAY NOT be included in ⎕ARG (this depends on the platform used).
In contrast, options that follow @code{-} in the script command line
WILL BE shown since this is controlled by GNU APL and not by the platform.
There is normally no point in passing any application options in the shebang
line to APL because such parameters can more easily be provided directly
further down in the script. However, in cases where the script has only
application options it may be convenient to make @b{@verb{|--|}} the last
option of the shebang so that the user of the script need not specify it
manually.

Using @code{0-} in the shebang line of the script file usually prevents
subsequent options in the shebang line from being interpreted as APL command
line options (and, more importantly, from causing @b{unknown option} errors
when GNU APL starts). At least under GNU/Linux with @b{bash} the subsequent
options will be passed to the APL application via ⎕ARG.

Depending on this exact behavior cannot be recommended. A cleaner approach
is to use options starting with - or @code{-} only for the interpreter
options, and options starting with + for application options. This should
work on all platforms because for strings, say, APLOPT, ARG1, ARG2, ...,
which contain no whitespace:

@itemize
@item @b{+APLOPT} is ignored, although shown in ⎕ARG,
@item @b{++APLOPT ARG1} is ignored, although shown in ⎕ARG,
@item @b{+++APLOPT ARG1 ARG2} is ignored, although shown in ⎕ARG,
@item @b{...}
@end itemize

@c ............................................................................
@node Helpful Features for Scripting
@subsection Helpful Features for Scripting

GNU APL provides three system variables and functions that are particularly
useful for scripting:

@itemize
@item ⎕ARG: access to the command line arguments given to the interpreter,
@item ⎕ENV: access to the environment variables of the process running the
        interpreter, and
@item ⎕INP: Here-Document-alike function for creating large text variables
@end itemize

@node ⎕ARG
@subsubsection ⎕ARG

The system variable ⎕ARG contains all command line arguments passed to the
interpreter. In the initial example above this would be a nested 3-element
vector:

@example
    /usr/local/bin/apl --id 1010
@end example

@node ⎕ENV
@subsubsection ⎕ENV

There are usually at least two ways of passing parameters to a script:

@itemize
@item as command line options, and/or
@item by means of environment variables
@end itemize

Just as the system variable ⎕ARG makes the command line options used
available to APL, so does the system function ⎕ENV make the environment
variables available to APL. Since there normally exist far more
environment variables than command line options, ⎕ENV is a monadic function
whose argument is a filter for the name(s) of the variable(s) to be retrieved.

⎕ENV B returns all environment variables whose name starts with B.
If ⍴B is 0, like:

@example
      ⎕ENV ''
@end example

then @b{all} environment variables of the process running GNU APL are
returned.

The result of ⎕ENV is a (possibly empty) N×2 matrix. The first column
contains the name(s) of the environment variable(s), while the second
column contains their value(s). The name and the value are both strings.

The number N of environment variables returned by ⎕ENV obviously depends on
the number of environment variables whose name begins with the prefix
given in B; shorter prefixes B therefore result in higher counts N. B is case
sensitive; by convention the names of environment variables are all uppercase.
Note the subtle difference between shell variables and environment variables
in shells like @b{bash}: ⎕ENV returns only environment variables (which
are set with command @b{setenv} in bash), but not shell variables (which
are set with command @b{set} in bash). For a shell variable to occur
in ⎕ENV it must be @b{export}ed into the environment of the process
before the GNU APL interpreter is started.

@node ⎕INP
@subsubsection ⎕INP

If you need to create a longer static text, like the body of a web page,
then you would normally construct it in APL like this:

@example
      BODY ← ⊂ 'First line'
      BODY ← BODY , ⊂ 'Second line'
      BODY ← BODY , ⊂ 'Third line'
   ...
@end example

The above is obviously not very handy for longer texts. Instead you may
write the following in a GNU APL script:

@example
      BODY←⎕INP 'END-OF-⎕INP'
First line
Second line
Third line
   ...
END-OF-⎕INP
@end example

This works like "Here documents" in @b{bash}. The right argument of ⎕INP is
the end of text marker.

⎕INP also has a dyadic form where the left argument specifies an escape
sequence from text to APL and back to the text, similar to PHP scripts.
For example:

@example
      Z←'<?apl' '?>' ⎕INP 'END-OF-⎕INP'
First line
Time is now: <?apl ⍕⎕TS ?>
Third line
...
END-OF-⎕INP
@end example

This creates a mainly static text with a small dynamic plug-in computed by
APL:

@example
      ⊃Z
First line                       
Time is now: 2022 8 4 15 15 2 177
Third line                       
...         
@end example

It should be noted that, while ⎕INP simplifies the construction of multi-line
strings compared to standard APL, it has come of age and these days an even
simpler feature - Multi-line strings as described below - is provided for the
same purpose. The monadic ⎕INP has therefore become obsolete.

@c ............................................................................
@node Double-quoted Strings
@subsection Double-quoted Strings

Standard APL literals (aka. strings) such as 'Hello' are single quoted. As a
matter of convenience, GNU APL also provides double-quoted strings which
differ from single quoted strings in two ways:

@itemize

@item A standard single quoted APL string is a vector if its length ≠ 1 but
a scalar if its length = 1. This is sometimes inconvenient and therefore
double-quoted strings are more consistent in always being vectors regardless
of their length. In other words, 'A' is a scalar while "A" is a vector.

@item Double-quoted strings understand the standard C escape sequences for
control characters, such as @b{\n} for ASCII line feed, @b{\t} for ASCII TAB,
etc.

@end itemize

@c ............................................................................
@node Multi-Line Strings
@subsection Multi-Line Strings

A substantial disadvantage of @b{⎕INP} is that it only works in immediate
execution mode (and consequently also in .apl scripts), but not the bodies of
defined functions. Multi-line strings fill this gap for string literals that
span several lines of text @b{inside} the body of defined functions (as well as
in immediate execution mode and in scripts).
Multi-Line strings have become more and more advanced (and elegant) over time.
For this reason they currently come in different flavors (of which the older
ones will eventually disappear).

@node Old-style Multi-Line Strings
@subsubsection Old-style Multi-Line Strings

Old-style multi-line strings were the first attempt to provide long strings
for defined functions. The idea for the old-style multi-line strings was
adopted from shells like @b{bash} where, for example,

@example
echo "Line1
   Line2"
@end example

prints:

@example
Line1
   Line2
@end example

In simple terms this means that as long as the number of @b{"} characters is
odd, the string remains open until the closing @b{"} is found (which makes
the number even again).

Old-style multi-line strings are not allowed in immediate execution mode
because they could easily be entered by mistake (by simply forgetting the
terminating @b{"}  or @b{'}  of a standard string) and that would have
obscured the familiar behavior of immediate execution mode. In function
definition mode, the total number of function lines is known and it is
therefore possible to check if a multi-line string inside the function
is properly terminated. Immediate execution mode, however, is open-ended
and therefore a forgotten closing @b{"} can lead to a fatal misinterpretation
of (intended) string content as (unintended) APL code, with lots of
nonsense error messages.

Like in @b{bash} and also other shells, an old-style multi-line string starts
at the last " on a line of a defined function that has an
@b{an odd number of " characters}. The string continues over the subsequent
function lines until the next " is found. The result is a nested
APL value containing one nested string per line involved.

In immediate execution mode an old-style multi-line string yields, like
in standard APL:

@example
No string end found+
@end example

For example:
@example
∇Z←FOO
 Z←"ABC
 DEF
 GHIJK"
∇

4 ⎕CR FOO
┏→━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━━┓ ┏→━━━━━┓┃
┃┃ABC┃ ┃ DEF┃ ┃ GHIJK┃┃
┃┗━━━┛ ┗━━━━┛ ┗━━━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━━━━┛
@end example

If the first string is empty then the nested value starts with an empty
string, but completely empty lines in defined functions are ignored:

@example
∇Z←FOO
 Z←"
ABC
 DEF

  GHIJK"
∇

4 ⎕CR FOO
┏→━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏⊖┓ ┏→━━┓ ┏→━━━┓ ┏→━━━━━━┓┃
┃┃ ┃ ┃ABC┃ ┃ DEF┃ ┃  GHIJK┃┃
┃┗━┛ ┗━━━┛ ┗━━━━┛ ┗━━━━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━━━━━━━━━┛
@end example

@node New-style Multi-Line Strings
@subsubsection New-style Multi-Line Strings

The combination of @b{⎕INP} for immediate execution mode and multi-line
strings for defined functions achieved, at least in principle, the goal of
specifying large amounts of text in a simpler manner than in standard APL.
However, having different methods for creating multi-line string literals
in different APL modes could not convince entirely.

The search for a more intuitive and more consistent solution then led to
new-style multi-line strings. These strings adopted the triple quote (""")
syntax known from e.g. the Python language. The syntax of new-style
multi-line strings is cleaner: empty lines are handled more consistently and
the string lines are completely separated from the APL code lines.
More importantly, new-style multi-line strings work alike in immediate
execution mode, in scripts, and in defined functions. New-style strings are a
little different and syntactically stricter than their old-style companions:

@itemize
@item the starting """ must be placed at the end of an APL code line, i.e.
there is no more mixing of APL code and string content in the same line.
An old-style string starts with APL code and ends with the first line of
the string, while a new-style string starts with an APL code line and
the string itself begins at the following line.

@item the ending """ must be placed at the end of a separate line with only
spaces allowed before the """ (to horizontally indent-align it with the
leading """ if so  desired). In old-style strings the text up to the
termination " is part of the string while in new-style strings it is not.

@item Neither the starting nor the ending line become nested strings of the
result, and every line can be clearly identified as being either APL code or
else string text.
@end itemize

Example:
@example
 4 ⎕CR """
ABC
 DEF

  GHIJK
       """
┏→━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━━┓ ┏⊖┓ ┏→━━━━━━┓┃
┃┃ABC┃ ┃ DEF┃ ┃ ┃ ┃  GHIJK┃┃
┃┗━━━┛ ┗━━━━┛ ┗━┛ ┗━━━━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━━━━━━━━━┛
@end example

When entering new-style multi-line strings, the prompt is prefixed by → to
indicate that a Multi-Line String is being entered.

⎕INP and old-style strings will coexist for a while, but may eventually
disappear at some point in time.

@node « » Strings
@subsubsection « » Strings

When new-style multi-line strings are used in scripts the related error
reporting (if the syntax is used incorrectly) can become cumbersome. For
example, if one forgets the closing """ of a multi-line string then the next
""" (which was meant as the start of a new multi-line string) is mistaken as
the (forgotten) end of the previous one. The often long content of
the new string is then parsed as APL code and will usually produce many
error messages, all of them nonsense. Things get worse if the script itself
is long because the odd/even nature of the quotes will then persist until
the end of the string (or at least until the next forgotten quote, if any).

To deal with this problem, GNU APL also allows strings whose contents are
enclosed in double angle quotation marks « and ». In inline strings '...'
and "...", as well as in multi-line strings """ ... """, there is no general
(and reliable) way to decide if a user has forgotten the ending quote(s),
which leads to the problems above.

For this reason, GNU APL now also allows inline and multi-line strings
like, for example:

@example
      «Hello world»
Hello world

      8 ⎕CR «««
→      Line 1
→        Line 2
→      »»»
┌→──────────────────┐
│┌→─────┐ ┌→───────┐│
││Line 1│ │  Line 2││
│└──────┘ └────────┘│
└ϵ──────────────────┘

      ⍝ provoke an error...
      ⍝
      8 ⎕CR «««
→      Line 1
→         Line 2
→            «««

*** WARNING: unexpected triplet ««« in mult-line string.
    Expected triplet: »»» (literal depth: 0).
    The triplet was added to the multi-line string.

→            »»»
┌→───────────────────────────────┐
│┌→─────┐ ┌→────────┐ ┌→────────┐│
││Line 1│ │   Line 2│ │      «««││
│└──────┘ └─────────┘ └─────────┘│
└ϵ───────────────────────────────┘
@end example

Another advantage of @b{«««} ... @b{»»»} over @b{"""} ... @b{"""} is that
a decent text editor (read: @b{vi/vim}) may jump back and forth between
the opening @b{«} and the corresponding closing @b{»} with a single key
stroke (character @b{%} in vi/vim).  For this to work, add the following
line to  @b{/etc/vim/vimrc} (or to @b{~/.vimrc}):

@example
set matchpairs+=«:»
@end example

@node Multi-Line Comments
@subsubsection Multi-Line Comments

One of the badly missing features of standard APL are comments that span
multiple lines. However, in GNU APL one can use multi-line strings in
combination with monadic primitive @b{⊣} for that purpose:

@example
      ⊣««« ⍝⍝⍝
       Multi-line
       Comment
       ...
       »»» ⍝⍝⍝
@end example

The triplet @b{⍝⍝⍝} in the example above is optional and has no effect (since
it is part of the multi-line string, but is then suppressed by ⊣), but is
nevertheless recommended as an indication that the text between the @b{⍝⍝⍝}
is a comment.

GNU APL is aware of this construct and optimizes it away entirely. Therefore
even long multi-line comments inside loops have, like normal APL comments,
no performance penalty at runtime.

@c ............................................................................
@node Multi-Line Literals
@subsection Multi-Line Literals

A limitation from the early days of APL is that literals like @b{1 2 3}
or @b{'abc'} can not stretch over several lines. As a consequence, the rank
of a literal can not be more than 1, and values with higher ranks require
the use of APL primitive (most likely dyadic @b{⍴}). A single line of input
can produce multiple lines of output, but these multiple lines of output can
not be fed back into the interpreter as literals.

This limitation is rather natural when APL is used interactively (and screen
oriented editors like @b{vi} or @b{emacs}) were not very common at the time
when APL was invented. At that time, nobody (and in particular not APL
programmers who had the reputation of being lazy) would have entered, say, a
numeric 10 by 10 matrix interactively.

However, for scripting the situation is quite different. An APL script is
typically created beforehand with a screen oriented editor, and screen
oriented editors like @b{vi} are available everywhere while its line oriented
predecessor (named @b{ex}) has died out in the meantime (more precisely,
@b{ex} has become a sub-mode of @b{vi}).

GNU APL extends the idea of multi-line strings to @i{multi-line literals}.
The syntax that governs the output of APL values (i.e. in the APL → terminal
direction) is simply applied to the input of APL literals (i.e. in the
terminal → APL direction) as well:

@verbatim
                                         ┌─────────┐          
                       ┌─────────┐       │  MULTI  │    (text,
      (text,           │ LITERAL │       │  -LINE  │     multi line)
       single line)    └────┬────┘       │ LITERAL │
                            │            └────┬────┘
                            │                 │
                            │     input       │
                            ↓     rules       ↓
                            │                 │
                         ╔══╧══╗         ╔════╧════╗
      (vectors,          ║ APL ║         ║ GNU APL ║
       values)           ╚══╤══╝         ╚════╤════╝
                            │                 │
                            │     output      │
                            ↓     rules       ↓
                            │                 │
                         ┌──┴──┐           ┌──┴──┐
      (text,             │     │           │     │    (text,
       multi-line)       │ OUT │           │ OUT │     milti-line)
                         │     │           │     │
                         └─────┘           └─────┘
@end verbatim

The relevant formatting rules for APL output are:

@itemize
@item one line of a APL output corresponds to one or more scalar items 
      of the APL value,
@item the items in the @b{same} output line are the scalars along the
      @b{last axis} of the APL value, where
    @itemize
    @item a single quoted character is a character scalar,
    @item several characters inside a single pair of quotes are a character
          vector (aka. string),
    @item unquoted characters are numeric scalars,
    @item nested items are two or more items enclosed in parentheses, and
    @item the APL scalars of an APL vector are separated by one or more blanks
    @end itemize
@item the items in @b{different} output lines are the scalars along
      @b{all except the last axis} of the APL value, where
    @itemize
    @item two-dimensional values (i.e. matrices) are vertically adjacent
          vectors.  @item three-dimensional values are vertically adjacent
          matrices, separated by one blank line,
    @item four-dimensional values are vertically adjacent three-dimensional
          values, separated by two blank lines,
    @item ...
    @item N-dimensional values are vertically adjacent N-1-dimensional values
          separated by N-2 blank lines,
    @end itemize
@end itemize

Now, the existing formatting rules for (single line) literals are:

@itemize
@item one literal corresponds to one APL scalar or APL vector, where:
    @itemize
    @item a single quoted character is one APL character scalar,
    @item several quoted characters are an APL character vector (aka. string),
    @item nested items are two or more items enclosed in blanks, and
    @item vertically adjacent scalars above are separated by blanks to create
          vectors; this is called @i{vector notation} in Kenneth Iverson's 
          APL book and in IBM APL2 documents and @i{strand notation} in other
          places,
    @end itemize
@end itemize

From these rules for single line literals, one can easily derive the rules
for multi line literals in a completely consistent fashion:

@itemize
@item consistency: the rules for every line of a multi-line literal shall be
      the same as for single line literals,
@item matrix: (directly) vertically adjacent literals become
      two-dimensional APL values,
@item vertically adjacent two-dimensional literals, separated by one blank
      line, become three-dimensional APL values,
@item vertically adjacent three-dimensional literals, separated by two blank
      line, become four-dimensional APL values,
@item ...
@item vertically adjacent @b{N-1}-dimensional literals separated by @b{N-2}
      blank line become @b{N}-dimensional APL values,
@end itemize

Similar to multi-line strings, a multi-line literal starts with the marker
@b{<<<} at the end of one line and ends with  the marker @b{>>>} on a
subsequent line. The value of the multi line literal is contained in the lines
between these two markers (excluding the lines with the markers). In contrast
to multi-line strings, can multi line literals be used recursively. That is,
one multi line literal can contain other multi line literals (as well as
multi-line strings) to created nested sub-values of ranks > 1.

Some examples may be worthwhile to explain the concept:

@example
      ⍝ two-dimensional multi-line literal
      ⍝
      ⎕ ← <<<
          1 2 3
          4 5 6
          7 8 9
          >>>
1 2 3
4 5 6
7 8 9
@end example

One blank line separates the matrix of a three dimensional value:

@example
      ⍝ three-dimensional multi-line literal
      ⍝
      ⎕ ← <<<
          1 2 3
          4 5 6
          7 8 9

          11 12 13
          14 15 16
          17 18 19
          >>>
 1  2  3
 4  5  6
 7  8  9

11 12 13
14 15 16
17 18 19
@end example

The start line of a literal may contain a comment; the comment does not
become part of the literal value:

@example
      ⍝ two-dimensional multi-line literal with comment
      ⍝
      ⎕ ← <<<   ⍝ same as 3 3⍴⍳9
          1 2 3
          4 5 6
          7 8 9
          >>>
1 2 3
4 5 6
7 8 9
@end example

The items of a multi-line literal can be nested:

@example
      8 ⎕CR <<<   ⍝ two-dimensional with nested 'abc'
            1 2     3
            4 'abc' 6
            7 8     9
            >>>
┌→────────┐
↓1     2 3│
│4 ┌→──┐ 6│
│  │abc│  │
│  └───┘  │
│7     8 9│
└ϵ────────┘

      8 ⎕CR <<<   ⍝ two-dimensional with nested 10 11 12
            1 2          3
            4 (10 11 12) 6
            7 8          9
            >>>
┌→─────────────┐
↓1          2 3│
│4 ┌→───────┐ 6│
│  │10 11 12│  │
│  └────────┘  │
│7          8 9│
└ϵ─────────────┘
@end example

Multi-line literal can contain other Multi-line literals or strings:

@example
      8 ⎕CR <<<   ⍝ two-dimensional with sub-literal:
            1 2     3
            4 <<< ⍝ sub-literal
              10 11 12
              13 14 15
              16 17 18
              >>>   6
            7 8     9
            >>>
┌→─────────────┐
↓1          2 3│
│              │
│4 ┌→───────┐ 6│
│  ↓10 11 12│  │
│  │13 14 15│  │
│  │16 17 18│  │
│  └────────┘  │
│              │
│7          8 9│
└ϵ─────────────┘

      8 ⎕CR <<<   ⍝ two-dimensional with sub-string:
            1 2     3
            4 «««
              Hello,
              World
              »»»   6
            7 8     9
            >>>
┌→─────────────────────────────────────────────┐
↓1                                          2 3│
│4 ┌→───────────────────────────────────────┐ 6│
│  │┌→─────────────────┐ ┌→────────────────┐│  │
│  ││            Hello,│ │            World││  │
│  │└──────────────────┘ └─────────────────┘│  │
│  └ϵ───────────────────────────────────────┘  │
│7                                          8 9│
└ϵϵ────────────────────────────────────────────┘
@end example

Nested multi-line literals can quickly become unreadable. It is therefore
recommended to use a clear indentation policy for the different nesting
levels.

@c ............................................................................
@node Automatic )MORE
@subsection Automatic )MORE

The GNU APL command )MORE provides, in some cases, additional information
about a prior APL error. The availability of such additional information
is indicated by a + at the end of the error message. The additional
information is automatically discarded when the next non-empty line
is entered. For example:

@example
      "ABC"[4]
INDEX ERROR+
      'ABC'[4]
      ^    ^
      )MORE
min index=⎕IO (=1), offending index=4, max index=⎕IO+2 (=3)
      ◊ ⍝ clears )MORE

      )MORE
NO )MORE ERROR INFO
@end example

This is rather useful in purely interactive mode where the user can issue
the )MORE command directly after an APL error has occurred to help figure
out what exactly went wrong. In a script, however, the standard )MORE command
(as inherited from IBM APL2) is fairly useless for two reasons:

@itemize
@item A human user has a chance to issue the )MORE command interactively
after an error has occurred, while a script continues after displaying the
error message, and
@item It is rather difficult to predict where the next errors will occur,
i.e. where to place the )MORE command in the script.
@end itemize

To make the )MORE command more useful for scripts, in GNU APL it can be
augmented with an optional argument AUTO like this:

@example
      )MORE AUTO ON
Automatic )MORE is now: ON
      
      "ABC"[4]
min index=⎕IO (=1), offending index=4, max index=⎕IO+2 (=3)
INDEX ERROR+
      'ABC'[4]
      ^    ^
@end example

If ON/OFF is omitted then the automatic )MORE mode is toggled. The additional
)MORE information is displayed before the standard 3-line APL error message.

@c ............................................................................
@node Script Example
@subsection Script Example

Note that the two different ways of running an APL script have an impact on
how ⎕ARG looks like. If stdin is redirected then there is only one (possibly
empty) set of command line options. Otherwise there are two sets of command
line options: command line options for the apl interpreter and command line
options for the script.

Consider the following simple script called SCRIPT.apl in directory workspaces:

@example
#! /usr/local/bin/apl --script

⊃⎕ARG   ⍝ show command line options
)OFF    ⍝ leave the interpreter
@end example

If SCRIPT.apl is redirected to stdin of the APL interpreter:

@example
 /usr/local/bin/apl --silent <  ../workspaces/SCRIPT.apl

 or

 /usr/local/bin/apl --silent -f ../workspaces/SCRIPT.apl
@end example

then the first line @option{#! /usr/local/bin/apl --script} of file SCRIPT.apl 
is merely a comment (GNU APL accepts both the traditional APL character
⍝ and the character # as start of a comment). The @code{-script} option
is therefore ignored and the following mix of input and output is
shown on the screen. The input from the script is indented by the
usual APL prompt of 6 blanks, while the output of the APL interpreter
is not indented.

@example
      #! /usr/local/bin/apl --script
      
      ⊃⎕ARG     ⍝ show command line options
      )OFF      ⍝ leave the interpreter
@end example

If we run the same script directly:

@example
../workspaces/SCRIPT.apl
@end example

Then we get:

@example
 /usr/local/bin/apl --script ../workspaces/SCRIPT.apl 
@end example

The @code{-script} implies @code{-noCIN} so that the
input lines for the interpreter are no longer echoed to the output.
This is most likely what you want when writing a script.

Also, the first line of the script is no longer ignored as a comment,
but controls the command line argument (and thus ⎕ARG) of the interpreter.
The additional command line argument @option{../workspaces/SCRIPT.apl}
comes from function execve (see 'man 2 execve').

If we provide an argument, say SCRIPTARG, to SCRIPT.apl:

@example
../workspaces/SCRIPT.apl SCRIPTARG
@end example

then it shows up at the end of ⎕ARG:

@example
 /usr/local/bin/apl --script ../workspaces/SCRIPT.apl SCRIPTARG 
@end example

A final note on scripting in GNU APL is that the ∇-editor works slightly
differently when it is used in a script. If a user edits an APL function
interactively then an attempt to open an existing function with a full
header gives a DEFN ERROR:

@example
      ∇Z←FOO B
[1] ∇
      
      ∇Z←FOO B
DEFN ERROR+
      ∇Z←FOO B
             ^
      )MORE
attempt to ∇-open existing function with new function header
@end example

In contrast to a user who can react to the DEFN ERROR, a script cannot detect
this situation and would continue to push lines (which were intended to be
the body of the defined function) into the APL interpreter. That would most
likely cause fairly undesirable behavior. For example, if the lines of a
defined function start with line numbers (like [1], [2], ...) then every
such line would give a SYNTAX ERROR, and other errors can be expected as
well.

For that reason, if the ∇-editor is used in a script and attempts to redefine
an existing defined function then the existing function is simply overwritten
with the new one and no DEFN ERROR is raised.


@c ----------------------------------------------------------------------------
@node Axis argument in defined functions
@section Axis argument in defined functions

Defined functions and operators (including lambdas) accept an axis argument.
For example:

@example
∇Z←Average[X] B
 Z←(+/[X]B) ÷ (⍴B)[X]
∇

      Average[1] 5 5⍴⍳25
11 12 13 14 15

      Average[2] 5 5⍴⍳25
3 8 13 18 23
@end example

Syntactically, the axis is used in the same way as for primitive functions
and operators.

There are no constraints on the axis such as being integers. Therefore you
can use an axis as a third function argument. Keep in mind, however, that
doing so will make your APL code incompatible with other APL interpreters.
Use this feature carefully!


@c ----------------------------------------------------------------------------
@node Function Groups
@section Function Groups

In the old APL1 every function of the interpreter was represented by a single
APL character. More precisely, the character represented 1, 2, or 3
sub-functions and the sub-function was selected by the number of arguments
provided:

@itemize
@item 1 for monadic functions,
@item 2 or monadic functions with axis,
@item 2 for dyadic functions (w/o axis), or
@item 3 for dyadic functions with axis.
@end itemize

The APL character itself was designed in a way that made it easy to
remember which functions it represented. 

Unfortunately this 1:1 relationship between APL characters and interpreter
functions did not scale very well. As the number of functions increased,
a new and, at that time, better scalable concept was introduced: ⎕-functions
(and variables). The APL standards define only a small number (less than 20)
of them, but different APL vendors were eagerly adding more system functions
to represent their proprietary extensions to th APL language. And so did
GNU APL. With ⎕-functions the original visual appearance that helped to
remember the many functions was replaced by abbreviations. That is, for
example, @b{⎕AV} stands for the @b{A}tomic @b{V}ector, @b{⎕TS} for
@b{T}ime @b{S}tamp, and so forth. 

As of this writing, the GNU APL interpreter has more than 100 non-standard
functions for various purposes. For such a large number the ⎕-function
approach is no longer scalable (try to remember 100 abbreviations and
think about the readability of the code using them!). Rather than adding a
new ⎕-function for every new function, GNU APL tries to keep the number
of ⎕-functions as small as reasonable and by grouping several interpreter
functions into sub-function of a single top-level ⎕-function.

We call such a top-level ⎕-function a @b{function group}. The sub-function
functions of a function group are somehow related by their
purpose. The function groups of GNU APL are:

@itemize
@item @b{⌹}: a collection of matrix factorizations (aka. decompositions)
@item @b{⎕CR}: a collection of data conversions (formerly known as monadic
  @b{C}harater @b{R}epresentations
@item @b{⎕FFT}: a collection of @b{F}ast @b{F}ourier @b{T}ransformations
@item @b{⎕FIO}: a collection of @b{F}ile @b{I}/O functions
@item @b{⎕MX}: a collection of @b{M}atri@b{x} functions
  functions
@item @b{⎕RVAL}: generation of @b{R}andom APL @b{Val}ues
@item @b{SQL}:  an interface to @b{SQL databases}
@end itemize

The syntax for function groups has evolved in 3 phases as elucidated in the
following.

@c ............................................................................
@node Phase 1: Numerical Axis Argument
@subsection Phase 1: Numerical Axis Argument

In the first phase, the number of functions per function group (at that time
@b{⎕CR} and @b{⎕FIO}) was fairly small. A sub-function of the function group
was selected as follows:

@itemize
@item If all functions in the function group were monadic (as is
      the case in @b{⎕CR}), then the sub-function was selected by
      the left argument of the function group.
@item Otherwise (as is the case in @b{⎕FIO}) the individual sub-function
      could be monadic, dyadic, or nomadic and the sub-function was selected
      by an axis argument of the function group.
@end itemize

For example:

@example
      4 ⎕CR B          ⍝ format B in a "boxed" fashion
      18 ⎕CR B         ⍝ convert APL string B to an UTF8-encoded byte vector

      ⎕FIO[3] B        ⍝ fopen file B for reading
      A ⎕FIO[3] B      ⍝ fopen file B with mode A
@end example

In both cases was the sub-function selector an integer scalar.

@c ............................................................................
@node Phase 2: String Axis Argument
@subsection Phase 2: String Axis Argument

Shortly after introducing the syntax of phase 1, GNU APL users began to develop
an APL workspace named @b{FILE_IO}. This workspace contains one APL wrapper
function for each numerical @b{⎕FIO} sub-function selector. These wrappers
were trivial one-liners like these:

@example
      ∇Zh ← As FIO∆fopen Bs
      ⍝⍝ fopen(Bs, As) filename Bs
       Zh ← As ⎕FIO[3] Bs
      ∇

      ∇Zh ← FIO∆fopen_ro Bs
      ⍝⍝ fopen(Bs, "r") filename Bs
       Zh ← ⎕FIO[3] Bs
      ∇
@end example

APL programs would @b{)COPY} this workspace and then use its normal defined
functions instead of @b{⎕FIO} with an axis argument or @b{⎕CR} with a left
argument.  The development of the @b{FILE_IO} suggested that APL users have
a preference of names over numbers to select a sub-function.

GNU APL accommodated this preference by allowing sub-function names (APL strings)
as sub-function selectors. These names were usually the same as the C library 
function that implemented them.

The phase 1 example above would now become the slightly better readable:

@example
      'style_boxed' ⎕CR B        ⍝ format B in a "boxed" fashion
      'string_to_UTF8' ⎕CR B     ⍝ convert APL string B to UTF8

      ⎕FIO['fopen'] B            ⍝ fopen file B for reading
      A ⎕FIO['fopen'] B          ⍝ fopen file B with mode A
@end example

@c ............................................................................
@node Phase 3: Member Notation
@subsection Phase 3: Member Notation

The third phase aims at further improving the readability. It replaces the
axis argument with a member notation. The phase 1 and 2 examples above would
now become:

@example
      ⎕CR.style_boxed B       ⍝ format B in a "boxed" fashion
      ⎕CR.string_to_UTF8 B    ⍝ convert APL string B to UTF8

      ⎕FIO.fopen B            ⍝ fopen file B for reading
      A ⎕FIO.fopen B          ⍝ fopen file B with mode A
@end example

@node Subfunction Lists
@subsection Subfunction Lists

If the number of sub-functions becomes large, then remembering all sub-function
names of a function group is still a problem. To avoid the need to consult
other documentation while programming, all function groups provide lists
that show their sub-functions and their syntax.

Let @b{FOO} be a function group. Then:

@itemize
@item @b{FOO ""} shows a list of sub-functions, and
@item @b{FOO ⍬} shows the syntaces (as per phase 1, 2, or 3) of them.
@end itemize

@c ----------------------------------------------------------------------------
@node Colored Output
@section Colored Output

The APL interpreter gets its input from the standard input (stdin),
which is normally connected to the user's keyboard,
but can also be a file if APL scripting, the @code{f} option,
or the @code{T} option is used.

The APL interpreter prints its results on either the standard output (stdout)
for normal APL output, or to the error output (stderr) for additional
trouble-shooting information.

You can print the 3 channels stdin, stdout, and stderr in different colors
by means of the debug command ]XTERM. Command ]XTERM ON enables colored
output while ]XTERM OFF disables it (for example to avoid annoying ANSI
Escape sequences when forwarding stdout or stderr to a file).

Default colored output assumes a terminal (-emulation) that understands
ANSI (or VT100) Escape sequences. The xterm that comes with most recent
GNU/Linux distributions is a perfect choice supporting both colors and
UTF-8 (Unicode) encoded character I/O.

Non-ANSI terminals, as well as other colors than the default ones, can be
configured in the 'preferences' file. The 'preferences' file also
contains a description of all possible color settings.


@c ----------------------------------------------------------------------------
@node Comparison Rules
@section Comparison Rules

Both IBM APL2 and the ISO standard require that the arguments of <, ≤, ≥,
and > (but not of = or ≠) are integer or real numbers. As a consequence,
the argument(s) of ⍋ or ⍒ (which require comparison) must also be a vector
of integer or real numbers.

In contrast, GNU APL also allows the comparison of characters and numbers or
the comparison of complex numbers according to the following, more general,
rules.

Let A and B be two APL values to be compared. The final result of comparing
A and B is the first verdict (i.e. either A < B, or A > B, or A = B)
obtained when following the rules below in the indicated order:

@enumerate
@item
Comparison by rank: if (⍴⍴A) < (⍴⍴B) then A < B and vice versa.

@item
Comparison by shape: if (⍴A) < (⍴B) then A < B and vice versa. The
first differing shape item (from the left) decides.

@item
Comparison of ravel elements: at this point (⍴A) ≡ (⍴B). If all
corresponding ravel elements of A and B are equal (i.e. tolerantly equal
within ⎕CT as defined in the ISO standard) then A = B.

Otherwise let A1 and B1 be the first corresponding ravel elements of A and B
with A1 ≠ B1. If A1 < B1 then A < B and vice versa. The comparison A1 < B1
is made according to the following rules 4 - 8 below.

@item
Comparison by depth:

@itemize
@item
If A1 and B1 are both nested: the rules 1 - 3 above are (recursively)
applied to corresponding ravel elements of ⊃A1 and ⊃B1 until a verdict is
obtained.

@item if A1 is simple and B1 is nested then A < B and vice versa.

@item
otherwise (i.e. A1 and B1 are both simple): A < B if A1 < B1 according to
rules 5 - 8 below and vice versa.
@end itemize

@item
Comparison by Unicode: if A1 and B1 are both character values and
(⎕UCS A1) < (⎕UCS B1) then A < B and vice versa.

@item
Comparison by type: If A1 is a character and B1 is numeric, then A < B
and vice versa.

@item
Comparison by numeric value: if A1 and B1 are both numeric values then:

@itemize
@item
Comparison by real part: if (9○A1) < (9○B1) then A < B and vice versa.

@item
Comparison by imaginary part: otherwise if (11○A1) < (11○B1) then A < B
and vice versa.
@end itemize

@item
Otherwise: A = B.
@end enumerate

Another way of describing the rules above is that the comparison of two values
is comprised of sub-comparisons of certain properties of the values in the
following order:

@itemize
@item the ranks of the values,
@item the shapes of the values,
@item the first differing ravel element (in row-major order) of the values,
@item the depths of the differing ravel elements,
@item the types (character vs. numeric) of the differing ravel elements,
@item the Unicodes of the differing ravel elements (if applicable)
@item the real parts of numeric values,
@item the imaginary parts of numeric values,
@end itemize

Note: Rules 1 and 2 above are only relevant for comparisons made in the
context of sorting (i.e. for ⍋ or ⍒). This is because for =, ≠, <, ≤, ≥,
or > either a RANK ERROR or a LENGTH ERROR is raised if the ranks or shapes
of A and B do not match:

@example
      (9 8) < (1 2 3)
LENGTH ERROR
      9 8<1 2 3
      ^  ^

      ⍋(9 8) (1 2 3)
1 2
@end example

The reason for comparing complex numbers first by their real parts and then
by their imaginary part and not, for example, first by their magnitude and
then by their angle is that the chosen order gives more consistent results
when comparing near-complex numbers or their true real companions. For
example, a magnitude first comparison of complex numbers would make
¯2 < ¯1 < ¯2J1E¯20 for the near-complex number ¯2J1E¯20.

@b{CAUTION}: The comparison of two strings (i.e. nested character
vectors) may give unexpected results because shorter strings come before
longer strings. For example, 'Zoo' comes @b{before} 'Adam' even
though one might expect the opposite.

@example
      Z[⍋Z ← 'Adam' 'Zoo']
 Zoo Adam 

      Z[⍋Z ← 'Adam' 'Zora']
 Adam Zora
@end example

This pitfall can be avoided by enforcing the same length for all strings
being compared or sorted. A simple way to achieve that is the use of
@b{⊂[2]⊃} like this (assuming ⎕IO←1):

@example
      Z[⍋Z ← ⊂[2]⊃ 'Adam' 'Zoo']
 Adam Zoo  

      Z[⍋Z ← ⊂[2]⊃ 'Adam' 'Zora']
 Adam Zora
@end example


@c ----------------------------------------------------------------------------
@node Complex Numbers
@section Complex Numbers

Complex numbers are fully supported.


@c ----------------------------------------------------------------------------
@node Debug Commands
@section Debug Commands

In addition to the classical APL commands like )LOAD or )SAVE, GNU APL
has a number of debug commands for debugging purposes.
Regular APL commands start with ) and print their output on stdout.
Debug commands start with ] and print their output on stderr. Normally
you cannot easily distinguish between stdout and stderr, but another
GNU APL feature, colored output, uses different colors for stdout and stderr.

Type )HELP or ]HELP in the interpreter for a list of all commands available.


@c ----------------------------------------------------------------------------
@node Direct Functions (Lambdas)
@section Direct Functions (Lambdas)

GNU APL supports direct functions (aka. lambdas), but only in a rather
limited form.

@c ............................................................................
@node Named Lambdas
@subsection Named Lambdas

A statement of the form

@example
      FUN ← @{ body_statement @} 
@end example

creates a @i{named lambda}. The single @i{body_statement} may contain the
special variable names @b{⍺} and @b{⍵} as well as the special function
names @b{⍶} and @b{⍹} which are replaced by the actual arguments of the
lambda when the  lambda is executed. The valence of the lambda is determined
as follows:

@itemize
@item If @b{⍶} and @b{⍹} are present then the lambda is a @i{dyadic operator}.
@item If only @b{⍶} is present then the lambda is a @i{monadic operator}.
@item If neither @b{⍶} nor @b{⍹} is present then the lambda is a normal
      defined function.
@item If both @b{⍺} and @b{⍵} are present in the body_statement then the
      operator or defined function is dyadic.
@item If only @b{⍵} is present then the operator or defined is monadic.
@item If neither @b{⍺} nor @b{⍵} is present then the lambda is niladic.
@item Lambdas with the left value argument @b{⍺} but without the right
      value argument @b{⍵} are invalid (SYNTAX ERROR).
@end itemize

There is a noteworthy pitfall related to niladic named lambdas. Recalling
to the fundamental APL evaluation rule:

@i{All functions execute according to their position in an expression.
The rightmost function whose arguments are available is evaluated
first}

implies that a niladic function is evaluated first (because it has no
arguments and consequently no arguments that are not available). Therefore
in an expression such as:

@example
NAME ← @{ statement @}
@end example

will the niladic lambda @b{@{ statement @}} be evaluated before the
assignment, The name @b{NAME} is therefore the result of the niladic lambda
and not the niladic lambda itself. IOW: the example above @i{does not}
create a niladic named lambda, but rather a normal variable @b{NAME} whose
value is the result of evaluating @b{statement}. Creating niladic named
lambdas is impossible.

GNU APL supports an axis argument in normal defined functions and
operators. In lambda expressions the Greek letter @b{χ} (Chi) is the
variable name for an axis argument.

The way a named lambda is implemented in GNU APL is that the assignment of
the form:

@example
      FUN ← @{ body_expression @}
@end example

is translated to a two-line function ⎕FX 'lambda_header' 'body_expression'.

For example:

@example
      )FNS

      SUM ← @{ ⍺ + ⍵ @}
@end example

is translated to a two-line function ⎕FX 'lambda_header' 'body_expression'.

For example:

@example
      )FNS

      SUM ← @{ ⍺ + ⍵ @}
      )FNS
SUM

      ∇SUM[⎕]∇
DEFN ERROR+
      ∇SUM[⎕]∇
             ^
      )MORE
function is a lambda
@end example

Named lambdas are included in function lists like @b{)FNS} along with other
defined functions. They can not be edited with the ∇-edidor, but can be
displayed using @b{⎕CR 'name'}:

@example
      ⎕CR 'SUM'
λ←⍺ λ1 ⍵
λ← ⍺ + ⍵ 
@end example

The lambda_header is automatically deduced from the presence or absence
of the variable names (⍺, ⍵, and χ) and function names (⍶ and ⍹) in the
body_expression and from whether the body_expression is empty (no λ←) or
not (i.e. with λ←).

It is possible to specify local variables that work exactly like their
companions in normal defined functions. Please note that this is different
from some other APL interpreters which treat all variables in lambda bodies
as local variables. The syntax for specifying local variables is the same
as for the header in normal defined functions: they are added at the end
with semicolons as separators. In the above example one could add local
variables C and D like this:

@example
      SUM ← @{ ⍺ + ⍵ ;C;D @}

      ⎕CR 'SUM'
λ←⍺ λ1 ⍵;C;D 
λ← ⍺ + ⍵     
@end example

If a named lambda is created inside a defined function, then the name of the
lambda (i.e. the name left of ← @{ ... @}) can be made a local variable of
the function in which the named lambda is created. This creates a lambda with
local scope.

Please note that the above only describes the current implementation of named
lambda in GNU APL. A consequence of that implementation is that the symbols
⍺, ⍵, χ, ⍶, ⍹, and λ are pretty much behaving like user defined variables. In
particular, they can be used outside named or unnamed lambdas. However, that
may change in the future and therefore such use of ⍺, ⍵, χ, ⍶, ⍹, and λ outside
is certainly a bad idea.

As of this writing, GNU APL lambdas allow multiple statements, separated
by @b{◊}. This feature is experimental, not portable, and should be avoided
where possible. In GNU APL, the value of a multi-statement lambda is the
value of the last statement and this differs from the semantics of
multi-statements in other APL interpreters!

@c ............................................................................
@node Unnamed Lambdas
@subsection Unnamed Lambdas

An unnamed lambda is an expression inside @{ and @} but without assigning
it to a name. This is often used together with the EACH operator.
For example:

@example
      @{ ⍴ , ⍵ @} ¨ 'a' 'ab' 'abc'
 1  2  3 
@end example

Unnamed lambdas are automatically local in scope (similar to labels). They can
be passed as function arguments to operators. However, unnamed lambdas are NOT
inserted into the symbol table of the interpreter. They are therefore not
visible to functions like ⎕CR or by the ∇-editor. Occasionally the names λ1,
λ2, ... may show up in commands like )SIS. These names are automatically
generated for unnamed lambdas in order to provide a name in places where a
function name is needed.

@c ............................................................................
@node Limitations of GNU APL Lambdas
@subsection Limitations of GNU APL Lambdas

There are a number of features related to lambdas that are present in other
APL interpreters but that are @b{NOT} implemented in GNU APL. This includes
multiple statements, guards, lexical scoping, and probably more.

The primary reason for these limitations in not the lazyness of the GNU APL
author, but the fact that the lambdas in other APL dialects introduce a 
execution semantics that is very different from the execution semantics of
@i{properly} defined functions (i.e. those created with the ∇-editor or
with @b{⎕FX}. For example, a properly defined function returns with @b{→0}
or after its last line was executed, while the lambdas in other dialects
also returns after the first non-committed value was computed (i.e. in the
middle of a multi-line function and without any @b{→0} (and without any 
warnings). The GNU APL author is an old-fashioned programmer from the APL1
aera who considers it outright dangerous to have two kinds of functions that
very much look the same (the only difference being the argument names @b{⍺}
and @b{⍵}), but that behave fundamentally different.

In contrast, GNU APL tries to keep the same execution semantics for lambdas
and for properly defined functions. As one of the consequences, you
@i{should not} create or modify a lambdas with @b{⎕FX} or with the ∇-editor
(it may be possible for debugging purposes, but then it is asking for trouble).


@c ----------------------------------------------------------------------------
@node Commands )COPY_ONCE )DUMP and DUMP-HTML
@section Commands )COPY_ONCE, )DUMP, and DUMP-HTML

In standard APL, workspaces are processed with the standard commands )LOAD,
)COPY, and )SAVE. GNU APL provides additional commands to process workspaces.

@c ............................................................................
@node )DUMP Command
@subsection )DUMP Command

GNU APL has a command )DUMP that saves a workspace to disk, similar to the
)OUT command. The difference between them is the file format being produced.
While )OUT produces a file in IBM's workspace interchange format (aka. an
.atf file), )DUMP produces a file in GNU APL's script format (i.e. readable
APL statements). Files written with )DUMP can be edited with normal text
editors (vi, Emacs), read back with "apl -f", or made executable (see
scripting).

@c ............................................................................
@node )DUMP-HTML Command
@subsection )DUMP-HTML Command

The )DUMP-HTML command is similar to the )DUMP command and the output format
is similar as well. The difference is that those characters that need
HTML-escaping (e.g. '<' becomes &lt; and '>' becomes '&gt; etc.) are
HTML-escaped. The output of the )DUMP-HTML command can therefore be directly
used by a web browser to display workspace listings (to share the code).

The files written with the )DUMP-HTML command have the extension @b{.html}.

In addition to the APL code, the .html files produced have a short HTML
<head> section template. The user should replace the fields marked
with @b{??????} with proper values (for the author, the copyright owner,
and a short description) before publishing the page.

@c ............................................................................
@node )COPY_ONCE Command
@subsection )COPY_ONCE Command

Command )COPY_ONCE copies all objects (variables, functions and operators)
from some other workspace into the current workspace similar to )COPY.
However, )COPY_ONCE does this only once; a second invocation of
)COPY_ONCE with the same workspace (and the same library reference number)
is silently ignored. This speeds up the )COPYing of libraries that occur
in several )DUMP-ed or )SAVE-ed workspaces.


@c ----------------------------------------------------------------------------
@node ]DOXY Command
@section ]DOXY Command

A particularly useful debug command is ]DOXY. It dumps the current workspace
in brows-able HTML format with listings of defined functions and hyperlinks
between them.

@smallexample
]DOXY                   ⍝ write documentation to /tmp/WSNAME/*
]DOXY dest              ⍝ write documentation to dest/WSNAME/*
@end smallexample

The starting point for browsing the documentation are the files:

@smallexample
/tmp/WSNAME/index.html         ⍝ for ]DOXY without arguments, or
dest/WSNAME/index.html         ⍝ for e.g. ]DOXY dest
@end smallexample

The index.html files above usually correspond to the following URIs in your
browser:

@smallexample
file:///tmp/WSNAME or
file:///absolute-path-to-dest/WSNAME respectively.
@end smallexample

In the above examples WSNAME is the )WSID of the workspace in which the ]DOXY
command was executed.

One can (and should make it a habit to) insert special comments into defined
functions which are copied into proper places inside the documentation that is
generated by the ]DOXY command. These "doxy" comments begin with ⍝⍝ (as opposed
to "normal" APL comments that start with a single ⍝. Doxy comments are
typically one-liners that briefly explain what a function is supposed to do.

For example:

@example
∇Z←A SUM B
 ⍝⍝ Return the sum of A and B          ← ]DOXY comment:  (double ⍝)
 ⍝  A: numeric                         ← "normal" APL comments (single ⍝) ...
 ⍝  B: numeric
 Z←A + B
∇
@end example

But please @b{note the following}: in the original @b{doxygen} program
(for C, C++, or other languages) source code lines may be a mix of code
and comments (including doxygen comments). In the @b{]DOXY} command of GNU
APL a line of a defined function shall be either APL code or else a doxygen
comment (i.e. starting with ⍝⍝). Doxygen comments indented with blanks are
allowed (for the sake of readability).

Since a doxygen comment is also a valid APL comment, it is perfectly legal
to put a doxygen comment at the end of a line that starts with APL code.
However, such lines will be ignored by the @b{]DOXY} command.

The parser generates no extra code from the doxygen comments, and therefore
the use of many doxygen comments causes no performance problem (and neither
do regular APL comments).


@c ----------------------------------------------------------------------------
@node ]KEYB Command
@section ]KEYB Command

In the early days of APL the user was typically faced with three problems:
APL keyboards, APL screens, and APL printers.

The advent of Unicode has soved the last two problems rather eleqantly:
the Uncode character set contains all APL characters, and screens (i.e CRT tubes
in the old days and screen windows these days) and printers display them
properly. What remains is the keyboard problem. The keyboard problem has two
sub-problems: Keyboard caps that show APL symbols in addition to the normal
ASCII characters, and generating proiper Unicodes when the keys are being
pressed (typically with the ALT or CTRL keys held down).

GNU APL provides the debug command @b{]KEYB} to help resolving these issues.
There are two cases where @b{]KEYB} may be useful:

@itemize
@item One can buy so called 'APL keyboards' or APL stickers that can be
glued on the key caps of a regular keyboard. As a matter of fact, APL
keyboards are simply regular keyboards with differen key caps. The keycodes
sent by APL keyboards do not differ from the keycodes of regular keyboards.
One the other hand, many programmers type blindly; for them the keycaps are
mostly irrelevant, and the substantial surcharge for APL keyboards may be
unreasonable. A related problem is national keyboards: most APL keyboards
have a US layout for their keycaps although a user may prefer a national
layout. Most of this can be helped with a keuboard layout that is printed
on the screen. A user that types bindly most of the time, but cannot
remember the position of all APL symbols (in particular of rarely used ones)
may find it useful to display the keyboard layout on the screen.

@item The other use case for @b{]KEYB} is troubleshooting the transformation
of keys pressed on the keyboard to Unicodes sent to the APL interpreter. This
transformation (which is explained in more detail below) is highly
conficurable. The good news is that these days any keyboard can be configured
to send any Unicode on any key as preferered by the user. The bad news is
that this configuration differs a lot between different platforms and can
be faily cumbersome. The @b{]KEYB} command is intended to help figuring
why pressing a partiicular key does not produce the APL character that it
should. It maybe wothwhile to note that even an APL keyboard can (and most
likely will) not produce the characters shown on its keypad if the
transformation from key presses to Unicdes is misconfigured. In that respect
APL keyboards are no better than keybiard stickers: Glueing a @b{⍴} sticker
onto the @b{R} does not make the @b{R} send the Unicode for @b{⍴}. Likewise,
an APL keyboard with @b{⍴} printed in addition to @b{R} does not make that
keyboard send the Unicode for @b{⍴}. Both cases only work as expected if
the transformation from keys to Unicdes is configured properly.
@end itemize

@c ----------------------------------------------------------------------------
@node The transformation of Keycodes to Unicodes
@subsection The transformation of Keycodes to Unicodes

In a nutshell, on a GNU/Linux (or other X-based) platform,
a key pressed or realeased is processed as follows:

@enumerate
@item The user presses or releases a key on the keyboard. This creates a
      @b{key-event} in the operating system. This key-event contains three
      items:
      @itemize
      @item a @b{keycode} which is a number between 0 and 255 that uniquely
            identidies the key. The keys in the main keyboard area (i.e.
            where all the alkphabetic keys are located), have keycodes between
            8 and 64,
      @item a @b{state} which tells if the key was pressed or released, and
      @item a @b{timestanp} which tells when key was pressed or released.
      @end itemize
@item The operating system processes the event:
      @itemize
      @item It remembers the new state of the key. If the key is a
            @b{modifier key} such as SHIFT, CTRL, or ALT, the the processing
            is done.
      @item Otherwise the key is a normal key and the operating system 
       computes a @b{keysymbol} from the @b{keycode}. In this computation
       the modifier keys whose states were remembered in the previous steps
       are taken into account. For that reason, e.g. pressing @b{R} when 
       (only) the SHIFT modifier key is pressed produces @b{R} and if SHIFT
       is released then it produces @b{r}. Every Unicode has a keysymbol (with
       the same value) but some keysymbols doe not correspond to a Unicode
       but o an action that the oprating system shall perform (e.g. cursor
       home in a termina window). A particularly important keysymbol is
       @b{0x0000} aka. @b{NoSymbol} which indicates that (for a given
       combination of modifer keys) no Unicode shall be generated.
      @end itemize

@item The operating system forwards the keysymbol towards the application
      (in our case GNU APL). On the way from the operating system to the
      application there may be other mappings applied to the keysymbol. For
      example, the window manager may bind functions to function the keys
      of the keyboard and call the function when the keysymbol for the
      function key was received from the operating system. Every such
      mapping can be a cause of trouble if keys needed by GNU APL are
      involved.
@end enumerate

@node xmodmap and friends
@subsection xmodmap and friends

The installation of an operating system installs a Keycodes to Unicodes
mapping according to the language choices made by the user during the 
installation. for example, in the US keycode @b{29} will be mapped to @b{y},
while in Germany keycode @b{29} will be mapped to @b{z}. At this point most
mappings for modifier ALT pressed, and most settings for CTRL pressed are
set to @b{NoSymbol} so that pressing these keys together with CTRL or ALT 
have no effect. To generate APL characteres instead of nothing,
the user has to modify the mapping as to produce APL Unicodes. This
modification is lost after rebooting the computer; therefore the mapping
is usually performed in a script that is run at login time of the user (i.e.
in ~/.bashrc).

The modification is made with a program. Unfortunately there exist several
such programs, e.g. @b{xmodmap}, @b{setxkbmap}, or @b{loadkeys}. See
@b{README-3-keyboard} for details. More is not always better
and the different programs for the same purpose have undesirable side effects
when used in parallel. An important lesson after experimenting with several
of these programs was: @i{Use one and only one of these programs on your
platform}.

Each of these programs has its own set of pitfalls. For example with
@b{xmodmap}, pressing @b{ALT-SHIFT-L} (in that order produces ⌷. Pressing
@b{SHIFT-ALT-L} produces nothing.

Despite of this, @b{xmodmap} seems to be the simplest and most reliable
of the modidication programs. It may have less capabilities than the others,
but has everything needed to run GNU APL. For this reason, GNU APL is
shipped with a configuration file for US keyboards
(@b{support-files/Juergen-Mint-21/xmodmap.apl}) and a file to undo the
mapping (@b{support-files/Juergen-Mint-21/xmodmap.ascii}). The latter is
redundant since @b{xmodmap} provides a way to save the current mapping
(see @b{man xmodmap}). There was a short period where @b{xmodmap} did not
work in Linux Mint several yeras ago, but this has been fixed in the meantime.

It is impossible for the author to explain all available mapping programs
and therefore this manual focuses on @b{xmodmap} and, as an alternative,
@b{setxkbmap/xkbcomp}. @b{xmodmap} is considerably simpler to configure
than @b{setxkbmap/xkbcomp} but not available on all platforms.

Suppose we run:

@b{$ xmodmap xmodmap.apl}

and suppose pressing key @b{R} does @i{@b{not}} produce @b{⍴}. This can have
the following reasons:

@enumerate
@item incorrect keycode for key @b{R},
@item correct keycode, but incorrect keysymbol(s), or
@end enumerate

Both cases can be identified with the @b{]KEYB}  command.

@node The ]KEYB command
@subsection The ]KEYB command

@node ]KEYB Syntax
@subsubsection ]KEYB Syntax

The syntax of the @b{]KEYB} command is:

@example
      @b{]KEYB [mode] [arguments]}
@end example

where the optional @b{mode} determines how the output shall be generated, and
the optional arguments specify additional details for the output. There are
3 different modes:

@itemize
@item Mode @b{XMOD} uses the program @b{xmodmap} to figure the keyboard layout.
      This mode, if it works, produces an output that is closest to reality.
@item Mode @b{XKBD} uses the X extension @b{Xkbd} which is typically configured
      with the programs @b{setxkbmap} and @b{xkbcomp}. GNU APL reads the result
      of that configuration back with function @b{XkbKeycodeToKeysym}. This
      mode, if it works, produces an output that is also closest to reality.
@item Mode @b{USER} display a keyboard layout provided by the user. The output
      of this mode is as correct as its input.
@item mode @b{GUESS} displays a typical APL keyboard layout (which may differ
      considerably from the true keyboard layout).
@end itemize

@b{WARNING}: In theory one could map some keys with @b{xmodmap} and others
with @b{xkbcomp}. Even though one might be able to achieve a desired mapping
in this way (and maybe quicker than with either of the methods alone),
@b{]KEYB} will not be able to display it properly. This is because the
@b{xmodmap -pke } does not see the @b{xkbcomp} mapping and the @b{X} function
@b{XkbKeycodeToKeysym()} does not see the @b{xmodmap} mapping. 

@node ]KEYB without a mode
@subsubsection ]KEYB without a mode

Command i@b{]KEYB} without a mode is for the normal day to day use (as
opposed to troubleshooting). It displays an APL keyboard on the screen,
which is very useful if the keyboard has no APL stickers or key caps.

@b{]KEYB} without a mode tries (in that order):

@enumerate
@item @b{]KEYB XMOD}
@item @b{]KEYB USER}
@item @b{]KEYB GUESS}
@end enumerate

The command stops after the first mode that was successful. Mode @b{GUESS}
always succeeds, therefore @b{]KEYB} will always produce an output. If that
output is correct is a different question, though.

The first line of the keyboard displayed shows which of the 3 methods above
was used to generate the output.

As may have become clear from the explanations above and/or from reading
@b{README-3-keyboard} in the top-level GNU APL directory, @b{]KEYB XMOD}
will fail if either @b{xmodmap} is not used at all, or if @b{xmodmap} is
combined with other keyboard mapping programs. For these cases, the user can
disable the automatic use of @b{xmodmap} in one of her preferences files.
The option to do so is @b{NO-XMODMAP} and ii is effective only of
]KEYB is calle  without a mode. That is, @b{]KEYB XMOD} will use @b{xmodmap}
even if @b{xmodmap} was disabled by @b{NO-XMODMAP} in a preferences file.


@node ]KEYB XMOD
@subsubsection ]KEYB XMOD

@b{]KEYB XMOD} runs, under the hood, @b{xmodmap -pke} to read the current
keycode to keysymbol mapping and displays the result in form of a
keyboard layout on the screen. For example:

@example
      ]KEYB XMOD
US Keyboard Layout.    Source: xmodmap -pke

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ ~  ║ !⌶ ║ @@⍫ ║ #⍒ ║ $⍋ ║ %⌽ ║ ^⍉ ║ &⊖ ║ *⍟ ║ (⍱ ║ )⍲ ║ _! ║ +⌹ ║         ║
║ `◊ ║ 1¨ ║ 2¯ ║ 3< ║ 4≤ ║ 5= ║ 6≥ ║ 7> ║ 8≠ ║ 9∨ ║ 0∧ ║ -× ║ =÷ ║ BACKSPC ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║       ║ Q  ║ W⍹ ║ E⋸ ║ R  ║ T⍨ ║ Y¥ ║ U  ║ I⍸ ║ O⍥ ║ P⍣ ║ @{⍞ ║ @}⍬ ║ |⊣   ║
║  TAB  ║ q? ║ w⍵ ║ e∈ ║ r⍴ ║ t∼ ║ y↑ ║ u↓ ║ i⍳ ║ o○ ║ p⋆ ║ [← ║ ]→ ║ \⊢   ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║ (CAPS   ║ A⍶ ║ S  ║ D  ║ F  ║ G  ║ H⍙ ║ J⍤ ║ K  ║ L⌷ ║ :≡ ║ "≢ ║         ║
║ LOCK)   ║ a⍺ ║ s⌈ ║ d⌊ ║ f_ ║ g∇ ║ h∆ ║ j∘ ║ kλ ║ l⎕ ║ ;⍎ ║ '⍕ ║ RETURN  ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║             ║ Z  ║ Xχ ║ C¢ ║ V  ║ B£ ║ N  ║ Mμ ║ <⍪ ║ >⍙ ║ ?⍠ ║          ║
║  SHIFT      ║ z⊂ ║ x⊃ ║ c∩ ║ v∪ ║ b⊥ ║ n⊤ ║ m∣ ║ ,⍝ ║ .⍀ ║ /⌿ ║  SHIFT   ║
╚═════════════╩════╩════╩════╩════╩════╩════╩════╩════╩════╩════╩══════════╝
@end example

Naturally, this only works, iff:

@itemize
@item @b{xmodmap} is installed and working,
@item the current keyboard mapping was installed with @b{xmodmap},
@item no other keyboard mappings are in effect, and
@item the keycodes are correct (see below).
@end itemize

IOW: @b{]KEYB XMOD} troubleshoots the case
@i{correct keycode, but incorrect keysymbol(s)} above.

@node ]KEYB USER
@subsubsection ]KEYB USER

@b{]KEYB USER} does not try to figure the current keyboard mapping but rather
displays a static file provided by the user. This mode is a fallback for the 
case where the other modes are not able to produce a correct keyboard layout.
The user can take the output of @b{]KEYB GUESS} as a starting point and correct
the deviations from the actual keyboard layout.

The name of file shown with @b{]KEYB USER} can be configured in one of the
@b{preferences} files.

@node ]KEYB GUESS
@subsubsection ]KEYB GUESS

@b{]KEYB GUESS} does not try to figure the current keyboard mapping but rather
displays the layout that the various keyboard configuration files that are
shipped with GNU APL are likely to produce. The layout is a standard US APL
keyboard.

@node Optional ]KEYB arguments
@subsubsection Optional ]KEYB arguments

@b{]KEYB XMOD KEYS} runs @b{xmodmap -pke}, but first shows the assumed
physical layout of the keyboard. By comparing the first half of the output
with the second half, one can see how @b{xmodmap} mapes keycodes to
Unicode characters. For example:

@example
      ]KEYB XMOD KEYS
Physical Keyboard:      Source: GNU APL builtin

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║
║ 49 ║ 10 ║ 11 ║ 12 ║ 13 ║ 14 ║ 15 ║ 16 ║ 17 ║ 18 ║ 19 ║ 20 ║ 21 ║ 22      ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║  Kc   ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║  Kc  ║
║  23   ║ 24 ║ 25 ║ 26 ║ 27 ║ 28 ║ 29 ║ 30 ║ 31 ║ 32 ║ 33 ║ 34 ║ 35 ║  51  ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║  Kc     ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║
║  66     ║ 38 ║ 39 ║ 40 ║ 41 ║ 42 ║ 43 ║ 44 ║ 45 ║ 46 ║ 47 ║ 48 ║ 36      ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║    Kc       ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║   Kc     ║
║    50       ║ 52 ║ 53 ║ 54 ║ 55 ║ 56 ║ 57 ║ 58 ║ 59 ║ 60 ║ 61 ║   62     ║
╠════╦═════╦══╩══╦═╩════╩════╩════╩════╩════╩════╩══╦═╩═══╦╩════╬═════╦════╣
║ Kc ║ Kc  ║ Kc  ║                Kc                ║ Kc  ║ Kc  ║ Kc  ║ Kc ║
║ 37 ║115  ║ 64  ║                65                ║113  ║116  ║ 109 ║105 ║
╚════╩═════╩═════╩══════════════════════════════════╩═════╩═════╩═════╩════╝

Keyboard Layout.    Source: xmodmap -pke

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ ~  ║ !⌶ ║ "  ║ #⍒ ║ $⍋ ║ %⌽ ║ ^⍉ ║ &⊖ ║ *⍟ ║ (⍱ ║ )⍲ ║ _! ║ +⌹ ║         ║
║ `◊ ║ 1¨ ║ 2⍫ ║ 3< ║ 4≤ ║ 5= ║ 6≥ ║ 7> ║ 8≠ ║ 9∨ ║ 0∧ ║ -× ║ =÷ ║ BACKSP  ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║       ║ Q  ║ W⍹ ║ E⋸ ║ R  ║ T⍨ ║ Y¥ ║ U  ║ I⍸ ║ O⍥ ║ P⍣ ║ @{⍞ ║ @}⍬ ║  |⊣  ║
║  TAB  ║ q? ║ w⍵ ║ e∈ ║ r⍴ ║ t∼ ║ y↑ ║ u↓ ║ i⍳ ║ o○ ║ p⋆ ║ [← ║ ]→ ║  \⊢  ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║  (CAPS  ║ A⍶ ║ S  ║ D  ║ F  ║ G  ║ H⍙ ║ J⍤ ║ K  ║ L⌷ ║ *≡ ║ "≢ ║         ║
║  LOCK)  ║ a⍺ ║ s⌈ ║ d⌊ ║ f_ ║ g∇ ║ h∆ ║ j∘ ║ kλ ║ l⎕ ║ +⍎ ║ '⍕ ║ RETURN  ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║             ║ Z  ║ Xχ ║ C¢ ║ V  ║ B£ ║ N  ║ Mμ ║ <⍪ ║ >⍙ ║ ?⍠ ║          ║
║    SHIFT    ║ z⊂ ║ x⊃ ║ c∩ ║ v∪ ║ b⊥ ║ n⊤ ║ m∣ ║ ,⍝ ║ .⍀ ║ /⌿ ║   SHIFT  ║
╠════╦═════╦══╩══╦═╩════╩════╩════╩════╩════╩════╩══╦═╩═══╦╩════╬═════╦════╣
║    ║     ║     ║                                  ║     ║     ║ !⌶  ║    ║
║CTRL║ Win ║ ALT ║              SPACE               ║ ALT ║ Win ║ 1¨9 ║CTRL║
╚════╩═════╩═════╩══════════════════════════════════╩═════╩═════╩═════╩════╝
@end example

Argument @b{KPAD} also shows the keypad of the keyboard. The keypad is normally
hidden so that the output fits into a 80-column terminal window. With @b{KPAD}
the terminal should have at least 101 columns. The standard GNU APL layout puts
line graphics characters onto the keypad, as well as « and » and some
currency symbols.

Argument @b{FUNK} also shows the function key row of the keyboard, but
only for @b{]KEYB XMOD KEYS}:

@example
      ]KEYB XMOD KEYS FUNK KPAD
US Keyboard Layout.    Source: GNU APL builtin

╔════╗    ╔════╦════╦════╦════╗    ╔════╦════╦════╦════╗    ╔════╦════╦════╦════╗    ╔════╦════╦════╗
║ Kc ║    ║ Kc ║ Kc ║ Kc ║ Kc ║    ║ Kc ║ Kc ║ Kc ║ Kc ║    ║ Kc ║ Kc ║ Kc ║ Kc ║    ║    ║ Kc ║ Kc ║
║ 09 ║    ║ 67 ║ 68 ║ 69 ║ 70 ║    ║ 71 ║ 72 ║ 73 ║ 74 ║    ║ 75 ║ 76 ║ 95 ║ 96 ║    ║    ║ 78 ║ 77 ║
╚════╝    ╚════╩════╩════╩════╝    ╚════╩════╩════╩════╝    ╚════╩════╩════╩════╝    ╚════╩════╩════╝

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗    ╔════╦════╦════╦════╗
║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║    ║ Kc ║    ║ Kc ║ Kc ║
║ 49 ║ 10 ║ 11 ║ 12 ║ 13 ║ 14 ║ 15 ║ 16 ║ 17 ║ 18 ║ 19 ║ 20 ║ 21 ║ 22      ║    ║ 77 ║    ║ 82 ║ 82 ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣    ╠════╬════╬════╬════╣
║  Kc   ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc   ║    ║ Kc ║ Kc ║ Kc ║    ║
║  23   ║ 24 ║ 25 ║ 26 ║ 27 ║ 28 ║ 29 ║ 30 ║ 31 ║ 32 ║ 33 ║ 34 ║ 35 ║ 51   ║    ║ 79 ║ 80 ║ 81 ║ Kc ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣    ╠════╬════╬════╣ 86 ║
║ Kc      ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║    ║ Kc ║ Kc ║ Kc ║    ║
║ 66      ║ 38 ║ 39 ║ 40 ║ 41 ║ 42 ║ 43 ║ 44 ║ 45 ║ 46 ║ 47 ║ 48 ║ 36      ║    ║ 83 ║ 84 ║ 85 ║    ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣    ╠════╬════╬════╬════╣
║  Kc         ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║  Kc      ║    ║ Kc ║ Kc ║ Kc ║    ║
║  50         ║ 52 ║ 53 ║ 54 ║ 55 ║ 56 ║ 57 ║ 58 ║ 59 ║ 60 ║ 61 ║  62      ║    ║ 87 ║ 88 ║ 89 ║ Kc ║
╠════╦═════╦══╩══╦═╩════╩════╩════╩════╩════╩════╩══╦═╩═══╦╩════╬═════╦════╣    ╠════╩════╬════╣108 ║
║ Kc ║ Kc  ║ Kc  ║              Kc                  ║ Kc  ║ Kc  ║ Kc  ║ Kc ║    ║ Kc      ║    ║    ║
║ 37 ║115  ║ 64  ║              65                  ║113  ║116  ║ 109 ║105 ║    ║ 90      ║    ║    ║
╚════╩═════╩═════╩══════════════════════════════════╩═════╩═════╩═════╩════╝    ╚═════════╩════╩════╝

@end example

As the @b{Source: GNU APL builtin} in the example above indicates, this is
the layout assumed by GNU APL, which may differ from the true layout of your
keybord. For example, say key @b{R} with ALT pressed does not produce @b{⍴}.
The Unicode for @b{⍴} is :

@example
      ⎕UCS "⍴"                    ⍝ Unicode for ⍴
9076
      ,5 ⎕CR 256 256 ⊤ ⎕UCS "⍴"   ⍝ Unicode for ⍴ in hex
2374
@end example

One can check if the keycodes displayed by @b{]KEYB XMOD KEYS}  are
correct with a programm called @b{xev} (which is normally pre-installed in
most GNU/Linux distributions. After starting @b{xev} and hitting the @b{R}
key several times (because it also shows many other unrelated events),
one can see the corresponding key events:

@example
      ...
KeyPress event, serial 38, synthetic NO, window 0x3c00001,
    root 0x75c, subw 0x0, time 35629183, (-590,275), root:(284,737),
    state 0x0, keycode 27 (keysym 0x72, r), same_screen YES,
    XLookupString gives 1 bytes: (72) "r"
    XmbLookupString gives 1 bytes: (72) "r"
    XFilterEvent returns: False
@end example

So we see that the @b{R} key does indeed have the keycode @b{27}.
Next we look at the line for key keycode @b{27} in the @b{xmodmap.apl}
file:

@example
keycode  27 =  r               R               U2374
@end example

This line tells us that, when the @b{R} is hit, the following
Unicodes shall be emitted:

@itemize
@item the Unicode for (lowercase) @b{r} if neither SHIFT nor ALT is held down,
@item the Unicode for (uppercase) @b{R} if SHIFT is pressed and ALT is not,
@item the Unicode for @b{⍴} = $2374  if ALT is pressed and SHIFT is not.
@end itemize

The line could have had a fourth entry (for SHIFT and ALT pressed); if that
entry is, like above,  missing then the same Unicode is emitted for both
ALT and SHIFT+ALT. Likewise, if the second entry is missing then the same 
code is used for SHIFT and no SHIFT. This is, for example, the case for
keycode 36 with keysym @b{Return} (the key that is sometimes labelled
@b{Enter} and sometimes @b{Return} and whose keysym is independent of the
state of the SHIFT and ALT keys. In the xmodmap the keysyms can be specified
in multiple ways: as ASCII characters (like @b{r} and @b{R} above, as
Unicodes like @b{U2374} above, or as symbols like @b{NoSymbol} or @b{Return},
which are defined in files @b{/usr/include/X11/X.h} and
 @b{/usr/include/X11/keysymdef.h}:

@example
#define NoSymbol         0L  /* special KeySym */
#define XK_Return    0xff0d  /* Return, enter */
@end example

Given the explanations above, the troubleshooting of keyboard mappings
becomes simple (though somewhat tedious in the details):

@enumerate
@item do @b{]KEYB XMOD KEYS} in GNU APL to display the keycodes, then
@item  run @b{xev} in a different window and verify, for every key on
       the keyboard, that the keycode shown by @b{xev} is the one shown by
       @b{]KEYB XMOD KEYS}, then
@item do @b{]KEYB XMOD} in GNU APL to display the keysyms, then
      @item edit the @b{xmodmap.apl} file (or whatever other file ws used
      as the argument for the @b{xmondmap} command) to see if the Unicodes
      for every keycode line are correct. In that test:
      @itemize
      @item the first keysym is used for the lowercase character, i.e. if
            neither SHIFT nor ALT are pressed,
      @item the second keysym is used for the uppercase character, i.e. if
            SHIFT is pressed and ALT is not. If no second keysym is present,
            then the first keysym is used instead,
      @item the third keysym is used for the character that shall be emitted
            when the ALT key is pressed and the SHIFT is not. If no third
             keysym is present then nothing is emitted, and
      @item the fourth keysym is used for the character that shall be emitted
            when both the ALT key and the SHIFT are pressed. If no fourth
            keysym is present then the behavior is the same as for ALT
            without SHIFT.
      @end itemize
@end enumerate

@c ----------------------------------------------------------------------------
@node ]NEXTFILE and ]PUSHFILE Commands
@section ]NEXTFILE and ]PUSHFILE Commands

@c ............................................................................
@node ]NEXTFILE
@subsection ]NEXTFILE

The debug command ]NEXTFILE, when used in an APL script file, terminates
the processing of that file and continues processing in the next script file
(if any) or else enters immediate execution mode. Unlike )OFF (which terminates
the interpreter), ]NEXTFILE does not terminate the interpreter but only changes
its input source to the next script file (if there is one remaining) and enters
immediate execution only if all scripts were processed.

The user may, for example, want to use the space near the end of the script
to add longer comments (as to what the script does, how it works, how it is
used, etc.) without the need to prepend every line with ⍝ or #.

For example:

@example
#!/usr/local/bin/apl

⍝ see documentation at the end of this file

<APL CODE...>

]NEXTFILE

This workspace does the following...
@end example


@c ............................................................................
@node ]PUSHFILE
@subsection ]PUSHFILE

The debug command ]PUSHFILE is similar to ]NEXTFILE in that it terminates
the processing of the current script file. Unlike ]NEXTFILE, ]PUSHFILE does
not proceed to the next script file but enters a new immediate execution
context in which the user can interact with the interpreter. This context
processes user input until a ]NEXTFILE command is given. After a ]NEXTFILE
command, the immediate execution context processing proceeds at the next
line after the ]PUSHFILE command.

@c ----------------------------------------------------------------------------
@node History and TAB completion
@section History and TAB completion

Until GNU APL 1.4 / SVN 465, GNU APL used libreadline for interactive user
input. libreadline did provide two useful features: tab expansion (the tab
key would expand file names) and history (the cursor up/down keys would recall
previously entered lines).

Since SVN 465 libreadline was removed and the standard TAB expansion and
history of libreadline were replaced by more context sensitive (i.e. APL
aware) implementations:

1. Instead of simply recalling the last line entered by the user, there are
now different histories for different input contexts:

1a. The input history in immediate execution recalls the last line entered
in immediate execution (and not, for example, lines entered in function
editing mode or ⍞ input.

1b. Likewise, ⍞ recalls the last line entered for ⍞-input

1c. ⎕ recalls the last line entered for ⎕-input

1d. In the ∇-editor, the other function lines of the function being edited
can be recalled. This is far more handy than the ∇-editor commands for
recalling function lines (which are not fully supported in GNU APL).

2. Instead of always TAB-completing file names, the tab character now
understands different TAB-completion contexts:

2a. Input starting with . or / is completed as a filename like readline did.

2b. Input starting with ) or ] is completed as command-name name or, to some
extent, as command arguments.

2c. Input starting with ⎕ is completed as a system function name or a
system variable name.

2d. Input starting with letters, ∆, or ⍙ is completed as a user defined
function or variable name.


@c ----------------------------------------------------------------------------
@node Logging Facilities
@section Logging Facilities

The APL interpreter has over 30 logging facilities. Each logging facility
can be ON (and then produces some logging output on stderr) or OFF. The
decision which logging facility shall be ON and which shall be OFF can be
made at compile time (of the APL interpreter) or at run-time. 

If the decision is made at compile time - we call that static logging - then
it cannot be changed later on. Otherwise - we call that dynamic logging -
there is a debug command ]LOG that allows logging facilities to be turned
ON or OFF.

@c ............................................................................
@node Static Logging
@subsection Static Logging

By default the logging facilities that shall be turned ON are defined
statically. To change the logging facilities that shall be turned ON,
you can edit the file src/Logging.def which defines the different logging
facilities. The first argument of macro log_def() specifies that the
indicated logging facility shall be ON (1) or OFF (0).

Static logging results in a faster interpreter than dynamic logging because
the decision to log something is made at compile time and not at run-time.

If you benchmark the APL interpreter, then ./configure Static Logging by
NOT setting DYNAMIC_LOG_WANTED=yes.

@c ............................................................................
@node Dynamic Logging
@subsection Dynamic Logging

Dynamic Logging is intended for trouble-shooters of GNU APL, but also for
those who are interested in the internals of GNU APL. Dynamic Logging is
enabled by setting DYNAMIC_LOG_WANTED=yes when running ./configure.

If Dynamic Logging is enabled, then the already mentioned file
src/Logging.def determines the initial setting of each logging facility.

The command

 ]LOG

(without arguments) then shows all logging facilities and their current
state. The command

]LOG N

toggles the state of logging facility N from OFF to ON and back.


@c ----------------------------------------------------------------------------
@node Rational Numbers
@section Rational Numbers

GNU APL has limited support for rational numbers. Instead of dividing integers
(and possibly causing rounding errors), integer quotients are kept undivided
internally until some function requires a conversion to a floating point
(double) value.

Currently only +, -, ×, and ÷ preserve rational numbers where possible, but
this list may grow in the future. Monadic + (a no-op for non-complex numbers)
explicitly converts rational numbers to floating point numbers.

A quotient is internally stored as a 64-bit numerator and a 64-bit denominator.
In some cases arithmetic with rational numbers is faster than with doubles, but
in most cases it is slower.

For that reason support for rational numbers is disabled by default and must
be enabled via ./configure (see README-2-configure).


@c ----------------------------------------------------------------------------
@node Hex Numbers
@section Hex Numbers

GNU APL supports sedecimal numbers. They start with $ and can be uppercase or
lowercase:

@example
      $2a
42

      $2A
42
@end example


@c ----------------------------------------------------------------------------
@node User-defined Commands
@section User-defined Commands

There is a simple mechanism to define additional APL commands. This mechanism
is intended to introduce new commands by APL libraries. Like system commands,
user-define commands can only be executed in immediate execution mode and not
from user-defined functions or from ⍎. It is not intended to extend the
functionality of user-defined commands beyond what is being described in the
following.

A user-defined command ]NEW_COMMAND is created with the debug command
]USERCMD like this:

@example
      ]USERCMD ]NEW_COMMAND APL_FUNCTION [mode]
@end example

APL_FUNCTION is an APL function that will be called when the command is
entered in immediate execution mode. The entire line entered by the user,
starting at ]NEW_COMMAND, is the right argument of APL_FUNCTION. If mode is
missing (or 0) then APL_FUNCTION is called monadically. If mode is 1 then
APL_FUNCTION is called dyadically; the left argument is a vector of strings
that is the left argument broken down into individual argument strings.

The function APL_FUNCTION that implements a command need not exist when the
command is created.

A single user-defined command ]UCMD, or all user-defined commands can
be deleted like this:

@example
      ]USERCMD REMOVE ]UCMD
      ]USERCMD REMOVE-ALL
@end example

@c ----------------------------------------------------------------------------
@node Structured Variables and Associative Arrays
@section Structured Variables and Associative Arrays

GNU APL has implemented two features that are closely related because, under
the hood, they share the same implementation: @b{structured variables} and
@b{associative arrays}.

@c ............................................................................
@node Structured Variables
@subsection Structured Variables

A structured variable is an APL variable that contains several related but
otherwise independent sub-variables aka. @b{members}. Such structured variables
can be convenient for passing many related arguments to a function, or
for returning multiple results from a function.

The different sub-variables of a structured variable are accessed by the
(top-level) name of the variable followed by a non-empty sequence of member
names, separated by '.'. In the following we will use uppercase names for
structured variables and lowercase names for their members. For example:

@example
      PERSON.address.street
@end example

is a structured variable @b{PERSON}, which has a member @b{address},
and @b{PERSON.address} is a structured (sub-)variable of PERSON which has
a member @b{street}.

For a structured variable and all its sub-variables, the usual rules for
normal variables apply. They can be created, erased, and even passed as
arguments to functions. @b{Nota bene}: Some GNU APL operators are implemented
as macros (i.e. internal defined APL functions). The macros may call APL
primitives that invalidate the fact that a value is structured and, as a
consequence, the operator results may become plain APL values. If this happens,
the structured nature of a plain APL value can be restored with 38 ⎕CR.

A structured variable is created by assigning a value to one of its members.
For example:

@example
      PERSON.firstname ← 'Jane' ⍝ create variable PERSON with member 'firstname'
      PERSON.lastname  ← 'Doe'  ⍝ add a second member 'lastname' to PERSON
@end example

The depth of a newly created member can be more than one. In this case the
intermediate members are created automatically. For example:

@example
      PERSON.address.street ← '42 Main Street' ⍝ implicitly creates PERSON.address
@end example

Empty structured variables can be created with @b{38 ⎕CR}:

@example
      EMPTY ← 38 ⎕CR CAPACITY ← 32
@end example

Note: A structured variable is automatically expanded when new members are
added to it. From time to time this expansion exceeds the space allocated
for the members of the variable and then the existing members need to be
copied into a new, larger structure. The overhead caused by this reorganization
can be avoided by specifying a sufficiently large capacity when the structure
is created (with @b{38 ⎕CR}).

An entire structured variable can be erased with @b{⎕EX} or with
@b{)ERASE} just like other variables. They erase the variable along
with all its members. In addition to erasing an entire structured variable,
individual members at any depth can also be erased with @b{⎕EX} or
@b{)ERASE}:

@example
      )ERASE PERSON.address      ⍝ OK, PERSON.address exists
      )ERASE PERSON.address      ⍝ error: PERSON.address does not exist anymore
NOT ERASED: PERSON.address
@end example

After having been created, the members of a structured variable can be
referenced and overwritten just like normal variables:

@example
      PERSON.address.street ←  '42 Main Street'   ⍝ create member address.street
      PERSON.address.street   ⍝ reference member address.street of PERSON
42 Main Street

      PERSON.address.street ←  '44 Main Street'   ⍝ overwrite address.street
      PERSON.address.street
44 Main Street
@end example

The members of a structured variable form a tree of (sub-) variables, similar
to the file system on a computer. This tree has:

@itemize
@item one root (which is the structured variable itself),

@item zero or more non-leafs (like sub-directories of the top-level root
  directory), and

@item zero or more leafs. The case of zero non-leafs occurs, for example,
directly after a new empty variable was created with @b{38 ⎕CR}.
@end itemize

Due to their tree-like structure, structured variables need to be printed
somewhat differently than normal APL variables. For example:

@example
      PERSON
.firstname: ┌→───┐
            │Jane│
            └────┘

.lastname: ┌→──┐
           │Doe│
           └───┘

.address: 
.address.street: ┌→─────────────┐
                 │44 Main Street│
                 └──────────────┘
@end example

Only the leafs of a deeply structured variable can have values and, as a
precaution, assigning a value to a non-leaf (including the root) raises
a DOMAIN ERROR:

@example
      B.b.c←'leaf-Abc'        ⍝ OK, since B.b.c will be a leaf
      B.b←42                  ⍝ will fail since B.b is not a leaf
DOMAIN ERROR+
      B.b←42
       ^ ^
      )MORE
member access: cannot override non-leaf member A.b
)ERASE or ⎕EX that member first.

      )ERASE B.b
      B.b ← 'leaf-Ab'         ⍝ now OK, since B.b will now become a (new) leaf

      )SIC
@end example

On the other hand, assigning a structured variable to the leaf of another
structured variable is valid and concatenates the members:

@example
      )ERASE A
      A.b.c ← 'leaf-Abc'      ⍝ variable A with leaf A.b.c
      C.d.e ← 'leaf-cde'      ⍝ variable C with leaf C.d.e
      A.b.c ← C               ⍝ overwrite leaf A.b.c of A
      A.b.c.d.e
leaf-cde
@end example

The file @b{src/testcases/Structured_variable.tc} contains the examples
above (and more) 

@c ............................................................................
@node Associative Arrays
@subsection Associative Arrays

APL arrays are primarily indexed with numbers (or arrays of numbers), which
is also the most efficient method. Many other languages provide, either in
addition or else alternatively, a method to index arrays by @b{keys}, where
the keys are frequently character strings. Such arrays are commonly referred
to as @b{associative arrays}.

APL has no associative array per se, but the structured variables in GNU APL
can be used for the same purpose. The only limitation is (currently) that the
keys need to be character strings (as opposed to arbitrary APL values).

That is, in GNU APL an associative array

@itemize
@item is a structured variable, and
@item can be indexed with arbitrary (!) strings
@end itemize

As long as the strings that are used as keys follow the same rules as APL
variable names (no leading digit, no '.' etc.), a structured variable can
also be indexed with a string. That can be useful if keys are being computed
or passed as function arguments. However, unlike indexing of APL arrays with
numbers, only one key per (bracket-) index is permitted.

@example
   A.key ← 42
   A.key
42
   A['key']
42
   A['key']←24
   A.key
24
   A['key' 'key']   ⍝ not allowed even though 'key' is a valid member
DOMAIN ERROR
      D['key' 'key']
@end example

Strings that do not follow the rules for variable names can be used with
bracket index, but not with the .member syntax:

@example
      ASSOC ← 38 ⎕CR 8
      KEY←'key.dot'   ⍝ works, but avoid such keys
      ASSOC[KEY]←42
      ASSOC[KEY]
42
      ASSOC[KEY]←43
      ASSOC[KEY]
43
      ASSOC.key.dot   ⍝ won't work: 'key.dot' is a single key, but key.dot is 2 keys
VALUE ERROR+
      ASSOC.key.dot
               ^
      )MORE
member access: structure ASSOC has no member key

      ASSOC.key.dot←44   ⍝ works: 2 (nested) keys
      ASSOC.key.dot
44
      ASSOC['key.dot']   ⍝ works: one key containing '.'
43
@end example

This is because ASSOC.key.dot above is being tokenized by the APL parser into
[ASSOC] [key] [dot], while 'key.dot' is a single key. The example above (with
different values for ASSOC.key.dot and for ASSOC['key.dot'] demonstrates why
such keys are better avoided.

Even though (one) bracket index can only access the top-level members of a
structured variable, repeated bracket index (or dyadic PICK) can be used to
access deeper nesting levels:

@example
      D.b.c←42
      D['b.c']   ⍝ will fail
INDEX ERROR+
      D['b.c']
      ^^
      )MORE
member access: member b.c was not found. The valid members are:
      b

      D['b']['c']   ⍝ will work
42

      ⍝ alternatively: use PICK
      ⍝
      'b' 'c' ⊃ D   ⍝ fails: 'b' 'c' is 'bc'
      "b" "c" ⊃ D   ⍝ works
42
@end example

In the above examples each bracket index (or each element of PICK) discards the
current top-level structure and descends into the structured sub-variable of
the chosen member. In this case PICK is more efficient since repeated bracket
index creates copies of the intermediate structured sub-variables.

If a variable is used as a structured variable then the number of its members
is typically small (since every member name occurs explicitly in the APL code
that uses it). If a variable is used as an associative array, then the member
names are typically being computed and the number of members can become rather
large (like the different key values in a database). When such a variable
(i.e. with many members) reaches its capacity, then finding a member (when
the array is indexed), or finding an unused place in the variable (when a new
member is added) becomes slower and slower. The GNU APL implementation has
addressed this by hashing into the variable (based on the member name)
instead of, for example, sequentially allocating the members. This is very
fast under normal circumstances, but becomes as slow as sequential allocation
when the number of members in the variable reaches its current capacity.
For this reason, if a structured variable is used as an associative array,
e.g. as a database, then it is better to not rely on the (automatic)
reorganization of the variable, but to create a large enough variable
(with 38 ⎕CR) from the beginning.

If that is not possible (because the (performance-) problem has occurred after
the variable was created, then one can manually increase the capacity like this:

@example
      TMP ← 39 ⎕CR ASSOC_ARRAY   ⍝ save ASSOC_ARRAY as normal APL array
      ⊣ ⎕EX 'ASSOC_ARRAY'        ⍝ erase it so that it can be assigned
      ASSOC_ARRAY ← 38 ⎕CR TMP   ⍝ new associative array with ≥ twice the size
@end example

As a rule of thumb, associative arrays will be fast as long their capacity is
more than twice the number of their members. The expression ASSOC_ARRAY[;1]
returns all keys (= members) of ASSOC_ARRAY, therefore the number of members
is ⍴ASSOC_ARRAY[;1] and the capacity is ↑⍴ASSOC_ARRAY.


@c ----------------------------------------------------------------------------
@node Monadic ⊢ and ⊣; dyadic ⊢ with Axis
@section Monadic ⊢ and ⊣; dyadic ⊢ with Axis

Monadic ⊢ is the identity function. It returns its (committed or non-committed)
right argument as a non-committed value.

Conversely, monadic ⊣ (called @b{Hide} in GNU APL) discards its (committed or
non-committed) right argument and returns a committed integer scalar 0. 

For the most part there is no difference between a committed value (= a value
that was assigned to a variable, including ⎕ and ⍞) and a non-committed value.
The point where it does make a difference is when the value is the final result
of a statement (as opposed to an intermediate result inside a statement). In
that situation (and only there) a non-committed value is printed but a
committed value is not.

You can use ⊢ in a similar fashion as ⎕← at the left end of a statement,
to print a value even though it was previously assigned to a variable.

The main motivation for ⊣ is that, at least in GNU APL, lambdas always return
a value. However, if a lambda is used only for the sake of its side effects,
say to print something, then the value returned by the lambda is often of no
interest and only messes up the APL output. In that situation ⊣ can be used
to suppress the printing of undesired return values from lambdas.

In earlier GNU APL versions, ⊣B and ⊢B would both return B; with ⊣ as committed
value and with ⊢ as non-committed value. But since the only real-life purpose
of ⊣ is to suppress the printing of B, the implementation of ⊣ was changed to
returning a committed integer scalar 0 instead of committed B. That reduced
the run-time of ⊣B from O(,B) to O(1). Also, ⊢B is marginally faster than ⎕←B.

Dyadic ⊢ with axis is a selection function that generalizes ⊣ and ⊢.

Let @b{Z←A ⊢[X] B}. Then:

@itemize
@item if X is a one-element vector and ↑X is 0 then Z ≡ A,
@item if X is a one-element vector and ↑X is 1 then Z ≡ B,
@item otherwise X selects items of A or B if the corresponding elements of
X are 0 or 1 respectively. In that case, the shapes of A and B must match
the shape of X, but one-element A or B are scalar extended to the shape of X.
@end itemize

Example:
@example
      A←2 3⍴'abcdef'
      B←2 3⍴⍳6
      X←2 3⍴0 1 0 1 0 1
      A ⊢[X] B
a 2 c
4 e 6

   A ⊢[X] '*'
a*c
*e*

  '*' ⊢[X] B
* 2 *
4 * 6
@end example


@c ----------------------------------------------------------------------------
@node Bit-wise Logical Functions: ⊤∧ ⊤∨ ⊤⍲ ⊤⍱ ⊤≠ and ⊤=
@section Bit-wise Logical Functions ⊤∧, ⊤∨, ⊤⍲, ⊤⍱, ⊤≠, and ⊤=

The APL functions And (∧), Or (∨), Nand (⍲), and Nor (⍱) operate primarily on
Boolean integers. Primarily means that the LCM variant for ∧ and the GCD
variant for ∨ are not considered in this context. (The LCM and GCD variants
are defined in the ISO standard and supported in GNU APL but not in IBM APL2).

However, probably more often than not one needs to compute Boolean functions
between the bits of arbitrary (non-Boolean) integers and not between entire
Boolean integers 0 or 1. Although that is possible to do in standard APL,
the procedure is fairly awkward and, more importantly, inefficient:

@itemize
@item
convert every integer argument X to a 64-item Boolean vector X←(64⍴2)⊤X,
@item
call the Boolean function ∧, ∨, ⍲, ⍱, =, or ≠ with the converted arguments,
and
@item
convert the Boolean result vector R back to the integer result Z←2⊥R
@end itemize

Note: for Boolean arguments the APL functions ≠ and = can be used to compute
the more customary Boolean functions XOR and XNOR respectively. In this context
= and ≠ are treated as Boolean functions even though they accept non-Boolean
arguments,

For example, using  5 ⎕CR (4⍴256)⊤X to display X in hex:

@example
      5 ⎕CR (4⍴256)⊤   A←$ABBADEAD
ABBADEAD
      5 ⎕CR (4⍴256)⊤   B←$00FF00FF
00FF00FF

      5 ⎕CR (4⍴256)⊤   2⊥ ((64⍴2)⊤A) ∧ (64⍴2)⊤B
00BA00AD
@end example

With the bit-wise @b{And} (⊤∧) the same can be achieved in a simpler fashion
and far more efficiently:

@example
      ⍝ Traditional AND
      5 ⎕CR (4⍴256)⊤   A←$ABBADEAD
ABBADEAD
      5 ⎕CR (4⍴256)⊤   B←$00FF00FF
00FF00FF

      ⍝ bit-wise AND
      5 ⎕CR (4⍴256)⊤   A ⊤∧ B
00BA00AD
@end example

@c ............................................................................
@node Dyadic ⊤∧ ⊤∨ ⊤⍲ ⊤⍱ ⊤= and ⊤≠
@subsection Dyadic ⊤∧, ⊤∨, ⊤⍲, ⊤⍱, ⊤=, and ⊤≠

The dyadic forms of @b{⊤∧}, @b{⊤∨}, @b{⊤⍲}, and @b{⊤⍱} are simply the bit-wise
variants of their Boolean counterparts:

@example
      A ⊤∧ B    ←→    2⊥ ((64⍴2)⊤A) ∧ (64⍴2)⊤B      ⍝ aka. AND
      A ⊤∨ B    ←→    2⊥ ((64⍴2)⊤A) ∨ (64⍴2)⊤B      ⍝ aka. OR
      A ⊤⍲ B    ←→    2⊥ ((64⍴2)⊤A) ⍲ (64⍴2)⊤B      ⍝ aka. NAND
      A ⊤⍱ B    ←→    2⊥ ((64⍴2)⊤A) ⍱ (64⍴2)⊤B      ⍝ aka. NOR
      A ⊤≠ B    ←→    2⊥ ((64⍴2)⊤A) ≠ (64⍴2)⊤B      ⍝ aka. XOR
      A ⊤= B    ←→    2⊥ ((64⍴2)⊤A) = (64⍴2)⊤B      ⍝ aka. NXOR or XNOR
@end example

@c ............................................................................
@node Monadic ⊤∨ and ⊤⍱
@subsection Monadic ⊤∨ and ⊤⍱

@b{⊤=}, @b{⊤≠}, and @b{⊤⍲} do not have a monadic form, but @b{⊤∧}, @b{⊤∨},
and @b{⊤⍱} do.
Formally monadic @b{⊤∨} and @b{⊤⍱} are dyadic @b{⊤∨} and @b{⊤⍱} with
a left argument of 0:

@example
      ⊤∨ B    ←→    0 ⊤∨ B      ⍝ real B to nearby integer
      ⊤⍱ B    ←→    0 ⊤⍱ B      ⍝ bit-wise Not
@end example

Even though Or-ing a Boolean vector with 0 itself has no effect, the
monadic ⊤∨ and ⊤⍱ are still useful due to their side effects: conversion
from near-integer float values to integers (monadic @b{⊤∨} and inversion
of the bits (monadic @b{⊤⍱}, The latter is needed since ∼ has a monadic
form and, as a consequence, ⊤∼ was not an option for a bit-wise @b{Not}
function.

Monadic @b{⊤⍱ B} is the bit-wise @b{Not} function of its argument:

      ⊤⍱ B    ←→    2⊥ ~(64⍴2)⊤B

Monadic @b{⊤∨ B} converts near-integer values B into true integers:

@example
      26 ⎕CR    1            ⍝ integer
16
      26 ⎕CR    1.1          ⍝ real
32
      26 ⎕CR    1÷1          ⍝ integer
16
      26 ⎕CR    1.1÷1.1      ⍝ real
32
      26 ⎕CR    ⊤∨ 1.1÷1.1   ⍝ integer
16
@end example

That also works for complex numbers with a near-zero imaginary part:

@example
      26 ⎕CR    1.1J0÷1.1      ⍝ real
32
      26 ⎕CR    ⊤∨ 1.1J0÷1.1      ⍝ integer
16
@end example

As a matter of fact, @b{⊤∨} works for all near-Gaussian complex numbers,
but the effect is not visible with 26 ⎕CR since Gaussian and non-Gaussian
complex numbers have the same cell type in GNU APL.

@c ............................................................................
@node Monadic ⊤∧
@subsection Monadic ⊤∧

The ISO standard defines two different concepts to decide if a real number R
is close to an integer I,
@itemize
@item
the real number R is said to be @b{near} to integer I if the absolute
value of I-R is smaller than some small constant called the @b{integer
tolerance} in ISO and @b{system tolerance} in IBM APL2. The @b{integer
tolerance} is typically platform dependent (1E¯10 in GNU APL).

@item
the real number R is said to be @b{equal to integer I within ⎕CT} if the
absolute value of (I-R)/I is smaller than ⎕CT.
@end itemize

The first concept is an absolute distance from a real R to a nearby integer I
while the second concept is a relative distance (larger numbers are allowed to
be farther away from a nearby integer than smaller numbers). The two concepts
differ in the same way that absolute and relative errors do.

Now, all bit-wise functions described so far use the first concept, i,e,
if their arguments are real or complex, then the smallest distance to
a (nearby) integer must not exceed the @b{integer tolerance}. In some contexts
that could be difficult to achieve and for that reason GNU APL provides
monadic ⊤∧ which works like ⊤∨ except that the permitted distance from a
nearby integer is now controlled by ⎕CT rather than by the integer tolerance.

All bit-wise logical functions throw a DOMAIN ERROR if an argument is not close
enough to an integer. The @b{integer tolerance} is typically smaller than ⎕CT
and in that case ⊤∧ is a little more tolerant than ⊤∨ regarding the distance
to nearby integers. In addition ⊤∧ gives the user (via ⎕CT) more control over
the permitted tolerance for real numbers in the neighborhood of integers.

@c ............................................................................
@node Character Arguments for Monadic ⊤⍱ and Dyadic ⊤∧ ⊤∨ == and ≠≠
@subsection Character Arguments for Monadic ⊤⍱ and Dyadic ⊤∧, ⊤∨, ==, and ≠≠

Some of the bit-wise operations are allowed to have characters as their right
argument. If the right argument is a character then the result is a character
as well. This can be useful for masking purposes such as extracting the lower
7 bits of 8-bit bytes containing ASCII characters.

The characters involved are treated as 32 bit quantities:

@example
      5⎕CR 'A'
41
      5⎕CR ⊤⍱'A'            ⍝ Note that 5⎕CR aka. ⎕CR.to_HEX ANDs with $FF
BE

      256 256 256 256⊤ ⎕UCS ⊤⍱ 'A'      ⍝ ⊤⍱ 'A' is FFFFFFBE
255 255 255 190
      256 256 256 256⊤$FFFFFFBE
255 255 255 190
@end example


@c ----------------------------------------------------------------------------
@node ⊤ (Encode) with Axis
@section ⊤ (Encode) with Axis

The dyadic primitive function @b{A ⊥ B} (Decode) computes the @b{values}
of @b{B} in a number system with radices @b{A}. The most common case is
a homogeneous radix, i.e. all items of @b{A} are the same. In that case
the radix @b{A} can be specified as a scalar @b{A} which will be
scalar-extended to the length of @b{B} (more precisely: to @b{↑⍴B}).
For example:

@example
      2 2 2 2 ⊥ 1 1 0 1
13

      2⊥1 1 0 1              ⍝ scalar extension of 2 to length 4
13

      ((⍴B)⍴2) ⊤ B←1 1 0 1   ⍝ scalar extension of 2 to ↑⍴B
13
@end example

The dyadic primitive function @b{A ⊤ B} (Encode) computes the @i{representation}
of @b{B} in a number system whose radices are @b{A}. The most common case is
again a homogeneous radix @b{A}. In this sense, Encode is the inverse function
of Decode. Unfortunately Encode can not scalar extend the radix @b{A}. This
is because the shapes of its arguments do not provide sufficient information
to determine the radix length that shall be used. As a consequence:

@itemize
@item the user needs to provide the radix @b{A} as a vector with identical
      items and with a length that is suitable for the items in @b{B},
@item if more radix items than needed are provided, then the result contains
      leading 0s (which is sort of OK because no information is lost), but
@item if less radix items than needed are provided, then an oberflow
      occurs and the result is truncated.  This is awkward because, for example
      
@example
      2 2 2 2 2 2 2 2 ⊤ 1000     ⍝ radix too short for 1000
1 1 1 0 1 0 0 0

      2 ⊥ 1 1 1 0 1 0 0 0        ⍝ therefore not 1000 !!!
232
      232 + (2⋆8) + (2⋆9)        ⍝ add missing items
1000
@end example

@item The IBM APL2 language reference manual proposes to use the APL expression
@b{⌊1∣+A⍟(∣B)+A=0} for the optimal length of the radix @b{A}.
@end itemize

We call a radix length @b{⍴A} @i{optimal} if it is neither too long nor to short.
In that case the result has no leading zeros and @b{B ≡ A⊥A⊤N}. For an optimal
radix length are @b{A⊤} and @b{A⊥} the inverse functions of each other.

In order to improve on this, GNU APL offers the possibility to automatically
(and without using @b{A⍟B}) compute the optimal radix length from @b{A}
and @b{B}. The solution implemented in GNU APL is to provide the optimal
radix length as an axis argument. There are two different cases for the axis
argument:

@itemize
@item If the optimal radix length is known, say @b{N > 0}, then the radix length
is given as axis argument @b{N>0}. In this case the left argument of @b{A⊤B} is
slightly simplified:

@example
      (N⍴↑A) ⊤ B   ←→   A ⊤[N] B   ⍝ optimal radix length N>0 is known
@end example

This case frequently occurs when integer(s) @b{B} are encoded into hexadecimal
8-bit (i.e. radix length = 2 hex digits), 16-bit words (i.e. radix
length = 4 hex digits), 32-bit longs (i.e. radix length = 8 hex digits),
or 64-bit long longs (i.e. radix length = 16 hex digits).

@item Otherwise the radix length is not known; this case is specified as axis
argument @b{0}. This is also the case where @b{⌊1∣+A⍟(∣B)+A=0} would be needed,
and it simplifies @b{A⊤B} considerably:

@example
      ((⌊1∣+A⍟(∣B)+A=0)⍴↑A) ⊤ B   ←→   A ⊤ [0]B   ⍝ unknown radix length
@end example
@end itemize

Please note the following:

@itemize
@item Obviously the case @b{N>0} is faster than the case @b{N=0} because the
      user has taken over radix length computation. However, @b{A⊤[N]B} is still
      faster than the classical @b{(N⍴↑A)⊤B}.

@item If all items in @b{B} are non-negative, then the case @b{N=0} uses
      the optimal radix length for unsigned numbers, i.e. @i{without a sign}.
      The result can be used in what is known as @i{unsigned base-A arithmetic}.

@item If some item of @b{B} is negative, then the case @b{N=0} uses a radix
      length that @i{includes} a sign (aka. @i{two's complement arithmetic} if
      @b{A=2}).  The sign is @b{0} for the non-negative items of @b{B} and
      @b{A-1} for the negative items of @b{B} (= 1 for binary encodings).
@end itemize

Examples:
@example
      2⊤[4] 13           ⍝ 4 digit radix, positive B
1 1 0 1

      2⊥1 1 0 1          ⍝ proof
13

      2⊤[5] ¯13          ⍝ 5 digit radix, negative B
1 0 0 1 1

      32 - 2⊥1 0 0 1 1   ⍝ proof
13

@end example


@c ----------------------------------------------------------------------------
@node Generalized ⍳
@section Generalized ⍳

@c ............................................................................
@node Generalized monadic ⍳
@subsection Generalized monadic ⍳

The standard function @b{Interval} (Z←⍳B) requires its argument B to be a
positive integer scalar or a length 1 integer vector. The result Z←⍳B then
contains all possible consecutive integer values having shape B.

Generalized monadic ⍳ extends this concept to integer vectors B with more
than 1 element. The result Z←⍳B again contains all possible indices of a
value which has shape B, but the rank of B can now be more than 1. The items
of Z are nested integer vectors consecutive along each axis of B as opposed
to integer scalars in the standard case.

@c ............................................................................
@node Generalized dyadic ⍳
@subsection Generalized dyadic ⍳

The standard function @b{Index Of} (Z←A⍳B) requires its left argument A to be
a scalar or a vector of rank 1. The items of the result Z are the (first)
positions (= indices) of the matching items of B in A (for the items found
in A) or else ⎕IO+⍴A (which is an invalid index of A) for the items of B that
are not found in A.

Generalized dyadic ⍳ extends this concept to find the (first) indices of the
corresponding items of B in A (in ravel order) where the rank of A is > 1.
Each item of the result Z is then either a nested integer vector (if the item
was found) or @b{- different from the standard case (!) -} the empty vector ⍬
if the item was not found).


@c ----------------------------------------------------------------------------
@node ⌹[X] - Miscellaneous Matrix and Vector Operations
@section ⌹[X] - Miscellaneous Matrix and Vector Operations

@c ............................................................................
@node ⌹[1] and ⌹[2] - QR Factorizations
@subsection ⌹[1] and ⌹[2] - QR Factorizations

@b{Z←⌹[1] B} and @b{Z←⌹[2] B} compute a QR factorization of the real or
complex matrix B. The results of @b{Z←⌹[1] B} and @b{Z←⌹[2] B} are
conceptually identical, while the algorithms used to compute the results
differ. Also, @b{Z←⌹[1] B} always exist, while @b{Z←⌹[2] B} requires
@b{libgsl} to be installed when GNU APL is compiled.

The axis argument @b{X} is used to either clear near-0 matrix items to
exactly 0.0 during the computation (in Garry Helzer's algorithm), or else
to select a different algorithm for computing the factorization. Set
@b{X←⎕CT} if unsure.

The result Z is a triple (Q R Ri)←Z with the following properties.
Let ⍴B=(M, N).

* @b{Q} is an orthogonal M×M matrix, i.e Q@sup{-1} = Q@sup{T}, and Q +.× ⍉Q
 is the M×M identity matrix I@sub{M} (with I@sub{M}←∘.=⍨⍳M).

* @b{R} is an upper triangular N×N matrix, i.e. R[m;n] = 0 for all n < m.
  Note that if M>N then, following a convention in other libraries (read:
  libgsl), the triangular matrices R and Ri have shape (N N) and not (M N).
  This is because the rows below row N are always 0 in R and in Ri, and
  are therefore not of interest. As a consequence, in order to restore
  B from Q and R one has to either use @b{Q+.×M N↑R} or else Q∘R (the latter
  accepts arguments R with fewer rows than columns of Q.

* @b{Ri} is the inverse of @b{R}, i.e. @b{Ri +.× R} is the N×N identity
  matrix I@sub{N}.

* @b{B = Q +.× M N↑R}.

Example 1 (real B)
@example
      ⊢B←3 3⍴ 1 1 3 2 4 2 4 8 7
1 1 3
2 4 2
4 8 7

      (Q R Ri)←⌹[⎕CT]B

      4⎕CR 4⍕Q
┏→━━━━━━━━━━━━━━━━━━━┓
↓ .2182  .9759  .0000┃
┃ .4364 ¯.0976 ¯.8944┃
┃ .8729 ¯.1952  .4472┃
┗━━━━━━━━━━━━━━━━━━━━┛

      4⎕CR 4⍕Q+.×⍉Q  ⍝ verify that Q is orthogonal
┏→━━━━━━━━━━━━━━━━━━━━┓
↓ 1.0000  .0000  .0000┃
┃  .0000 1.0000  .0000┃
┃  .0000  .0000 1.0000┃
┗━━━━━━━━━━━━━━━━━━━━━┛

      4⎕CR0 4⍕R       ⍝ verify that R is upper triangle
┏→━━━━━━━━━━━━━━━━━━━━┓
↓ 4.5826 8.9469 7.6376┃
┃  .0000 ¯.9759 1.3663┃
┃  .0000  .0000 1.3416┃
┗━━━━━━━━━━━━━━━━━━━━━┛

       Q+.×R         ⍝ verify that B is Q+.×R (i.e. B is (⍉Q)+.×R)
1 1 3
2 4 2
4 8 7

      4⎕CR0 4⍕Ri+.×R  ⍝ verify that Ri is the inverse of R
┏→━━━━━━━━━━━━━━━━━━━━┓
↓ 1.0000  .0000  .0000┃
┃  .0000 1.0000  .0000┃
┃  .0000  .0000 1.0000┃
┗━━━━━━━━━━━━━━━━━━━━━┛
@end example

Example 2 (complex over-determined B)

@example
      B←5 4⍴4J6 6J3 5J10 3J2 8J10 3J4 5J10 5J8 3J1 2J3 4J5 1J3 1J4 9J9 9J6 2J7 2J10 7J6 9J8 10J10

      (Q R Ri)←⌹[⎕CT]B

      4⎕CR 4⍕Q∘⍉Q  ⍝ verify that Q is orthogonal
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
↓ 1.0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000┃
┃  .0000J.0000 1.0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000┃
┃  .0000J.0000  .0000J.0000 1.0000J.0000  .0000J.0000  .0000J.0000┃
┃  .0000J.0000  .0000J.0000  .0000J.0000 1.0000J.0000  .0000J.0000┃
┃  .0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000 1.0000J.0000┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      4⎕CR 4⍕R       ⍝ verify that R is upper triangle
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
↓ 8.5870J15.2557 10.1035J10.9309 13.5055J18.7452 10.0961J15.2530┃
┃  .0000J.0000    8.7353J5.5589   9.2429J.4120    2.5086J.7539  ┃
┃  .0000J.0000     .0000J.0000   ¯2.4869J4.4115  ¯5.9788J¯6.0801┃
┃  .0000J.0000     .0000J.0000     .0000J.0000    7.4038J¯3.9492┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

       Q∘5 4↑R         ⍝ verify that B is Q+.×R (i.e. B is (⍉Q)+.×R)
4J6  6J3 5J10  3J2
8J10 3J4 5J10  5J8
3J1  2J3 4J5   1J3
1J4  9J9 9J6   2J7
2J10 7J6 9J8  10J10
@end example

Please note as well that QR factorization is currently experimental. So
please double-check its results before using it in production code.

@c ............................................................................
@node The Impact of ⎕CT for ⌹
@subsection The Impact of ⎕CT for ⌹

Due to the lack of a better place we describe here how @b{⎕CT} is used
in the various @b{⌹} functions.

@node A⌹B and ⌹B
@subsubsection A⌹B and ⌹B

The first step in the computation of @b{A⌹B} or @b{⌹B} is the estimation
of the number of linearly independent columns of @b{B}. If that number is
smaller than the total number of columns (IOW some columns of @b{B} are
linearly dependent) then a @b{DOMAIN ERROR} is raised before the computation
of @b{⌹B} starts. Although @b{B} may be over-determined (so it may have more
rows than columns and in that case the rows of @b{B} cannot be linearly
independent) it may @i{not} have linearly dependent columns.

Now, for every real or complex number @b{⍺} does @b{⎕CT} define a range
of nearby numbers that are considered equal to @b{⍺} even though they are,
computationally speaking, different?

In the same fashion, let @b{B} be a matrix with linearly dependent columns.
For every such @b{B} does @b{⎕CT} define a range of nearby matrices that
are also considered linearly dependent even though they are, strictly
speaking, linearly independent? As a matter of fact, the probability
that the columns of @b{B} are strictly linearly independent decreases
as the size of the matrix grows. If a matrix has linearly dependent
columns, then every new row added to the matrix can make it linearly
independent, but not the other way around.

The consequence for the user of GNU APL is the following. If GNU APL raises
a @b{DOMAIN ERROR} caused by the supposedly linearly dependent columns of some
matrix @b{B} then there are two possibilities:

@enumerate
@item the columns of @b{B} are indeed linearly dependent, e.g. @b{⌹B←2 2⍴3 4}.
      In this case there is no way to compute @b{⌹B}.
@item the columns of @b{B} are only close (as controlled by @b{⎕CT}) to
      some other strictly linearly dependent matrix. In this case (which
      is far more likely than the other), one may decrease @b{⎕CT}
      until the matrix becomes linearly independent. For example, @b{⌹B}
      may raise a DOMAIN ERROR with the default @b{⎕CT←1E¯13} but not
      with @b{⎕CT←1E¯15}.
      However, one should use this technique with great care because
      matrices that are nearly linearly dependent tend to produce
      significant rounding errors in the result.
@end enumerate

@node ⌹[X] B: Various Factorizations of B
@subsubsection ⌹[X] B: Various Factorizations of B

The numeric scalar @b{X} chooses one of several algorithms that computes a 
factorization (aka. decomposition) of a matrix @b{B}:

@itemize
@item @b{⌹[1]B} computes a QR-factorization of @b{B}. It uses the algorithm
      published by Garry Helzer in APL Quote Quad in 1990. This algorithm is
      always available.

@item @b{⌹[2]B} also computes a QR-factorization of @b{B}. However, it uses a
      @b{libgsl} based algorithm and is therefore only available
      if @b{libgsl} was installed when GNU APL was ./configure'd..

@item @b{⌹[3]B} computes a RQ-factoriztion of @b{B}.
  It also requires @b{libgsl}.

@item @b{⌹[4]B} computes a LQ-factoriztion of @b{B}.
  It also requires @b{libgsl}.

@item @b{⌹[5]B} computes a QL-factoriztion of @b{B}.
  It also requires @b{libgsl}.

@item @b{⌹[6]B} computes a LU-factoriztion of @b{B}.
  It also requires @b{libgsl}.

@item @b{⌹[TOL]B} with a small real @b{TOL}. This variant is obsolete and
only provided for backward compatibility with older versions of GNU APL
(and is subject to being removed at some point in time). It uses the Helzer
algorithm with @b{TOL} instead of @b{⎕CT}. New code should not use @b{⌹[TOL]}
 but rather @b{⌹[1]} or @b{⌹[2]} after setting @b{⎕CT←TOL}.
@end itemize

The Helzer algorithm recursively factors a N×N matrix, a (N-1)×(N-1) matrix,
a (N-2)×(N-2) matrix, and so forth. After every step of that recursion,
all matrix items that are @i{close} to @b{0.0} are set to exactly @b{0.0}.
The meaning of @i{close} is defined by @b{⎕CT} in @b{⌹[1]B} (resp. @b{TOL} in
the obsolete case above).

In the Helzer algorithm, GNU APL uses @b{⎕CT} as the tolerance @b{TOL}
which decides if a matrix item close to @b{0.0} shall be set to @b{0.0}
or not (APL functions @b{A TOL X} and @b{A CPR B} in the Helzer paper).
The Helzer paper leaves it open how to implement @b{A TOL X} and @b{A CPR B}).
GNU APL uses the example implementation given in the Helzer paper.
The @b{TOL} and @b{CPR} functions seem to be related to what is elsewhere
known as the @i{condition number} of the matrix that is being factorized.
In the libgsl based algotithms is @b{⎕CT} not used.

The factorizations @b{⌹[1]}, @b{⌹[2]}, @b{⌹[3]}, @b{⌹[4]}, and @b{⌹[5]}
return a (nested) vector @b{(Q T Ti)} where @b{Q} is orthogonal,
@b{T} is (upper or lower) triangular, and @b{Ti} is @b{T@sup{-1}}.

Let @b{⍴B = (M N)} and let @b{min_MN = min(M, N)}. Then the
factorization @b{⌹[6]} returns a nested vector @b{(P U L)} where:
@itemize
  @item vector @b{P} is a permutation of @b{⍳M},
  @item matrix @b{L} is lower triangular with shape @b{(M min_MN)} and all 
    of its diagonal items equal to @b{1.0}.,
  @item matrix @b{U} is upper triangular with shape @b{(min_MN N)}, and,
    most importantly,
  @item @b{B[P;] = L∘U}.
@end itemize

@node Matrix Inverse
@subsubsection Matrix Inverse

The matrix @b{Ti} returned by @b{⌹[1]} or @b{⌹[2]} provides an alternative
to @b{⌹B} for computing the inverse of a matrix @b{B}, in particular if the
QR-factorization of @b{B} was already computed.

Since @b{B = Q∘T}, it follows that @b{B⁻¹ = T⁻¹∘Q⁻¹ = Ti∘⍉Q}.

Indeed:

@example
      B←?5 5⍴100   ⍝ a random matrix B
      (Q T Ti)←⌹[2]B
      (⌹B) = Ti∘⍉Q
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
@end example

The computation of @b{⍉Q} and @b{Ti∘⍉Q} is considerably faster than the
computation of @b{⌹B} if @b{B} is large.


@c ----------------------------------------------------------------------------
@node ⌹[7] B: Print Polynomial B
@subsubsection ⌹[7] B: Print Polynomial B

Let @b{B ← b₀ b₁ b₂ ... bₙ}. The standard interpretation of this vector
@b{B} in APL is that @b{B} is an element of a vector space over the
field @b{ℝ} or @b{ℂ}. The elementary APL primitives @b{+}, @b{-}, @b{×}
and @b{÷}, but also the more powerful primitives @b{+.×} and @b{⌹},
and last but not least the scalar extension mechanism of several primitive
are tailored for this interpretation.

However, another and less common interpretation of @b{B} is this:
vector @b{B} is a polynomial of (a single) indeterminant @b{X}:

B ←→ b₀ + b₁X + b₂X² + ... + bₙXⁿ

This interpretation easily extends to APL values of higher rank, i.e.
matrices are polynomials of two indeterminants @b{X} and @b{Y}, values of
rank 3 are polynomials of three indeterminants @b{X}, @b{Y}, and @b{Z},
and so on. In this case, assuming @b{⎕IO←0}:

@itemize
@item An APL value @b{P} with rank @b{r←⍴⍴P} represents a polynomial with
      @b{r} indeterminants.
@item The item @b{P[k;m;n;...]} is the coefficient of the polynomial term
      that has the indeterminant powers @b{XᵏYᵐZⁿ}...
@item The vector @b{P[;0;0;...]} are the coefficients of the polynomial terms
with only powers of @b{X}, 
@item The vector @b{P[0;;0;...]} are the coefficients of the polynomial terms
      with only powers of @b{Y}, 
@item The vector @b{P[0;0;;...]} are the coefficients of the polynomial terms
      with only powers of @b{Z},
@item and so on. The other items of @b{P} are coeffiecients of the polynomial
      terms with mixed powers of the different indeterminants.
@item The item @b{P[[0;0;0;...} aka. @b{↑P} is the constant term of the
      polinomial.
@end itemize

Function @b{⌹[7] B}, aka. @b{⌹.poly_print B} returns a string with this
interpretation. In its monadic form @b{⌹[7] B} it uses the names
@b{x}, @b{y}, @b{z}, ... for the indeterminants of the polynomial.
In its dyadic form @b{A ⌹[7] B} does i@b{A} specify the names that shall
be used. Two forms are supported for @b{A}: the characters of a plain APL
string @b{(⍴A == ⍴⍴B)} is understood as a list of single character
indeterminants. iAlternatively A list of nested strings @b{A} (again with
@b{(⍴A == ⍴⍴B)}) is understood as N different indeterminant names. For example:

@example
      ⍝ monadic, 1 indeterminant
      ⌹.poly_print ⎕←POLY←3 2 1
3 2 1
x² + 2x + 3

      ⍝ monadic, 2 indeterminants
      ⌹.poly_print ⎕←POLY←2 3⍴⍳6
1 2 3
4 5 6
6x²y + 5xy + 4y + 3x² + 2x + 1

      ⍝ monadic, 3 indeterminants
      ⌹.poly_print ⎕←POLY←3 3⍴⍳9
1 2 3
4 5 6
7 8 9
9x²y² + 8xy² + 7y² + 6x²y + 5xy + 4y + 3x² + 2x + 1

      ⍝ dyadic, two 1-character indeterminants X and Y
      'XY' ⌹.poly_print ⎕←POLY←2 3⍴⍳6
1 2 3
4 5 6
6X²Y + 5XY + 4Y + 3X² + 2X + 1

      ⍝ dyadic, two named indeterminants _Foo and _Bar
      '_Foo' '_Bar' ⌹.poly_print ⎕←POLY←2 3⍴⍳6
1 2 3
4 5 6
6_Foo²_Bar + 5_Foo_Bar + 4_Bar + 3_Foo² + 2_Foo + 1

@end example
 
@c ----------------------------------------------------------------------------
@node A ⌹[8] B: Multiply Polynomials A and B
@subsubsection A ⌹[8] B: Multiply Polynomials A and B

The scalar APL primitives @b{+} and @b{-} can be used for vectors
and for polynomials alike. In contrast, the multiplication of two polynomials 
@b{A}  and @b{B} is different from the scalar multiplication @b{A×B} and is 
also different from the vector multiplication @b{A +.× B}.

Dyadic @b{A ⌹[8] B} aka. @b{A ⌹.poly_multiply B} fills this gap and computes
the product of two polynomials. For example:

@example
      ⍝ single indeterminant x
      ⍝
      "A:"    (⌹.poly_print A ← 4 0 0 6 3 1)
 A: x⁵ + 3x⁴ + 6x³ + 4 

      "B:"    (⌹.poly_print B ← 5 0 2)
 B: 2x² + 5 

      "PROD:" (⌹.poly_print PROD ← A ⌹.poly_multiply B)
 PROD: 2x⁷ + 6x⁶ + 17x⁵ + 15x⁴ + 30x³ + 8x² + 20 

      ⍝ two indeterminants x and y
      ⍝
      "A:"    (⌹.poly_print A ← 2 2⍴0 1 1 0)
 A: x + y 

      "B:"    (⌹.poly_print B ← 2 2⍴0 ¯1 1 0)
 B: x - y 

      "PROD:" (⌹.poly_print PROD ← A ⌹.poly_multiply B)
 PROD: x² - y² 
@end example

@c ----------------------------------------------------------------------------
@node A ⌹[9] B: Divide Polynomial A by Polynomial B (univariate)
@subsubsection A ⌹[9] B: Divide Polynomial A by Polynomial B (univariate)

The scalar APL primitives @b{+} and @b{-} can be used for vectors
and for polynomials alike. In contrast, the division of a polynomial
@b{A} by a polynomial @b{B} is different from the scalar division @b{A÷B}
and is  also different from the matrix division @b{A ⌹.× B}.

Dyadic @b{A ⌹[9] B} aka. @b{A ⌹.poly_divide B} fills this gap and computes
the quotient of two polynomials. Since polynomials form a ring (and not a
field) the quotient is not always a polynomial (at least not one with
non-negative exponents). For this reason @b{A ⌹.poly_divide B} performs
a division with remainder and the result is a pair @b{(Quotient, Remainder)}
 where @b{Remainder} is a polynomial with a degree smaller that the degree
of polynomial @b{B}. That is, @b{⍴Remainder = ¯1+⍴B}.

@b{NOTE:} @b{A ⌹[9] B} is limited to polynomials with 1 indeterminant.

@c ----------------------------------------------------------------------------
@node A ⌹[10] B: Divide Polynomial A by Polynomial B (multivariate)
@subsubsection A ⌹[10] B: Divide Polynomial A by Polynomial B (multivariate)

The limitiation of @b{A ⌹[10] B} to handle only polynomials with one 
indeterminan is lifted with @b{A ⌹[10]} aka. @b{A ⌹.poly_divideN B}.
The suffix @b{N} shall indicate that the number off indefinites is > 1.
However, some caution needs to be exercized when using this function,
as described in the following.

A @i{polynomial} is the sum of zero or more @i{terms}. The terms of a
polynomial are commonly called @i{monomials}.

@b{Tip}: to list, for example, all possible monomials of a polymomial with
three indeterminants x, y, and z and at most powers of 3:

@example
      ⍝ show all monomials of indeterminants x, y, z
       ⍝ and max.powers 0, 1, and 2 (in lexicographic order)
      ⍝
      ⌹.poly_print 3 3 3⍴1
x²y²z² + x²y²z + x²y² + x²yz² + x²yz + x²y + x²z² + x²z + x² + xy²z² + xy²z + 
      xy² + xyz² + xyz + xy + xz² + xz + x + y²z² + y²z + y² + yz² + yz + y + 
      z² + z + 1
@end example

In the set of all @i{monomials} an @i{order}, that is a relation @b{<} between
any two elements of the set needs to be defined. In the
simple cases of dividing integers or polynomials with only one indeterminant
the order @b{<} is canonical:

@itemize
@item order of integer: 0 < 1 < 2 < 3 ...
@item order of polynomials: x⁰ < x¹ < x² < x³ ...
@end itemize

An order between @i{monomials} need not only satisfy the general requirements
of an order (i.e. being reflexive, antisymmetric, and transitive), but also
the following:

@itemize
@item if x@sup{⍺} < x@sup{β} then x@sup{γ}x@sup{⍺} < x@sup{γ}x@sup{β}
      for all γ, and
@item M ≥ 0 for every monomial M
@end itemize

The general problem with multivariate polynomials (i.e. polynomials with more
than one indeterminant, such as @b{x² + y²}) is then that the quotient and 
remainder of two multivariate polinomials is not unique but depend on
the order defined between its monomials. The division of integers and
of polynomials with one indeterminant is based on their canonical order,
which results in a unique quotient and a unique remainder in these cases.
In contrast there can be several different orders between the monomials
of a multivariate polynomial, and these different orders leads to different
quotients and remainders.

For example:
@example
Let
      f = x2y + xy2 + y2,
      g₁ = xy − 1, and
      g₂ = y² − 1.

Dividing f by g₁ and then by g₂ gives:

      f = (x + y)(xy − 1)  +  (y² − 1)  +  (x + y + 1)
        = (x + y) g₁       +      g₂    +  (x + y + 1)

However, dividing f by g₂ and then by g₁ gives:

      f = x(xy − 1)  +  (x + 1)(y² − 1)  +  (2x + 1)
        = x g₁       +  ((x + 1) g₂      +  (2x + 1)

@end example

This ambiguity is resolved by always dividing a polynomial by
its @i{largest term}, where the largest term is in turn defined
by the order of the mononomials that was chosen. IOW: a monomial order
defines the which of the monomials of a polynomial is its largest monomials,
and in every iteration of the division the largest monomial is being
annihilated (by subtracting a multiple of the divisor from it).

Now, a commonly used order in the division of polynomials, and also the order
used by @b{A ⌹[10] B}, is the @i{lexicographic order} of the polynomials.
@xref{⌹[11] B Scan Polynomial B}.

For example: @b{x < x²}, @b{xy < xyz}, @b{y < x⁸}, and @b{y⁸ < x} in the
lexicographic order.

With these prerequisites, the division algorithm of GNU APL (which was
inspired by @url{https://lall.stanford.edu/ee464/lectures/groebner_bases.pdf})
is this:

@example
      Let LT(p) denote the largest term in polynomial p, and let       (1)

      q₁ = q₂ = ... = qₙ = 0   ⍝ quotient                              (1a)
      r  = 0                   ⍝ remainder                             (1b)
      p = f                    ⍝ dividend                              (1c)

      while p ≠ 0:                                                     (2)
            let i be the smallest i such that LT(gᵢ) divides LT(p).    (3)
            if such an i exists then:                                  (3a)
                  qᵢ = qᵢ + LT(p) ÷ LT(gᵢ)                             (4a)
                  p = p - gᵢ × LT(r) ÷ LT(gᵢ)                          (4b)
            otherwise:                                                 (3b)
                  r = r + LT(p)                                        (5a)
                  p = p - LT(gᵢ)                                       (5b)
@end example

The algorithm divides polynomial @b{f} by polynomial
@b{g = g₁ + g₂ + ... + gₙ} and yields a quotient @b{q = q₁ + q₂ + ... + qₘ}
and a remainder @b{r = r₁ + r₂ + ... + rₖ}. In every iteration the largest
term in @b{p} is annihilated by "moving" it to either i@b{q} (after
dividing it by @b{LT(gᵢ)} if that is possible) or else to @b{r}
(if not). Obviously:

@itemize
@item The divisor @b{g} does not change at all,
@item @b{p} gets shorter and shorter in each iteration and therefore the
      algorithm terminates after at most @b{n} iterations. In addition the
      subtraction in @b{4b} certainly annihilates the leading term of @b{p},
      but @i{could} also annihilate other monomials of @b{p} (which then also
      reduces the number of iterations performed).
@item The quotient @b{q} grows as new quotients are being added in
      step @b{4b}.
@item The remainder @b{r} also grows as new items are added in step @b{5a}.
      In the integer and polynomial with one indeterminant cases is the length
      of the remainder limited by the degree of the dividend (so the
      remainder of dividing e.g. @b{x³ + ax² + bx + c} by anything has always
      no more than 3 momomials). In contrast a polynomial with a max. power
      of, say, 3 but 4 indeterminants can have up to 4⁴ = 256 momomials.
@end itemize

@c ----------------------------------------------------------------------------
@node A ⌹[11] B: Divide Polynomial A by Polynomial B (with order)
@subsubsection A ⌹[11] B: Divide Polynomial A by Polynomial B (with order)

As explained above, different monomial orders of a polynomial can lead
to different quotients and remainders when dividing them. This iproblem
is addressed by @b{A ⌹[11] B} aka. @b{A ⌹.poly_divideNO B}.
The suffix @b{NO} shall indicate that the number off indefinites is > 1,
and that a user defined monomial order shall be used.

Let @b{P} be a polynimial. An monomial order @b{O} for @b{P} is then 
a matrix with the following properties:

@itemize
@item O has the same shape as P: (⍴P) ≡ (⍴O).
@item ↑O (the order value of the contant term of the polynomial) is 0
@end itemize

In addition, as already mentioned above, the order defined by @b{O} shall be
consistent: if x@sup{⍺} < x@sup{β} then x@sup{γ}x@sup{⍺} < x@sup{γ}x@sup{β}
for all γ in @b{O}. @b{A ⌹.poly_divideNO B} checks that @b{↑O = 0} but not
the consistency of @b{O}. If the consistency condition is violated, then
in the best case the results are incorrect, and in the worst case the division
algorithm may not terminate.

For example, given a Polynomial @b{P}, the lexicographic order of @b{P} can
be constrocted as follows:

@example
      ⍝ construct the lexicographical order of all possible monom in P
      ⍝
      ORDER←-⎕IO-(⍴P)⍴⍳×/⍴P
@end example

After an order @b{O} has been created for a polynomial @b{P}, both can be
appended to create an argument for @b{A ⌹.poly_divideNO B}:

@example
      ⍝ append the desired monomial order O to a polynomial P.
      ⍝
      PO ← (2,⍴P)⍴(,P),,O
@end example

Putting the pieces together:

@example
      ⍝ create the dividend
      ⍝
      "A:" A←⌹.poly_scan "x³ - x² - xy² - 2x + y² + 4"
 A:    4 0  1 
     ¯2 0 ¯1 
     ¯1 0  0 
      1 0  0 

      ⍝ create the divisor
      ⍝
      "B:" B←⌹.poly_scan "x - y"
 B:   0 ¯1 
      1  0 

      ⍝ create the monomial orders for A and B
      ⍝
      ORDER_A←-⎕IO-(⍴A)⍴⍳×/⍴A
      ORDER_B←-⎕IO-(⍴B)⍴⍳×/⍴B

      ⍝ append the orders to their polynomial
      ⍝
      ARG_A ← (2,⍴A)⍴(,A),,ORDER_A
      ARG_B ← (2,⍴B)⍴(,B),,ORDER_B

      ⍝ call ⌹.poly_divideN and poly_divideNO
      ⍝
      (Q1 R1) ←     A ⌹.poly_divideN B
      (Q2 R2) ← ARG_A ⌹.poly_divideNO ARG_B

      ⍝ the result should be the same, because
      ⍝ ORDER_A is what ⌹.poly_divideN uses.
      ⍝ 
      (Q1 R1) ≡ (Q2 R2)
1
@end example

@c ----------------------------------------------------------------------------
@node ⌹[11] B Scan Polynomial B
@subsubsection ⌹[11] B: Scan Polynomial B

@b{⌹[11] B} aka. @b{⌹.poly_scan B} is the inverse function of @b{⌹[7] B}.
It converts a string @b{B} (in the format produced by @b{⌹poly_print B} and
returns the corresponding APL value.

@example
      ⊢Z←⌹.poly_scan "1 + 2xy + 3y²"
1 0 3
0 2 0

      ⌹.poly_print Z
2xy + 3y² + 1
@end example

@b{NOTE} that the terms of a polynomial are commutative. The same set of terms,
summed up in different orders, defines the same polynomial. Therefore:

@b{M ≡ ⌹.poly_scan ⌹.poly_print M}

however:

@b{P ≡ ⌹.poly_print ⌹.poly_scan P}

holds only if the terms in @b{P} occur in the same order as produced
by @b{⌹.poly_print}.  The order in which @b{⌹.poly_print B} prints the terms
of a polynimial @b{B} is what is known as the @b{lexicographic order},
which can, in this context, be defined as follows:

A sequence of vectors is in lexicographic order (or: 
@i{ordered lexicographically}), if every smaller item i@b{a} in the sequence
comes before every larger item @b{b} in the sequence. Where @b{smaller}
resp, @b{larger} is defined as follows:

@itemize
@item Let @b{a = a₀ a₁ ... aₙ},
@item let @b{b = b₀ b₁ ... bₙ},
@item let @b{∆ = ∆₀ ∆₁ ... ∆ₙ} with @b{∆ⱼ = bⱼ - aⱼ}, and
@item let @b{j} be the smallest @b{j} such that @b{∆ⱼ ≠ 0}.
      That is: the first @b{j-1} items of @b{a} and @b{b} are equal)
      and @b{j} is the first position at which vectors @b{a} and @b{b} differ.
@item Then:
  @itemize
  @item If  @b{∆ⱼ > 0} then @b{a <@sub{lex} b} (i.e. @b{a} is smaller than @b{b}
        and @b{b} is larger than @b{a}).
  @item Otherwise @b{∆ⱼ < 0} and then @b{b <@sub{lex} a} (i.e. @b{a} is
        larger than @b{b} and @b{b} is smaller than @b{a}).
  @end itemize
@end itemize

The lexicographic order is also defined for vectors of different lengths,
but in this context all vectors compared have the same length (number of
indeterminants).

@c ----------------------------------------------------------------------------
@node A ⌹[20] B: Integral B
@subsubsection A ⌹[20] B: Integral B

Function @b{A ⌹[20] B} aka. @b{A ⌹.integral B} computes the integral of
expression @b{B}. For example:

@example
      ⍝ monadic (same as A=1)
      ⍝
      ⌹.integral "sqrt(x³)"
2*x*sqrt(x³)/5

      ⍝ A=1 : vector result
      ⍝
      1 ⌹.integral "sqrt(x³)"
2*x*sqrt(x³)/5

      ⍝ A=2 : matrix result
      ⍝
      2 ⌹.integral "sqrt(x³)"
      _____
2⋅x⋅╲╱  x³ 
───────────
     5     
@end example

@b{NOTE}: The implementation of @b{⌹.integral} depends on @b{python}. It also
depends on the python package @b{sympy} which does the actual computation
of the integral. Normally the GNU APL @b{./configure} script determines which
optional libraries for GNU APL are present and compiles GNU APL accordingly.
In the python case this is not possible because python and its packages
can be installed globally (by a privilreged user such as @b{root}) or by
a unprivilreged user (e.g. below her home directory). It could therefore
happen that e.g. python is available for the user that installs GNU APL, 
but not for a user that uses the installed GNU APL. The GNU APL
@b{./configure} has no way to detect such a situation. For this reason
GNU APL checks at runtime (i.e. when @b{⌹.integral} is used) if python and
@b{sympy}  are installed (and raises a @b{DOMAIN ERROR} if not).

Now to the arguments of @b{⌹.integral}. The right argument @b{B} is a string
with the expression for which the integral shall be computed. The syntax
of @b{B} ia pretty much defined by @b{sympy}. A maybe noteworthy difference
between strings understood by @b{⌹.integral} and strings understood by
@b{sympy} is that powers in @b{sympy} cwhave to be written as e.g. @b{a**n},
while @b{⌹.integral} also understands @b{aⁿ}.

The left argument @b{A} is a vector of options. Currently only one option
is defined which defines the format of the result:

@itemize
@item @b{A[1]=1} : the result is a string (see example above)
@item @b{A[1]=2} : the result is a character matrix (see example above). The
      matrix is primarily created by the sympy function @b{pretty()}
      (which is unfortunately not formally specified). For this reason
      is @b{A[1]=2} somewhat experimental and may produce a garbled output in
      some cases.
@item @b{A[1]=3} : the result is a string in a simple to parse
      format.
@item @b{A[1]=4} : the result is a string in LATEX format.
@item @b{A[1]=5} : the result is a character matrix in Math_ML format.
@item @b{A[1]=6} : the result is a character matrix in Dot format.
@end itemize

@b{MathML} is an XML-based industry standard for mathematical expressions.

@b{Dot} is a program for printing directed graphs.

@c ----------------------------------------------------------------------------
@node ⎕CC - Character Classes
@section  ⎕CC - Character Classes

Some older APL interpreters (e.g. APL68000) had system variables for
character classes, such as @b{⎕D} for digits 0..9 or @b{⎕A} for letters
A..Z. GNU APL provides a single system function @b{⎕CC} for that
purpose.

@node monadic ⎕CC B: Return character class
@subsection  monadic ⎕CC B: Return character class

@b{⎕CC B} returns all characters of character class(es) @b{B}. If @b{B} is an
integer scalar then the result is a character vector, If @b{B} is an integer
vector then the result is a nested vector of integer vectors, with one item
per character class.

@example
      ⍝ character class 1: digits (old ⎕D)
      ⍝
      ⎕CC 1
0123456789

      ⍝ character class 2: uppercase letters A..Z (old ⎕A)
ABCDEFGHIJKLMNOPQRSTUVWXYZ

      ⍝ character class 3: lowercase letters a..z
abcdefghijklmnopqrstuvwxyz

      ...

      ⍝ Line drawing characters
      ⎕CC 7
┌─┬─┐╒═╤═╕
├─┼─┤╞═╪═╡
└─┴─┘╘═╧═╛
╓─╥─╖╔═╦═╗
╟─╫─╢╠═╬═╣
╙─╨─╜╚═╩═╝

      ⍝ Some mathematical symbols
      ⎕CC 8
⎲⎛⎞⎡⎤⎧⎫
⎳⎜⎟⎢⎥⎨⎬
↔⎝⎠⎣⎦⎮⎮
ℕℤℚℝℂ⎩⎭
@end example

A special case is @b{⎕CC ⍬} (or @b{⎕CC ⍳0}: it prints the available
character classes for ⎕CC:

@example
      ⎕CC ⍬
      ⎕CC   1:  digits 0-9 (same as ⎕CC 10)
      ⎕CC   2:  uppercase characters A-Z (same as ⎕CC 26)
      ⎕CC   3:  lowercase characters a-z (same as ⎕CC ¯26)
      ⎕CC   4:  ASCII characters (same as ⎕CC 127)
      ⎕CC   5:  superscripts ( ²³¹ʲᵏᵐᵗ⁰ⁱ⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾ⁿ )
      ⎕CC   6:  subscripts ( ᵢ₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎ₖₘₙⱼ )
      ⎕CC   7:  line drawing chatacters
      ⎕CC   8:  octal digits 0-7
      ⎕CC   9:  some mathematical symbols
      ⎕CC  10:  decimal digits 0-9
      ⎕CC  16:  UPPERCASE HEXADECIMAL DIGITS 0-9 A-F
      ⎕CC ¯16:  lowercase hexadecimal digits 0-9 A-F a-f
      ⎕CC  17:  Hexadecimal Digits 0-9 A-F a-f
      ⎕CC  26:  UPPERCASE CHARACTERS A-Z
      ⎕CC ¯26:  lowercase characters A-Z
      ⎕CC  33:  base32 encoding (RFC 4648)
      ⎕CC  48:  greek characters A-Ω α-ω
      ⎕CC  52:  characters A-Z a-z
      ⎕CC  65:  base64 encoding (RFC 4648)
      ⎕CC  95:  printable characters 0x20 ... 0x7E
      ⎕CC 128:  ASCII characters
@end example

In many cases is the character class number easy to remember because:

@itemize
@item the character class number is equal to the number of characters in
      the class, e.g. @b{26} for @b{A-Z}, @b{52} for @b{A-Z and a-z},
@item negative numbers are the lowercase variants of the uppercase class,
      e.g. @b{26} for @b{A-Z} and @b{¯26}  for @b{a-z}, and
@item @b{2*class} is the mixed case of the uppercase @b{class}.
@end itemize

@c ----------------------------------------------------------------------------
@node dyadic A ⎕CC B: Test character class
@subsection  dyadic A ⎕CC B: Test character class

The dyadic @b{A ⎕CC B} returns, for every item in A, integer @b{1} if the item
is contained in one of the character classes @b{B} and @b{0} if not.

@example
      'a45Y' ⎕CC 1
0 1 1 0

      'a45Y' ⎕CC 1 2
0 1 1 1

       (⎕CC 7) ⎕CC 7
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
@end example

For most classes this lookup is an optimized constant time operation (for
every item of @b{A} and faster than e.g. @b{(⎕CC n) ⍳ A}.

@c ----------------------------------------------------------------------------
@node Dyadic ⎕CR
@section Dyadic ⎕CR

The @b{⎕CR} function has now an optional left argument that selects one of
several formatting styles and conversion functions in addition to the
well-known monadic form.

Calling @b{⎕CR} monadically without an axis and with an empty right argument
shows a list of all functions provided by @b{⎕CR}:

@example
      ⎕CR ''
@end example

@b{Definition}: A @i{byte vector} is an integer (!) vector with numbers
having a (signed or unsigned) 8-bit value (i.e. a value from -128 to 255
inclusive). Such byte vectors are frequently used arguments and results
of @b{⎕FIO} functions.

Let @b{Z←A ⎕CR B}.

Then the left argument @b{A} of @b{⎕CR} selects one of several sub-functions
of @b{⎕CR}:

@itemize
@item @b{A∈0 1 2 3 4 7 8 9 29}: various formatting styles (boxed, 
      APL input/output, etc.).  Just try them out.

@item @b{A∈5 6}: convert byte vector B to a string of uppercase or lowercase
i     hex digits respectively. Every byte in @b{B} becomes 2 characters
      in @b{Z}.

@item @b{A=10}: convert variable named in @b{B} to an APL expression
      producing it.

@item @b{A=11}: convert value @b{B} to byte vector @b{Z} in CDR ("Common Data
      Representation", an IBM standard) format (similar to @b{3 ⎕TF}).

@item @b{A=12}: convert byte vector @b{B} in CDR format to value @b{Z}.

@item @b{A=13}: convert hex string @b{B} to byte vector @b{Z}.

If a conversion has an inverse conversion (like 12 being the inverse of 11)
then the inverse conversion can be expressed as the negative of the conversion
number. For example, @b{12 ⎕CR B} is the same as @b{¯11 ⎕CR B}.

@item @b{A=14}: conversion 11 followed by conversion 13 (Value to hex string
      in CDR format)

@item @b{A=15}: conversion 13 followed by conversion 12 (hex string in CDR
      format to Value)

@item @b{A=16}: encode byte vector B into @b{Z} (base64 encoding, @b{RFC 4648})

@item @b{A=17}: decode base64 vector @b{B} into byte vector @b{Z} (base64
      encoding, @b{RFC 4648})

@item @b{A=18}: convert text vector @b{B} into byte vector @b{Z} (UTF8 encoding,
      @b{RFC 3629}) 

@item @b{A=19}: convert byte vector B into text vector @b{Z} (UTF8 encoding,
      @b{RFC 3629})

@item @b{A=20-25}: like 3,4,i or 7-9 but using a formatting similar to
      NARS APL @b{⎕FMT} (showing the axis lengths as numbers instead of
      @b{→} and @b{↓})

@item @b{A=26}: @b{Z} is the cell types of the ravel elements of @b{B}, i.e.
      @itemize
      @item @b{2}: character,
      @item @b{16}: integer,
      @item @b{32}: real number,
      @item @b{64}: complex number.
@end itemize

@item @b{A=27}: Z[I] is the primary data representation (for example the
      real part  of a complex number, or the numerator of a rational number)
      of B[I].

@item @b{A=28}: Z[I] is the additional data representation (for example
      the imaginary part of a complex number, or the denominator of a
      rational number) of B[I].

@item @b{A=30}: @b{Z} is B with all top-level elements conformed to a common
      rank and shape (as required by the @b{⍤} operator). This conversion is
      primarily used  internally by the GNU APL interpreter.

@item @b{A∈31 32}: These conversions are used internally by @b{⎕INP}.

@item @b{A=33}: convert tagged byte vector to a TLV (Tag/Length/Value)
      buffer. The TLV buffer can be sent over a byte stream (socket) and
      easily decoded at the other end. Say @b{B = B[1], B[2], ..., B[n]} such
     that @b{B1} is an Integer (the tag) and @b{B[j]} is a character in the
     range 0-255 for j > 1.
     Let @b{Z←33 ⎕CR B} with @b{Z = Z[1], Z[2], ... Z[m]}. Then @b{Z[1 2 3 4]}
     is the 4 byte tag, @b{Z[5 6 7 8]} is the 4 byte data length
     (n-1) = (m - 8), and @b{1↓B == 8 ↓ Z}. In other words, the first 4 bytes
     of @b{Z} are the tag in big endian byte order, the next 4 bytes are the
     length of @b{B} except the tag, and the rest of @b{Z} is @b{B} except
     the tag.

Example:
@example
      Tag←55   ⍝ hex 37
      5 ⎕CR 33 ⎕CR Tag,'Value'
000000370000000556616C7565
@end example

@item @b{A=34}: this is the inverse of @b{33 ⎕CR}. The intended use for
@b{33 ⎕CR} and @b{34 ⎕CR} is the transmission of a tagged byte vector over
e.g a TCP socket:

@verbatim
  Sender                                              Receiver
  ──────                                              ────────
  T,Data →→→ 33 ⎕CR T,Data   →→→ TCP connection →→→   34 ⎕CR T,Data →→→ T,Data
@end verbatim

Example:
@example
      34 ⎕CR ¯5 ⎕CR '000000370000000556616C7565'
55 Value
@end example

The TLVs constructed by @b{33 ⎕CR} by the sender and decoded with
@b{34 ⎕CR} by the receiver can be sent back-to-back over a TCP connection
or similar in such a way that the receiver knows exactly after which byte
a TLV ends (and possibly the next TLV, if any, starts). This is perfect
for connections over which data is sent sporadically. @b{33 ⎕CR} and
@b{34 ⎕CR} are particularly useful for encoding and decoding TLV byte
buffers exchanged between GNU APL and processes that were forked by
GNU APL with @b{⎕FIO[57]} (aka. @b{fork()} and @b{execve()}).

@item @b{A=35}: convert a string of lines (containing LF characters as line
      delimiters) to a nested vector of lines.

@item @b{A=36}: the inverse of @b{⎕CR[35]}. Convert a nested vector of lines
      to a string with lines terminated with LF characxters.

@item @b{A=37}: Same as @b{⎕CR B} but without removing any indentation.

@item @b{A=38}: Return an empty structured variable with (integer)
      capacity @b{Bi}, or convert an N×2 matrix @b{B} to a structured variable.

@item @b{A=39}: convert a strucured variable @b{B} to an N×2 variable.

@item @b{A=40}: pack a boolean variable Bi (experimental).

@item @b{A=41}: unpack a boolean variable Bi (experimental).

@item @b{A=42}: tokenize the APL statement Bs.

@item @b{A=43}: parse the APL statement Bs.

@item @b{A=44}: decode the (integer) token tag B or the (nested) token B.
@end itemize

Most dyadic @b{⎕CR} variants whose argument @b{B} is expected to be a byte
vector throw:

@itemize
@item RANK ERROR if 1≠⍴⍴B
@item DOMAIN ERROR if one of the B[j] is not a proper byte value
@end itemize

A proper byte value is either an integer in the range -128...255 inclusive,
or a (Unicode) character with a code point between U+FF80 and U+FFFF
(inclusive, corresponding to a negative signed char in C/C++) or between
U+0000...U+00FF (inclusive, corresponding to an unsigned char or to a signed
positive char in C/C++). Real, Complex, or rational numbers are never proper
byte values even if their value is close to an integer. Nor are nested APL
values or values being assigned.

@b{⎕CR} is a function group, therefore its sub-functions can be selected with
a name. For example: monadic @b{⎕CR.hex_to_bytes} is the same as monadic
@b{⎕CR[13]} or dyadic @b{13 ⎕CR}.

@c ----------------------------------------------------------------------------
@node Dyadic ⎕FX (Native Functions)
@section  Dyadic ⎕FX (Native Functions)

A Native Function is a function that can be called in APL like a normal user
defined APL function, but is implemented in C++.

A native function is created with A ⎕FX B. A is a string that is the
path of a shared library and B is the name of the function in APL.

The GNU APL package contains a shared library file_io.so that contains the
implementation of a native function for reading and writing files (fopen(),
fclose(), ...), For example:

@example
      ⍝ fix native function in lib_file_io.so as FILE_IO
      ⍝
      'lib_file_io.so' ⎕FX 'FILE_IO'
FILE_IO

      ⍝ show overview of sub-functions in FILE_IO
      ⍝
      FILE_IO ''
   Functions provided by this library.
   Assumes 'lib_file_io.so'  ⎕FX  'FUN'

   Legend: e - error code
           i - integer
           h - file handle (integer)
           s - string
           A1, A2, ...  nested vector with elements A1, A2, ...

           FUN     ''    print this text on stderr
        '' FUN     ''    print this text on stdout
           FUN[ 0] ''    print this text on stderr
        '' FUN[ 0] ''    print this text on stdout

   Zi ←    FUN[ 1] ''    errno (of last call)
   Zs ←    FUN[ 2] Be    strerror(Be)
   Zh ← As FUN[ 3] Bs    fopen(Bs, As) filename Bs mode As
   Zh ←    FUN[ 3] Bs    fopen(Bs, "r") filename Bs
      ...
@end example

Recent versions of GNU APL have replaced the native FILE_IO function above by
the system function ⎕FIO. ⎕FIO need not be ⎕FX'ed and is otherwise backward
compatible to the native function. New function numbers are, however, only
added to ⎕FIO and not to the old native function FILE_IO. The parameters of
the functions are described in the man pages for, e.g. strerror, fopen, ...
and are fairly obvious.

Many functions in FILE_IO have byte vectors as arguments or return byte 
vectors. A byte vector is an integer vector whose numbers fit into a byte
(so they are integers between -128 and 255). Often ⎕UCS and the functions in
dyadic ⎕CR are used to convert such byte vectors to/from, for example,
Unicode strings.

The GNU APL package also contains other shared libraries as templates
for your own native functions. Copy one of the files
src/native/template_F0.cc (for niladic native functions),
src/native/template_F12.cc (for nomadic native functions),
src/native/template_OP1.cc (for monadic native operators),
or src/native/template_OP2/cc (for dyadic native operators) to your own
.cc file and adjust src/native/Makefile.am accordingly.

Note: The )IN and )OUT commands of GNU APL support native functions, but
to do so they have to use dyadic ⎕FX. This renders the workspace
interchange file (.atf files) written by )OUT incompatible with all other
APL interpreters if the workspace contains native functions. The )OUT 
command prints a warning when it is used with a workspace that contains
native functions.

Note: As of GNU APL 1.6, the native function FILE_IO has been turned into the
system function ⎕FIO. The syntax of ⎕FIO is the same as for FILE_IO. The
)CLEAR workspace command will close all open files. 


@c ----------------------------------------------------------------------------
@node ⎕ARG - Interpreter command line arguments
@section ⎕ARG - Interpreter command line arguments

@b{⎕ARG} contains the command line arguments with which GNU APL was invoked.
@xref{APL Scripting}.

@c ----------------------------------------------------------------------------
@node ⎕DLX - Knuth's Dancing Links Algorithm
@section ⎕DLX - Knuth's Dancing Links Algorithm

⎕DLX is an implementation of Donald Knuth's Dancing Links Algorithm (called
DLX by Knuth himself, but is sometimes also referred to as Knuth's Algorithm X).
⎕DLX is a generic backtracking machine that can be used to dramatically
simplify problems like the 8 queens problem on a chess board or sudokus.

The monadic form of ⎕DLX, i.e. ⎕DLX B, is a shortcut for 0 ⎕DLX B. It
computes the first solution for the constraint matrix B.

The dyadic form of ⎕DLX, i.e. A ⎕DLX B, has an integer scalar A as left
argument which determines the details of the computation as follows:

A > 0: The algorithm tries to find all solutions, but stops when A solutions
have been found. This is handy while debugging code using ⎕DLX. The
result is a nested vector with one vector item per solution.

A = 0: The algorithm stops when the first solution was found. In this case
the solution is a simple (non-nested) numeric vector.

A = ¯1: like A > 0 but finding all solutions

A = ¯2: like A = 0 but instead of returning the first solution, the number of
solutions (i.e. 0 or 1), the number of backtracks, and the number of
link dances is returned as a 3-element numeric vector.

A = ¯3: like A = ¯1 but instead of returning all solutions, the number of
solutions, the number of backtracks, and the number of link dances is
returned as a 3-element numeric vector.

A = ¯4: A number of single steps in Knuth's Algorithm are performed. Let e.g.
A←¯4 r1 r2 r3. Then Z←A ⎕DLX B is the matrix B after 3 steps r1, r2, and r3
have been performed. r1, r2, and r3 are valid (as per ⎕IO) row numbers of B,
and a step with a given row changes B as follows:
@itemize
@item
the given row is set to 0 (this row becomes part of the final result),
@item
all other rows that have a 1 or a 2 in the same column as B are set to 0
(these 1s or 2s in the other rows conflict with the 1 or 2 in the given row
and are therefore removed from B), and
@item
all columns that have no 1s or 2s left are removed from B (these columns have
a 1 or 2 in B and the constraint is therefore satisfied).
@end itemize
The purpose of ¯4 ⎕DLX is:
@itemize
@item
To demonstrate how Knuth's algorithm works, and/or
@item
To preset some initial values in B. For example, one can first initialize
the constraints for an empty Sudoku (which is the same matrix for all Sudokus)
and then enter the initial digits of a particular Sudoku using 4 ⎕DLX. If there are, say, 20 initial digits in the Sudoku then A has 21 elements (¯4 plus the
20 rows representing the 20 initial digits. The result of ¯4 ⎕DLX B has the
same number of rows as B (with some rows now cleared), but fewer columns than B.
@end itemize

The right argument B of ⎕DLX B or A ⎕DLX B is a constraints matrix whose
columns consist of either 0s and 1s (called a primary column) or 0s and 2s
(called secondary columns). The 0s, 1s and 2s can be the integers 0, 1, or 2,
characters '0', '1', or '2' respectively, or ' ' meaning '0'. The character
representation is useful when B becomes large and shall be printed because
the spaces in the numerical variant will not be printed. In the following,
B is assumed to be numeric.

Let Z←A ⎕DLX B, and let R be a solution in Z, that is, Z itself (A = 0) or
R is Z[k] for some k if A ≠ 0. And let S←+⌿B[R].

Then S = s1 s2 s3, ... sN where N is the number of columns in B and sj=1
if j is a primary column of B and sj∈0 1 if j is a secondary column of B.

In other words, ⎕DLX B computes a subset of the rows of B in such a way that
for every column j of B exactly one (for primary column j) or at most one
(for a secondary column) of the rows in a solution has its j'th element set
to 1 and all other words set their j'th element set to 0.

In yet other words, for every solution returned by ⎕DLX B, a 1 in one row
prevents all other rows that also have a 1 in that column, and all rows
together have exactly one 1 in every primary column and at most one 1 in
every secondary column. In the absence of secondary columns, the problem
solved by ⎕DLX is also known as the "exact cover problem"

If all that sounds weird and useless, consider the following APL program
for finding all solutions of the 8 Queens problem on a chess board (which
probably every programmer has programmed at some point in time):

@example
      RC←8↑'1' ◊ D←15↑¯8↑'2'   ⍝ helpers for constructing Q8
      ⍝     rows   cols   diag1    diag2
      Q8←⊃@{(R⌽RC),(C⌽RC),((C-R)⌽D),((¯7-R+C)⌽D)⊣(R C)←-8 8⊤⍵-⎕IO@} ¨ ⍳64
      Z←¯1 ⎕DLX Q8
      @{⎕UCS (65+⌊⍵÷8)(49+8∣⍵←⍵-⎕IO)@} ¨ ⊃Z[1 2 3 92] ⍝ solutions 1, 2, 3, and 92
 A1 B5 C8 D6 G2 E3 F7 H4 
 A1 B6 C8 D3 E7 F4 G2 H5 
 A1 B7 C4 E8 D6 G5 F2 H3 
 A8 B4 C1 D3 G7 E6 F2 H5 

      8 8⍴("+" "Q")[⎕IO+(⍳64)∈⊃Z[1]]   ⍝ visualize solution Z[1]
Q + + + + + + + 
 + + + + Q + + + 
 + + + + + + + Q 
 + + + + + Q + + 
 + + Q + + + + + 
 + + + + + + Q + 
 + Q + + + + + + 
 + + + Q + + + + 
@end example

Obviously Z contains solutions for the 8 Queens problem; the total number of
solutions is well known to be 92 and we showed only the first three and the
last solution above.

The constraint matrix Q8 is the key to success. The matrix has 64 rows - one
row for every field of the chess board. And it has 8 + 8 + 15 + 15 columns.
The first 8 columns of Q8 are constraints that prevent more than one Queen
from being placed in the same row of the chess board (the argument ⍵ is the
field number counting from left to right and from bottom to top).
The next 8 columns of Q8 are constraints that prevent more than one Queen
from being placed in the same column of the chess board.

If we would call ⎕DLX with only these constraints, i.e. ⎕DLX T8←64 16↑Q8,
then we would get the solutions of the 8 tower problem. However, we continue
and add 15 more constraints for each of the two diagonals. The resulting
constraint matrix Q8 is this:

@example
      Q8

1       1              2                     2
1        1              2                   2 
1         1              2                 2  
1          1              2               2   
1           1              2             2    
1            1              2           2     
1             1              2         2      
1              1              2       2       
 1      1             2                     2 
 1       1             2                   2  
 1        1             2                 2   
 1         1             2               2    
 1          1             2             2     
 1           1             2           2      
 1            1             2         2       
 1             1             2       2        
  1     1            2                     2  
  1      1            2                   2   
  1       1            2                 2    
  1        1            2               2     
  1         1            2             2      
  1          1            2           2       
  1           1            2         2        
  1            1            2       2         
   1    1           2                     2   
   1     1           2                   2    
   1      1           2                 2     
   1       1           2               2      
   1        1           2             2       
   1         1           2           2        
   1          1           2         2         
   1           1           2       2          
    1   1          2                     2    
    1    1          2                   2     
    1     1          2                 2      
    1      1          2               2       
    1       1          2             2        
    1        1          2           2         
    1         1          2         2          
    1          1          2       2           
     1  1         2                     2     
     1   1         2                   2      
     1    1         2                 2       
     1     1         2               2        
     1      1         2             2         
     1       1         2           2          
     1        1         2         2           
     1         1         2       2            
      1 1        2                     2      
      1  1        2                   2       
      1   1        2                 2        
      1    1        2               2         
      1     1        2             2          
      1      1        2           2           
      1       1        2         2            
      1        1        2       2             
       11       2                     2       
       1 1       2                   2        
       1  1       2                 2         
       1   1       2               2          
       1    1       2             2           
       1     1       2           2            
       1      1       2         2             
       1       1       2       2              
@end example

To see what, for example, the first solution looks like and how it relates
to the constraints matrix Q8:

@example
      ⍝ the rows in Q8 of the first solution
      ⍝
      Z[1]
 1 14 24 27 39 44 50 61 

      ⍝ the first solution translated back into the problem domain
      ⍝
      @{⎕UCS (65+⌊⍵÷8)(49+8∣⍵←⍵-⎕IO)@} ¨ ⊃Z[1]
 A1 B6 C8 D3 E7 F4 G2 H5 

      ⍝ the constraints of the rows of the first solution
      ⍝
      Q8[⊃Z[1];]
1       1              2                     2
 1           1             2           2      
  1            1            2       2         
   1      1           2                 2     
    1         1          2         2          
     1     1         2               2        
      1  1        2                   2       
       1    1       2             2           

      ⍝ all primary constraints met?
      ⍝
      +⌿ ' '≠ Q8[⊃Z[1];] 
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 
      1 0 0 0 0 1
@end example

And that's it: call ⎕DLX to get the solution(s). In general all problems that
can be solved by ⎕DLX contain 3 steps:

@itemize @bullet
@item translate the problem into a constraints matrix B,
@item Z←⎕DLX B, and
@item translate the result Z back into the problem domain.
@end itemize

Another application of ⎕DLX is solving sudokus. The constraints matrix is
a little more complicated, but the principle is the same. GNU APL is shipped
with two workspaces: sudoku.apl (which solves sudokus without using ⎕DLX) and
sudoku_DLX.apl (which solves sudokus using ⎕DLX).


@c ----------------------------------------------------------------------------
@node ⎕ENV - Environment Variables
@section ⎕ENV - Environment Variables

⎕ENV contains the environment variables of the process that is running
GNU APL. @xref{APL Scripting}.

@c ----------------------------------------------------------------------------
@node ⎕FIO - File I/O Functions
@section ⎕FIO - File I/O Functions

As of GNU APL 1.6, the native function FILE_IO has been replaced by the
system function ⎕FIO. ⎕FIO normally takes a numeric axis argument which
selects one of many different functions, most of which are contained in
the standard C library. The arguments for these functions are usually the same
as the corresponding C functions and the man page for each function describes
the meaning of the arguments.

For example, ⎕FIO[3] corresponds to fopen() and 'man fopen' explains what fopen does (opening a file).

Calling ⎕FIO monadically without an axis and with an empty right argument
shows a list of all functions provided by ⎕FIO:

@example
      ⎕FIO ''
   Functions provided by ⎕FIO...

   Legend: a - address family, IPv4 address, port (or errno)
           d - table of dirent structs
           e - error code (integer as per errno.h)
           h - file handle (integer)
           i - integer
           n - names (nested vector of strings)
           s - string
           u - time divisor: 1       - second
                             1000    - milli second
                             1000000 - micro second
           y4 - seconds, wday, yday, dst (
           y67- year, mon, day, hour, minute, second, [dst]
           y9 - year, mon, day, hour, minute, second, wday, yday, dst
           A1, A2, ...  nested vector with elements A1, A2, ...

           ⎕FIO     ''    print this text on stderr
        '' ⎕FIO     ''    print this text on stdout
           ⎕FIO[ 0] ''    print this text on stderr
        '' ⎕FIO[ 0] ''    print this text on stdout

   Zi ←    ⎕FIO[ 1] ''    errno (of last call)
   Zs ←    ⎕FIO[ 2] Be    strerror(Be)
   Zh ← As ⎕FIO[ 3] Bs    fopen(Bs, As) filename Bs mode As
   Zh ←    ⎕FIO[ 3] Bs    fopen(Bs, "r") filename Bs

File I/O functions:

   Ze ←    ⎕FIO[ 4] Bh    fclose(Bh)
   Ze ←    ⎕FIO[ 5] Bh    errno (of last call on Bh)
   Zi ←    ⎕FIO[ 6] Bh    fread(Zi, 1, 5000, Bh) 1 byte per Zi
   Zi ← Ai ⎕FIO[ 6] Bh    fread(Zi, 1, Ai, Bh) 1 byte per Zi
   Zi ← Ai ⎕FIO[ 7] Bh    fwrite(Ai, 1, ⍴Ai, Bh) 1 byte per Ai
   Zi ←    ⎕FIO[ 8] Bh    fgets(Zi, 5000, Bh) 1 byte per Zi
   Zi ← Ai ⎕FIO[ 8] Bh    fgets(Zi, Ai, Bh) 1 byte per Zi
   Zi ←    ⎕FIO[ 9] Bh    fgetc(Zi, Bh) 1 byte
   Zi ←    ⎕FIO[10] Bh    feof(Bh)
   Zi ←    ⎕FIO[11] Bh    ferror(Bh)
   Zi ←    ⎕FIO[12] Bh    ftell(Bh)
   Zi ← Ai ⎕FIO[13] Bh    fseek(Bh, Ai, SEEK_SET)
   Zi ← Ai ⎕FIO[14] Bh    fseek(Bh, Ai, SEEK_CUR)
   Zi ← Ai ⎕FIO[15] Bh    fseek(Bh, Ai, SEEK_END)
   Zi ←    ⎕FIO[16] Bh    fflush(Bh)
   Zi ←    ⎕FIO[17] Bh    fsync(Bh)
   Zi ←    ⎕FIO[18] Bh    fstat(Bh)
   Zi ←    ⎕FIO[19] Bh    unlink(Bc)
   Zi ←    ⎕FIO[20] Bh    mkdir(Bc, 0777)
   Zi ← Ai ⎕FIO[20] Bh    mkdir(Bc, AI)
   Zi ←    ⎕FIO[21] Bh    rmdir(Bc)
   Zi ← A  ⎕FIO[22] 1     printf(         A1, A2...) format A1
   Zi ← A  ⎕FIO[22] 2     fprintf(stderr, A1, A2...) format A1
   Zi ← A  ⎕FIO[22] Bh    fprintf(Bh,     A1, A2...) format A1
   Zi ← Ac ⎕FIO[23] Bh    fwrite(Ac, 1, ⍴Ac, Bh) 1 Unicode per Ac, Output UTF8
   Zh ← As ⎕FIO[24] Bs    popen(Bs, As) command Bs mode As
   Zh ←    ⎕FIO[24] Bs    popen(Bs, "r") command Bs
   Ze ←    ⎕FIO[25] Bh    pclose(Bh)
   Zs ←    ⎕FIO[26] Bs    return entire file Bs as byte vector
   Zs ← As ⎕FIO[27] Bs    rename file As to Bs
   Zd ←    ⎕FIO[28] Bs    return content of directory Bs
   Zn ←    ⎕FIO[29] Bs    return file names in directory Bs
   Zs ←    ⎕FIO 30        getcwd()
   Zn ← As ⎕FIO[31] Bs    access(As, Bs) As ∈ 'RWXF'
   Zh ←    ⎕FIO[32] Bi    socket(Bi=AF_INET, SOCK_STREAM, 0)
   Ze ← Aa ⎕FIO[33] Bh    bind(Bh, Aa)
   Ze ←    ⎕FIO[34] Bh    listen(Bh, 10)
   Ze ← Ai ⎕FIO[34] Bh    listen(Bh, Ai)
   Za ←    ⎕FIO[35] Bh    accept(Bh)
   Ze ← Aa ⎕FIO[36] Bh    connect(Bh, Aa)
   Zi ←    ⎕FIO[37] Bh    recv(Bh, Zi, 5000, 0) 1 byte per Zi
   Zi ← Ai ⎕FIO[37] Bh    recv(Bh, Zi, Ai, 0) 1 byte per Zi
   Zi ← Ai ⎕FIO[38] Bh    send(Bh, Ai, ⍴Ai, 0) 1 byte per Ai
   Zi ← Ac ⎕FIO[39] Bh    send(Bh, Ac, ⍴Ac, 0) 1 Unicode per Ac, Output UTF8
   Zi ←    ⎕FIO[40] B     select(B_read, B_write, B_exception, B_timeout)
   Zi ←    ⎕FIO[41] Bh    read(Bh, Zi, 5000) 1 byte per Zi
   Zi ← Ai ⎕FIO[41] Bh    read(Bh, Zi, Ai) 1 byte per Zi
   Zi ← Ai ⎕FIO[42] Bh    write(Bh, Ai, ⍴Ai) 1 byte per Ai
   Zi ← Ac ⎕FIO[43] Bh    write(Bh, Ac, ⍴Ac) 1 Unicode per Ac, Output UTF8
   Za ←    ⎕FIO[44] Bh    getsockname(Bh)
   Za ←    ⎕FIO[45] Bh    getpeername(Bh)
   Zi ← Ai ⎕FIO[46] Bh    getsockopt(Bh, A_level, A_optname, Zi)
   Ze ← Ai ⎕FIO[47] Bh    setsockopt(Bh, A_level, A_optname, A_optval)
   Ze ← As ⎕FIO[48] Bh    fscanf(Bh, As)
   Zs ←    ⎕FIO[49] Bs    return entire file Bs as nested lines
   Zs ← LO ⎕FIO[49] Bs    ⎕FIO[49] Bs and pipe each line through LO.
   Zi ←    ⎕FIO[50] Bu    gettimeofday()
   Zy4←    ⎕FIO[51] By67  mktime(By67)  Note: Jan 2, 2017 is: 2017 1 2 ...
   Zy9←    ⎕FIO[52] Bi    localtime(Bi) Note: Jan 2, 2017 is: 2017 1 2 ...
   Zy9←    ⎕FIO[53] Bi    gmtime(Bi)    Note: Jan 2, 2017 is: 2017 1 2 ...
   Zi ←    ⎕FIO[54] Bs    chdir(Bs)
   Ze ← As ⎕FIO[55] Bh    sscanf(Bs, As) As is the format string
   Zs ← As ⎕FIO[56] Bs    write nested lines As to file named Bs

Benchmarking functions:

           ⎕FIO[200] Bi    clear statistics with ID Bi
   Zn ←    ⎕FIO[201] Bi    get statistics with ID Bi
           ⎕FIO[202] Bs    get monadic parallel threshold for primitive Bs
        Ai ⎕FIO[202] Bs    set monadic parallel threshold for primitive Bs
           ⎕FIO[203] Bs    get dyadic parallel threshold for primitive Bs
        Ai ⎕FIO[203] Bs    set dyadic parallel threshold for primitive Bs
@end example

A new feature of ⎕FIO (which is not available with the native function FILE_IO)
is ⎕FIO[49]. ⎕FIO[49] is a monadic operator which takes a monadic conversion
function as function argument. For example:

@smallexample
Z←F ⎕FIO[49] 'filename'
@end smallexample

reads the file named filename line by line. For every line read, the conversion
function F is called and the result returned by F is enclosed and stored in Z.
In other words,

@smallexample
Z←F ⎕FIO[49] 'filename'
@end smallexample

does:

@smallexample
Z F¨Z←⎕FIO[49] 'filename'
@end smallexample


@c ----------------------------------------------------------------------------
@node ⎕FFT - Fast Fourier Transform
@section ⎕FFT - Fast Fourier Transform

For those interested in signal processing and the like, GNU APL provides ⎕FFT:

The monadic form ⎕FFT B is a shortcut for the dyadic form 0 ⎕FFT B. It computes
the FFT of complex or real B without applying a window function.

The dyadic form A ⎕FFT B provides more control over what ⎕FFT computes. A is
an integer scalar which falls into one of three ranges.

The first range from ¯15 to ¯10 does not compute an FFT, but returns the result
of multiplying B with one of several window functions, that are frequently
used in the context of FFTs. The result has same shape as B and can be used for
analyzing or troubleshooting FFTs:

@itemize
@item A=¯10: no FFT, return the Hann window applied to B
@item A=¯11: no FFT, return the Hamming window applied to B
@item A=¯12: no FFT, return the Blackman window applied to B
@item A=¯13: no FFT, return the Blackman-Harris window applied to B
@item A=¯14: no FFT, return the Blackman-Nuttal window applied to B
@item A=¯15: no FFT, return the Flat-Top window applied to B
@end itemize

The second range around 0 contains the computation of the forward and inverse
FFTs:

@itemize
@item  0 ⎕FFT B returns the "normal" (aka. forward) FFT of the numeric array B.
@item ¯1 ⎕FFT B returns the inverse FFT of the numeric array B.
@end itemize

The third range from 10 to 15 corresponds to the first range and first
multiplies B with a window function and then computes the FFT:

@itemize
@item A=10: FFT(B × Hann window)
@item A=11: FFT(B × Hamming window)
@item A=12: FFT(B × Blackman window)
@item A=13: FFT(B × Blackman-Harris window)
@item A=14: FFT(B × Blackman-Nuttal window)
@item A=15: FFT(B × Flat-Top window)
@end itemize

⍴⍴B can be 1 (one-dimensional FFT, the most common case) or more. The
implementation of ⎕FFT uses libfftw3, aka. "The fastest Fourier Transform in
the West." GNU APL checks the presence of libfftw3 when it is ./configure'd.
If libfftw3 is present then ⎕FFT will hopefully return the expected result;
if not then a SYNTAX ERROR will be raised when ⎕FFT is used.

⎕FFT honors the presence of /etc/fftw/wisdom (see man fftw-wisdom) to speed
up the computations performed by ⎕FFT. Creating /etc/fftw/wisdom will take a
few hours, though, so that creating it will not pay off for most mortals.


@c ----------------------------------------------------------------------------
@node ⎕GTK - GTK Interface
@section ⎕GTK - GTK Interface

GTK (Gimp ToolKit) is a rather powerful library for creating graphical user
interfaces (GUIs). ⎕GTK makes a subset of the almost 10000 functions in the
different GTK libraries available to GNU APL programs. With ⎕GTK a GNU APL
program can, for example, replace the somewhat crude ⎕ or ⍞ input methods
of standard APL with a more intuitive GUI.

The details of using ⎕GTK would go far beyond the scope of this info manual
and has therefore been put into a separate document.
See: @b{HOWTOs/Quad-GTK.html}.

@c ----------------------------------------------------------------------------
@node ⎕JSON - JSON Parsing
@section ⎕JSON - JSON Parsing

GNU APL provides ⎕JSON for decoding and encoding JSON strings and files.
A (valid) JSON string is mapped to an APL value as follows:

@itemize
@item JSON value ←→ structured or non-structured APL value
@item JSON number ←→  APL number
@item JSON string ←→  APL string
@item JSON literal ←→  enclosed APL string (one of either ⊂'null', ⊂'true',
      or ⊂'false' )
@item JSON array ←→  APL vector
@item JSON object ←→  structured APL value (associative array)
@end itemize

@c ............................................................................
@node Monadic ⎕JSON
@subsection Monadic ⎕JSON

Z←⎕JSON B converts the string B into an APL value according to the mapping
above. If that fails then command )MORE provides information about the error
and where in B the error was detected.

@c ............................................................................
@node Dyadic ⎕JSON
@subsection Dyadic ⎕JSON

Z←A ⎕JSON B performs one of the following conversions:

@itemize
@item @b{Z←0 ⎕JSON B} is the same as monadic Z←⎕JSON B.

@item @b{Z←1 ⎕JSON B} is like Z←0 ⎕XML B, except that B is not the JSON
string itself but the name of a file from which a JSON string is read.

@item @b{Z←2 ⎕JSON B} is the inverse of Z←0 ⎕XML B. It converts an APL
value to a JSON string.

@item @b{Z←3 ⎕JSON B} is like 2 ⎕JSON B, except that the member names
(keys) of all JSON objects are sorted alphabetically. That is of little use
(but costs performance) if Z is not displayed but processed further, but
it could help troubleshooting when Z is displayed and the number of keys in Z
is large.
@end itemize


@c ----------------------------------------------------------------------------
@node ⎕MAP - Map Value
@section ⎕MAP  - Map Value

⎕MAP changes the ravel items of its right argument according to a mapping
table provided as its left argument. Let Z←A ⎕MAP B.

The left argument A of Z←A ⎕MAP B shall be a N×2 matrix. Each 2-element row of
A, say A[J;], specifies a separate mapping A[J;1] → A[J;2]. The result Z has
the same shape as the right argument B. The items Z[...] of their result are
constructed from their corresponding items B[...] in B as follows:

@itemize

@item if B[...] ≡ A[J;1] for some J then Z[...] is A[J;2].

@item otherwise B[...] is different from all A[1;J] and then Z[...] is B[...]

@end itemize

In other words, Z is B, but with items of B found in A[;1] replaced by their
mapped item A[;2].

In practice the left argument A is frequently created from a literal APL value
such as 5 2⍴'eEwWaAzZ92' in the example below. To simplify these cases,
A ⎕MAP B also accepts a vector instead of a N×2 matrix:

      A ⎕MAP B   ←→   ((N 2)⍴A) ⎕MAP B   if (2×N)   ←→    ⍴A

Examples:

@example
      ⍝ the map A
      ⊢A←5 2⍴'eEwWaAzZ92'
eE    ⍝ map 'e' → 'E'
wW    ⍝ map 'w' → 'W'
aA    ⍝ map 'a' → 'A'
zZ    ⍝ map 'z' → 'Z'
92    ⍝ map '9' → '2'

      ⍝ the value B being mapped
      ⊢B←'Halloween'
Halloween

      ⍝ the result of A ⎕MAP B
      A ⎕MAP B
HAlloWEEn

      A←'eEwWaAzZ92'   ⍝ vector A instead of N×2 matrix
      A ⎕MAP B
HAlloWEEn
@end example

NOTES:
@itemize @bullet
@item
The keys of the mapping A (i.e. the elements in column A[;1]) must be
unique. If they are not then the mapping is ambiguous and a DOMAIN ERROR
is raised.

@item
Nested items in B: if an item of B is nested, then it is either equal to some
(also nested) key, say A[N;1], (in that case it is being mapped to A[N;2] in
the result), or it is different from all keys in A[;1] (and in that case the
corresponding item in the result is the item in B.

@item
Nested items in A: if a key, say A[N;1], is nested and matches an equal (hence
also nested) item in B, then the corresponding item in the result will be
A[N;2] (which may or may not be nested). If a (nested or not nested) key A[N;1]
is equal to an item in B, then the corresponding item in the result will be
nested if A[N;2] is nested. If, for some N, the depths of A[N;1] and A[N;2]
differ, then depths of B and A ⎕MAP B may differ as well. That is, ⎕MAP
conserves the shape of B, but not necessarily the depth of B.

@item
By default the mapping of B is non-recursive, i.e. the top-level items of
B are compared with the top-level keys A[;1], and nested keys and items of B
are handled as described above.

Sometimes, however, it is desirable to recursively descend into the nested
sub-values of B (though never of A). This can be achieved by enclosing A as
shown in the examples below.
@end itemize

@example
      A←5 2⍴'eEwWaAzZ92'
      B←'Hal' 'low' 'een'   ⍝ nested B
      4 ⎕CR A ⎕MAP B        ⍝ non-recursive (none of the keys in A[;1] matches)
┏→━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━┓┃
┃┃Hal┃ ┃low┃ ┃een┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━━━┛

      4 ⎕CR (⊂A) ⎕MAP B   ⍝ recursive (some simple keys in A[;1] match)
┏→━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━┓┃
┃┃HAl┃ ┃loW┃ ┃EEn┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━━━┛
@end example


@c ----------------------------------------------------------------------------
@node ⎕MX - Matrix/Statistics
@section ⎕MX - Matrix/Statistics

⎕MX was kindly contributed by Henrik Moller

⎕MX[x] is a system function that provides a collection of matrix and
statistical operations such as covariance, eigensystems, and so on, along
with random number generation in several statistical distributions. Particular
operations are selected through the use of the axis specifier @b{x}:

@example
            ⎕MX[1] B       (monadic) Determinant
        (A) ⎕MX[2] B       (nomadic) Cross product
          A ⎕MX[3] B       (dyadic)  Vector angle
            ⎕MX[4] B       (monadic) Eigenvector
            ⎕MX[5] B       (monadic) Eigenvalue
            ⎕MX[6] B       (monadic) Ident
            ⎕MX[7] B       (monadic) Rotation matrix
          A ⎕MX[8] B       (dyadic)  Homogeneous transform
            ⎕MX[9] B       (monadic) Norm
        (A) ⎕MX[10] B      (nomadic) Randoms
        (A) ⎕MX[11] B      (nomadic) Covariance
          A ⎕MX[12] B      (dyadic)  Histogram
          A ⎕MX[13] B      (dyadic)  Print
            ⎕MX[14] B      (monadic) Set RNG seed
@end example

(These may by shown by entering ⎕MX '' or ⎕MX[0]0 on the APL command line.)
In the function list above, @b{A} indicates a mandatory left argument,
while (@b{A}) indicates an optional left argument. The right argument @b{B}
of @b{⎕MX[X]} is always mandatory.

@c ............................................................................
@node Determinant
@subsection Determinant

The determinant operation, ⎕MX[1], is monadic and is valid only for
numeric arguments (real or complex) where ⍴⍴B = 2 and (1↑⍴B) = (¯1↑⍴B)
(i.e. a square matrix). For Example:

@example
      mx[1] 3 3⍴5 3 1 9 7j2 6 2 8 4j5
¯164J76
@end example

@c ............................................................................
@node Cross Product
@subsection Cross Product

The cross product operation, ⎕MX[2], is nomadic. In monadic form, it
requires a numeric argument (real or complex) where ⍴⍴B = 2 and
(↑⍴B)+1 = (1↓⍴B), (i.e. a matrix of N-1 rows of length-N vectors).
For Example:

@example
      ⊢B←3 4⍴ 1 2 3j6 4 5 6 7 8 9 10 11 12
1  2  3J6  4
5  6  7    8
9 10 11   12
      ⎕MX[2] B
0J48 0J¯72 0 0J24
@end example

For reasons I'm not enough of a mathematician to understand, cross
products are only valid in 3-space and 7-space, so the only valid monadic
arguments are matrices of shapes [2 3] or [6 7]. However, ⎕MX, doesn't
check this and will happily give you a result in any dimensionality thus
leaving it to your imagination what it may mean. I'm also not enough of a
mathematician to know what the significance may be of cross products
involving complex numbers and thus I can't guarantee I'm using an
appropriate method to compute them.

In dyadic form @b{A ⎕MX[2] B}, cross-product requires left and right
arguments to be numeric and ⍴⍴A = 1, ⍴⍴B = 1, ⍴A = 3, and ⍴B = 3:

@example
      1 2 3j6 ⎕MX[2] 4 5 6
¯3J¯30 6J24 ¯3
@end example

@c ............................................................................
@node Vector Angle
@subsection Vector Angle

The vector angle operation ⎕MX[3] is dyadic and returns the angle between
two isomorphic real or complex vectors where ⍴⍴A = 1, ⍴⍴B = 1, and ⍴A = ⍴B:

@example
      (⍳8) ⎕MX[3] 4j4+⍳8
0.2515670406J0.09930859532
@end example

@c ............................................................................
@node Eigenvector
@subsection Eigenvector

The eigenvector operation, ⎕MX[4], is monadic and returns the complex
eigenvectors of real non-symetric square matrices:

@example
      B
¯1  1 ¯1 1
¯8  4 ¯2 1
27  9  3 1
64 16  4 1
      ⎕PP←4
      ⎕MX[4] B
  0.09988            0.1113          ¯0.2925          ¯0.9445
  0.04308J0.009687  ¯0.07091J0.1389   0.5166J¯0.01601  0.8396J0.04139
  0.04308J¯0.009687 ¯0.07091J¯0.1389  0.5166J0.01601   0.8396J¯0.04139
 ¯0.1449             0.3566           0.9194           0.08118
@end example

where each row represents an eigenvector:

@example
      E←⎕MX[4] B
      E[⎕IO;]                   ⍝ first vector
0.09988 0.1113 ¯0.2925 ¯0.9445
      E[⎕IO+1;]                 ⍝ second vector
0.04308J0.009687 ¯0.07091J0.1389 0.5166J¯0.01601 0.8396J0.04139
      etc.
@end example

The eigenvalues are computed concurrently and the eignenvector matrix
is sorted according to the descending magnitude of the corresponding
eigenvalues.

@c ............................................................................
@node Eigenvalue
@subsection Eigenvalue

The eigenvalue operation, ⎕MX[5], is also monadic with an argument of
the same shape as that of the argument of the eigenvector operation. It
returns a sorted complex vector of eigenvalues of the matrix argument:

@example
      ⊢F←⎕MX[5] B
¯6.414 5.546J3.085 5.546J¯3.085 2.323
@end example

The order of eigen vectors, @b{E} from above, is related to the order of
the eigenvalues, @b{F}, by the relationship:

@example
             E[k;] ⬄ F[k]
@end example

(Note that, from the above example:

@example
      |F
6.41391 6.346285607 6.346285607 2.3228
@end example

the order of the magnitude of the elements is descending, making the
initial eigenvector, @b{E[⎕IO;]}, the dominant eigenvector and each
successive eigenvector successively less significant.)

Accordingly:

@example
      4 ⎕CR (⊂⍉(1,⍴F)⍴F),(⊂E)
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓┃
┃↓¯6.414       ┃ ↓  0.09988           0.1113         ¯0.2925          ¯0.9445          ┃┃
┃┃ 5.546J3.085 ┃ ┃  0.04308J0.00969  ¯0.07091J0.139   0.5166J¯0.01601  0.8396J0.04139  ┃┃
┃┃ 5.546J¯3.085┃ ┃  0.04308J¯0.00969 ¯0.07091J¯0.139  0.5166J0.01601   0.8396J¯0.04139 ┃┃
┃┃ 2.323       ┃ ┃ ¯0.1449            0.3566          0.9194           0.08118         ┃┃
┃┗━━━━━━━━━━━━━┛ ┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛┃
┗ϵ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
   Values            Vectors
@end example

@c ............................................................................
@node Ident
@subsection Ident

The monadic ident operation, ⎕MX[6], takes a scalar real integer argument
and returns a square identity matrix of that size:

@example
      ⎕MX[6] 4  
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
@end example

@c ............................................................................
@node Rotation
@subsection Rotation

The rotation operation, ⎕MX[7], is monadic and takes a real or complex
scalar angle and returns a 2-space rotation matrix:

@example
      ⎕MX[7] ○30÷180
0.866 ¯0.5
0.5    0.866
@end example

@c ............................................................................
@node Homogeneous transform
@subsection Homogeneous transform

The homogeneous operation ⎕MX[8] is dyadic and takes as its left argument
a 3-space displacement vector and, as its right argument, a 3-space
elemental rotation (i.e., independent rotations around the X, Y, and Z axes
respectively) and returns the corresponding homogeneous matrix:

@example
      1 2 3 ⎕MX[8] ○(30 45 60)÷180
 0.6124 0.2803  0.7392 0
 0.3536 0.7392 ¯0.5732 0
¯0.7071 0.6124  0.3536 0
 1      2       3      1
@end example

@c ............................................................................
@node Norm
@subsection Norm

The monadic norm operation, ⎕MX[9], takes a real or complex non-scalar
numeric argument and returns a result of the same shape such that the sum
of the squares of the elements equals 1:

@example
      ⎕PP←10
      ⎕MX[9] 1 2 3 4
0.1825741858 0.3651483717 0.5477225575 0.7302967433
      (⎕MX[9] 1 2 3 4)*2
0.03333333333 0.1333333333 0.3 0.5333333333
      +/(⎕MX[9] 1 2 3 4)*2
1
     
      ⊢B←(3 3⍴⍳9)-⎕IO
0 1 2
3 4 5
6 7 8
      ⎕MX[9] B
0            0.0700140042 0.1400280084
0.2100420126 0.2800560168 0.350070021
0.4200840252 0.4900980294 0.5601120336
      +/,(⎕MX[9] B)*2
1
@end example

If the magnitude of the argument is zero, the operation returns an all-zero
result with the same shape as the argument.


@c ............................................................................
@node Randoms
@subsection Randoms

The nomadic randoms operation, @b{⎕MX[10]} or @b{⎕MX[10 x]} where @b{x}
is an integer specifying the distribution of the randoms generated.
At present, the distibutions supported are:

@itemize
@item @b{x=0}: Normal distribution (default)
@item @b{x=1}: Lognormal distribution
@item @b{x=2}: Chi Squared distribution
@item @b{x=3}: Student T distribution
@end itemize

If @b{x} is not specified then the normal distribution is used.

The right argument @b{B} of @b{⎕MX B} is a real or complex scalar
that specifies the desired standard deviation. If @b{B} is complex,
the value returned is also complex with random real and imaginary components.

If the optional left argument @b{A} is specified, then @b{A} must be a
scalar integer and @b{Z←A ⎕MX B}  and the operation returns a random vector
@b{Z} with @b{A = ⍴Z}.

If a mean value is  applicable for the selected distribution, then it is
set to @b{0}.

@example
      ⎕MX[10] 2
¯1.0504575511
      ⎕MX[10] 2
¯2.9337180373
      5 ⎕MX[10] 2
0.05154476627 3.80631021 2.455474045 3.165260688 0.6792784063
      ⎕PP←4
      3 ⎕MX[10] 1j1
¯1.134J¯1.453 1.111J¯0.1916 ¯1.106J¯0.9383
@end example

@image{MX_distributions}

(This image is a plot of three 200-bucket histograms of 200000 samples
each of a normal distribution, a chi-square distribution, and a
Student T distribution. See Histograms below.)

An initial seed may be specified with

@example
        ⎕MX[14] <seed>
@end example

where <seed> is a single integer. This allows repeatable random sequences:

@example
      ⊣⎕MX[14] 3
      4 ⎕MX[10] 1
¯1.358348261 ¯1.75175064 ¯2.009430519 ¯0.5167968281
      ⊣⎕MX[14] 3
      4 ⎕MX[10] 1
¯1.358348261 ¯1.75175064 ¯2.009430519 ¯0.5167968281
@end example

@c ............................................................................
@node Covariance
@subsection Covariance

The covariance operation, ⎕MX[11], is nomadic. In monadic form, it
interprets each row of a matrix argument as samples of a given
variable, computes the covariance of each sample set with respect to
every other sample set, and presents the covariances as a matrix. For
example:

@example
      ⊢B←3 4⍴?12⍴10
5 8  1 9
7 6  2 9
6 5 10 4
      ⎕PP←10
      ⎕MX[11] B
12.91666667   9.333333333 ¯9.25
 9.333333333  8.666666667 ¯7.333333333
¯9.25        ¯7.333333333  6.916666667
@end example

In dyadic form, it computes the covariance of the argument vectors:

@example
      1 2j6 3 4 ⎕MX[11] 4 3 2j5 1
¯1.666666667J¯2.5
@end example

In dyadic form, ⍴⍴A = 1, ⍴⍴B = 1, and ⍴A = ⍴B.

Sample variance can be computed by supplying the same argument on the
left and right:

@example
      A← 6 7 10 13 14 14 18 19 22 24
      A ⎕MX[11] A ⍝ or ⎕MX[11]⍨ A
36.67777778 
@end example

@c ............................................................................
@node Histograms
@subsection Histograms

The histogram operation, ⎕MX[12], is dyadic and requires a single integer
left argument (the number of buckets) and a real or complex vector as a
right argument (the samples). It then creates an equal-width-partition
histogram of the samples contained in the right-argument vector. The width
of each bucket is equal to the range of the samples, max - min, divided by
the number of buckets. If the samples are complex, the imaginary components
are ignored. As examples:

@example
      4 ⎕MX[12] (⍳101)-⎕IO
25 25 25 26
@end example

The distribution curves above were the result of expressions like:

@example
      200 ⎕MX[12] 200000 ⎕MX[10 0] 1
@end example

I.e. a 200-bucket histogram of a 200 thousand-member vector of 
normally distributed samples with a standard deviation of 1.

@c ............................................................................
@node Print
@subsection Print

The print operation, ⎕MX[13], is dyadic and provides an easy way to
dump large quantities of data to files. It's left argument is a
string containing the file name to be created or, if the name starts
with a ``>'', appended to. The right argument may be a scalar, a
vector, a matrix, or a string. For example:

@example
      "samples.txt" ⎕MX[13] 34
      ">samples.txt" ⎕MX[13] 3 4
      ">samples.txt" ⎕MX[13] 3 4⍴⍳12
      ">samples.txt" ⎕MX[13] "example line"
      ">samples.txt" ⎕MX[13] "example\nline"
@end example

will create a file ``samples.txt'' containing:

@example
34
3 4
 1  2  3  4
 5  6  7  8 
 9 10 11 12 
example line
example
line
@end example

Referring again to the distribution example, the plot data for that
was created by:

@example
    ∇
[0]   distro;v
[1]   v←⍉1 200⍴200 ⎕MX[12] 200000 ⎕MX[10 0] 1
[2]   v←v,⍉1 200⍴200 ⎕MX[12] 200000 ⎕MX[10 2] 8
[3]   v←v,⍉1 200⍴200 ⎕MX[12] 200000 ⎕MX[10 3] 8
[4]   v ⎕MX[13] 'MX_distribution.data'
    ∇
@end example

which created a 200-by-3 matrix, v, and printed it to the file
MX_distribution.data, that being the format required for use by
gnuplot:

@example
   1   48    0
   0  119    0 
   0  187    0 
   1  331    0
   0  500    0
   1  728    0
   0  977    0
   2 1161    0
   1 1475    0
      .
      .
      .
  for 200 lines
@end example

Instead of a filename, @b{⎕MX[13]} also accepts an integer scalar which should
be a file handle as returned by @b{⎕FIO[3]} aka. @b{⎕FIO.fopen}. This allows
e.g. for texts before and/or after the value(s) printed by @b{⎕MX[13]}.
For example:

@example
      Heading←⎕CR.string_to_UTF8 "Print: 10+⍳10\n\n"   ⍝ a heading
      Value←10 + ⍳10                                   ⍝ a value
      Handle←"w" ⎕FIO.fopen "/tmp/test"                ⍝ open file for writing
      Heading ⎕FIO.fwrite Handle                       ⍝ write the heading
      Handle ⎕MX.print Value                           ⍝ write the value
      ⎕FIO.fclose Handle                               ⍝ close the file
@end example

which produces file @b{/tmp/test} with the following 3-line content:

@example
Print: 10+⍳10

11 12 13 14 15 16 17 18 19 20 
@end example

@c ----------------------------------------------------------------------------
@node ⎕PLOT - Plot Data
@section ⎕PLOT - Plot Data

⎕PLOT is a function for visualizing numerical APL values. The values to be
plotted are provided as the right argument of ⎕PLOT while the optional
left argument controls details of the output, such as the plot window size,
colors for plot lines, points, and grids, etc.

The general syntax of ⎕PLOT is:

@example
   ⎕PLOT ⍬       ⍝ show a list of attributes and their default values
   H←⎕PLOT B     ⍝ plot B with all attributes set to their default values
   H←A ⎕PLOT B   ⍝ plot B with specified default attributes overridden
   ⎕PLOT H       ⍝ close the plot window with handle H
   ⎕PLOT  0      ⍝ verbosity: OFF (no debug output)
   ⎕PLOT ¯1      ⍝ verbosity: ON (print attribute values before plotting)
   ⎕PLOT ¯2      ⍝ verbosity: DEBUG (also print debug information)
   ⎕PLOT ¯3      ⍝ close all plot windows
   ⎕PLOT ¯6      ⍝ return all open plot window handles H
@end example

@c ............................................................................
@node The Plot Data B
@subsection The Plot Data B

The plot data @b{B} can be a vector (for a single plot line to be drawn)
or a matrix (in that case one plot line per matrix row is drawn, by default
in different colors). Each data item has to be numeric and represents one
point in the plot. The points that are adjacent in a row of the matrix are
connected by lines. By default plotted points are black and the lines
connecting them are green (for the first plot line). However, all colors
used, all diameters of points, and all thicknesses of the lines can be
fine-tuned by overriding the default values (see dyadic A ⎕PLOT B below).

@node Complex Plot Data B
@subsubsection Complex Plot Data B

If a data item B[N] or B[row;N] is @b{complex}, say B[N] = x + iy. then it is
placed (after some scaling) at position (x, y) of the plot. The X-range
[Xmin ... Xmax] of the plot is then determined by the real parts of B,
and the Y-range [Ymin ... Ymax] of the plot is determined by the imaginary
parts of B. That is:

@example
Xmin ← ⌊/,9○B    ⍝ smallest real part of B
Xmax ← ⌈/,9○B    ⍝ largest real part of B
Ymin ← ⌊/,11○B   ⍝ smallest imaginary part of B
Ymax ← ⌈/,11○B   ⍝ largest imaginary part of B
@end example

Example (plot a circle (actually: a regular 20-gon)):

@example
      ⎕PLOT +⌿1 0J1×[1]1 2 ∘.○ (0,⍳2×N) × ○÷N←10
@end example

produces this plot window (only visible in the HTML version of this document;
in text mode see file @b{doc/PLOT_circle.png}):

@image{PLOT_circle}

@c ............................................................................
@node Real Plot Data B
@subsubsection Real Plot Data B

On the other hand, if a data item B[N] or B[row;N] is @b{real}, then it is
placed at position (N, x) of the plot. The X-range is then
[⎕IO ... ⎕IO + ¯1↑⍴B] and the Y-range [Y@sub{min} ... Y@sub{max}] of the
plot is determined by the values of B. That is:

@example
Xmin ← ⎕IO             ⍝ smallest real part of B
Xmax ← ¯1 + ⎕IO + ⍴B   ⍝ largest real part of B
Ymin ← ⌊/,B            ⍝ smallest value in B
Ymax ← ⌈/,B            ⍝ largest value in B
@end example

Example:

@example
      ⎕PLOT 0 1 ¯1 2 ¯2 3 ¯3
@end example

produces this plot window (only visible in the HTML version of this document;
in text mode see file @b{doc/PLOT_zigzag.png}):

@image{PLOT_zigzag}

NOTE: In theory one can also mix real and complex values, even though doing so
makes little sense. If at least one item of the plot data B is complex, then
all real items in B are taken as complex with imaginary part 0.

@c ............................................................................
@node The Plot Attributes A
@subsection The Plot Attributes A

There are a number of attributes like colors, point sizes, line widths,
etc. that control how the resulting plot will look like.

Similar to function groups like @b{⎕CR} or @b{⎕FIO}, an empty argument
@b{B} in monadic @b{⎕PLOT B} displays a list of the arguments. The list of
arguments grows over time and therefore the list displayed by @b{⎕PLOT ⍬}
may be more up-to-date than this description. For example:

@example
      ⎕PLOT ''

   ⎕PLOT Usage:

   ⎕PLOT B     with ⍴⍴B > 0: plot B with default attribute values
   ⎕PLOT B     with integer scalar B: special ⎕PLOT functions
   A ⎕PLOT B   plot B with attribute overrides specified by A
           ├────────  0: verbosity OFF
           ├──────── ¯1: show X events
           ├──────── ¯2: show data
           ├──────── ¯3: close all windows
           ├──────── ¯4: show rendering
           ├─────── ¯6: show open handles
           └──── N > 0: close window N

   A is a nested vector of strings.
   Each string A[i] has the form "Attribute: Value"
   Colors are specified either as #RGB or as #RRGGBB or as RR GG BB)

   The attributes understood by ⎕PLOT and their default values are:

   1. Global (plot window) Attributes:

caption:            ⎕PLOT          (plot window caption)
output_filename:                   (output file name)
gui_driver:                        (GUI driver: GTK, XCB, or ASCII)
auto_close:         0              (= do not close X window automatically)
                   (1)             (= close if file was written successfully)
                   (2)             (= always close X window automatically)
with_border:        1              (= write plot area and window borders)
                   (0)             (= write only plot area to output file)
pw_pos_X:           50 pixel       (plot window position X)
pw_pos_Y:           50 pixel       (plot window position Y)
border_width:       10 pixel       (width of the window border)
pa_width:           600 pixel      (plotarea width)
pa_height:          400 pixel      (plotarea height)
pa_border_L:        50 pixel       (pixels left of the plotarea)
pa_border_R:        20 pixel       (pixels right of the plotarea)
pa_border_T:        25 pixel       (pixels above the plotarea)
pa_border_B:        25 pixel       (pixels below the plotarea)
gridX_style:        1              (X grid style = ──────── )
                   (2)             (             = ╴╴╴╴╴╴╴╴ )
                   (3)             (             = ─╴─╴─╴─╴ )
axisX_arrow:        0              (X-axis arrow)
axisX_label:        X              (X-axis label)
axisY_arrow:        0              (Yaxis arrow)
axisY_label:        Y              (Y-axis label)
axisZ_arrow:        0              (Z-axis arrow)
axisZ_label:        Z              (Z-axis label)
gridX_pixels:       44 pixel       (pixels between X grid lines)
gridX_variable:     0              (draw X grid (only) at plot points)
gridX_line_width:   1 pixel        (thickness of the X-grid lines)
gridX_color:        #000000        (color of the X-grid lines)
gridY_style:        1              (Y grid style, see gridX_style above)
gridY_pixels:       33 pixel       (pixels between Y grid lines)
gridY_line_width:   1 pixel        (thickness of the Y-grid lines)
gridY_color:        #000000        (color of the Y-grid lines)
gridZ_style:        1              (Z grid style, see gridX_style above)
gridZ_pixels:       33 pixel       (pixels between Z grid lines)
gridZ_line_width:   1 pixel        (thickness of the Z-grid lines)
gridZ_color:        #000000        (color of the Z-grid lines)
canvas_color:       #FFFFFF        (background color of the plot window)
legend_color:       #F0F0F0        (background color of the legend)
legend_X:           50 pixel       (the X position of the legend)
legend_Y:           50 pixel       (the Y position of the legend)
legend_dY:          15 pixel       (the distance between legend lines)
legend_lX:          50 pixel       (the length of the legend lines)
rangeX_min:         0.0            (the start of the X range to be plotted)
rangeX_max:         0.0            (the end of the X range to be plotted)
rangeY_min:         0.0            (the start of the Y range to be plotted)
rangeY_max:         0.0            (the end of the Y range to be plotted)
rangeZ_min:         0.0            (the start of the Z range to be plotted)
rangeZ_max:         0.0            (the end of the Z range to be plotted)
origin_X:           100 pixel      (X position offset of the origin)
origin_Y:           100 pixel      (Y position offset of the origin)
format_X:                          (format for X-axis ticks (GTK only))
                    %sT1%T2...%Tn  (static texts for ticks)
                    %G, %g         grid line number (starting at 1 or )
                    %v             value
                    %S             seconds (SS)
                    %I             minutes (MM)
                    %H, %h         hours (HH or h/hh)
                    %D, %d         day (DD or d/dd)
                    %M, %m         month (MM or m/mm)
                    %q             quarter (0..3)
                    %Q             quarter (1..4)
                    %Y, %y         year (YYYY or yy)
format_Y:                          (format for Y-axis (dito))
format_Z:                          (format for Z-axis (dito))

color_level-P:      (none)         (color gradient at P% (surface plots only))

   2. Local (plot line N) Attributes:

line_color-N:       #00FF00        (the color of plot line N)
line_style-N:       1              (line style, see gridX_style above)
line_width-N:       3 pixel        (the thickness of plot line N)
point_color-N:      #000000        (the color of the plot points)
point_style-N:      1              (= plot_points: ● )
                   (2)             (= plot_points: ▲ )
                   (3)             (= plot_points: ▼ )
                   (4)             (= plot_points: ◆ )
                   (5)             (= plot_points: ■ )
                   (6)             (= plot_points: ✚   (GTK only)
                   (7)             (= plot_points: ✖   (GTK only)
point_size-N:       8 pixel        (the outer diameter of the plot points)
point_size2-N:      0 pixel        (the inner diameter of the plot points)
legend_name-N:                     (the name of plot line N in the legend)
@end example

There are 2 kinds of attributes: @b{window attributes} that affect the entire
plot, and @b{line  attributes} that affect one of the plot lines
(which correspond to one row of @b{B}). For example, the background color
of the plot is a window attribute, while the line color is a line attribute.
Line attributes have a @b{-N} suffix, where @b{N} starts at @b{⎕IO}.

One way to specify
multiple attributes in one go is to use multi-line strings.

For example (plot 2 lines of 10 random numbers each, named "random row 1" and
"random row 2"):

@example
      Data ← ?2 10⍴10   ⍝ two rows of random data
      Attributes  ← """
legend_name-1: Random Row 1
legend_name-2: Random Row 2
                    """
      Attributes ⎕PLOT Data
@end example

While specifying the plot attributes A in one place is most convenient for
a single plot window or when the attributes are read from some external
test file, it is sometimes simpler to specify the plot attributes
individually. This can be done by using a structured APL variable as left
argument A:

@example
Attributes.legend_name_1 ← "Random Row 1"
Attributes.legend_name_2 ← "Random Row 2"
      Attributes ⎕PLOT Data
@end example

@b{PITFALL:} if @b{A} is specified as a structured APL Value then:
@itemize
@item attribute names must follow the GNU APL member name syntax, and therefore
@item the (line-) attribute name is separated from its line number by an
      @b{_} (underscore), and
@item string values must be quoted.
@end itemize

@itemize
@item the (line-) attribute name is separated from its line number by a
      @b{-} (minus), and
@item string values must not be quoted.
@end itemize

For example:
@example
      DATA ← 1○.1×⍳100
      ⍝ structured variable A1
      ⍝
      A1.point_style_1 ← 2             ⍝ separator: _ (underscore)
      A1.gui_driver    ← 'ASCII'       ⍝ string value quoted
      A1 ⎕PLOT DATA

   
      ⍝ multi-line string A2
      A2←«««
      point_style-1:    2             ⍝ separator: - (minus)
      gui_driver:       ASCII         ⍝ string value not quited
         »»»
      A2 ⎕PLOT DATA

      A2 ⎕PLOT 1○.1×⍳100
A ⎕PLOT 1○.1×⍳100
@end example

If the global attribute @b{output_filename} is given, then the plot is
not only displayed in a window on the screen, but also written to a file
with the name of the file provided as the value of the attribute.
@xref{Eigenvector} below.

For example:

@example
      ⊣ ( ⊂ "output_filename: /tmp/bitmap.png" ) ⎕PLOT 1 3 1 4 2
@end example

writes the plot of a a zigzag curve to file @b{/tmp/bitmap.png}.

@b{Caution}: ⎕PLOT with attribute @b{output_filename} overwrites existing
files without any warning). Errors, most likely lack of permissions to write
the output file, raises a DOMAIN ERROR with )MORE possibly providing details
as to why the file could not be written.

Some line attributes such as @b{line_width} or @b{point_size} are
frequently the same for several or even all plot lines. In that case one can
use the line attribute without a -N or _N suffix which is then used as the
default for all lines (unless overridden by the same attribute with -N or
_N suffix.

For example:

@example
Attributes.line_width   ← 2   ⍝ all plot lines:     2 pixels thick,
Attributes.line_width_2 ← 4   ⍝ except plot line 2: 4 pixels thick
@end example

As of GNU APL 1.9, ⎕PLOT legends may be dragged to other locations within a
plot window using the usual click and drag GUI technique.

@c @node Tick formatting (GTK only)
@subsubsection Tick formatting (GTK only)

By default, the ticks on the X, Y, and Z axes show the naked value
of their corresponding coordinates. Sometimes you may want them to look a
little different, e.g. by adding units to the numbers. This can be achieved
with window properties @b{format_X/Y/Z}, possibly accompanied by property
@b{gridX_variable}.

The properties @b{format_X/Y/Z} may be set to a format string that allows
the value that is being displayed at the ticks. The format is a string that
is a mix of static text and @b{tick items}. A @b{tick item} is computed
from the displayed value or from its position on the axis. There are
two different types of tick items: static and dynamic.

A @b{static tick item} starts with @b{%s}, followed by one or more texts,
separated by %. The texts are displayed in the same order starting from
the origin and proceeding along the axis. For example:

ATT.format_X ← '%sT1%T2%T3'

will name X-axis ticks @b{T1}, @b{T2}, and @b{T3}, regardless of the X
coordinate. @b{static tick items} are the fallbacks for those cases where
@b{dynamic tick items} cannot produce the desired result. In other
words, @b{static tick items} must be constructed in APL beforehand.

In contrast, @b{dynamic tick items} are computed from the value at the
axis, which is often simpler than computing them beforehand in APL. A
format string may contain 0 or more tick items which are used in a printf()
like manner.

Most of the @b{dynamic tick items} (see @b{⎕PLOT ⍬} for a complete list)
are related to points in time (years, quarters, months, days, hours, minutes,
and seconds). For them the X coordinate (time) is, according to the *nix time
conventions, seconds since midnight 1970. Since this number of seconds is not
easy to compute in APL, @b{⎕FIO.secs_epoch} comes to rescue:

@example
Q1  ← ⎕FIO.secs_epoch 2023 2 15   ⍝ Feb. 15 (middle of Q1) 00:00:00
@end example

An example for plotting the quarterly values @b{1 3 4 2} is:

@example
      )CLEAR

SPQ ← 91×24×60×60                 ⍝ seconds per quarter
Q1  ← ⎕FIO.secs_epoch 2023 2 15   ⍝ Feb. 15 (middle of Q1) 00:00:00

X ← Q1 + SPQ×0 1 2 3              ⍝ middles of Q1, Q2, Q3, and Q4
Y ← 1 3 2 4                       ⍝ values for Q1, Q2, Q3, and Q4

ATT.format_X ← "Q-%Q/%y"          ⍝ X-axis ticks: quarter and year

ATT ⎕PLOT X + 0J1×Y
@end example

Sometimes it is more convenient to display the exact X coordinates of the
plot points along the X axis instead of drawing an X-grid and visually
guessing the X-values between the vertical grid lines. This can be achieved
with setting @b{gridX_variable} to 1. Usually the X-coordinates are evenly
distributed along their axis while the Y and Z-coordinates are not. For this
reason there is no @b{gridY_variable} or @b{gridZ_variable}.


@c ............................................................................
@node Plot Window Handling
@subsection Plot Window Handling

Each call of ⎕PLOT with some plot data @b{B} opens a separate window in the X
server (which implies that ⎕PLOT requires X). That window runs in its own
thread until it is closed. A plot window can be closed in different ways:

@itemize
@item interactively by clicking on the close button (often located near the
      top-right corner) of the plot window,
@item interactively by opening the drop down menu (if any) of the window
      (near the top left-of the window) and selecting the menu option named
      @b{Close},
@item programmatically from APL by calling @b{⎕PLOT ¯3} (which closes @b{all}
     open plot windows), or
@item programmatically from APL by calling @b{⎕PLOT H} (which closes a
      specific plot window), where H is the plot window handle returned by
      monadic @b{⎕PLOT B}, or by dyadic @b{A ⎕PLOT B} with plot attributes
      @b{A} and plot data @b{B}.
@end itemize

Note that the exact method for closing a plot window interactively with the
mouse or the keyboard (as opposed to programmatically with @b{⎕PLOT ¯3} or
@b{⎕PLOT H} from APL) differs between different window managers. The above
corresponds to the typical method(s) for closing windows on most systems.

@c ............................................................................
@node Output File Format
@subsection Output File Format

Early versions of ⎕PLOT were using @b{libxcb} to display the results of
⎕PLOT while recent versions use @b{libgtk-3}. The primary driver for this
change was the lack of full Unicode support in @b{libxcb} which made it,
for example, difficult to properly display APL characters in window captions
or in plot line legends.

Writing the pixels of a plot to a file uses functions provided by either 
@b{libxcb} or @b{libgtk-3}, and these functions generate different
file formats.

@b{libxcb} is the older of the two libraries and it therefore produces the
older file format @b{.bmp} (bitmaps). @b{libgtk-3} is the newer library and
it produces the somewhat newer file format @b{.png} (portable network
graphics). There are tools like @b{convert} from package @b{imagemagick}
that can be used to convert different image file formats into each other
and therefore this incompatibility should not cause too much trouble.

For backward compatibility, the @b{libxcb} option for ⎕PLOT will continue
to exist for some time, but bug fixes and new features will only be made for
the @b{libgtk-3} option.


@c ............................................................................
@node GUI driver
@subsection GUI driver

@b{⎕PLOT} provides 3 different ways to generate its output, called
 @i{GUI drivers}. Two of these drivers (@b{GTK and XCB}) are optional,
while third third driver @b{ASCII} is not.  Optional means that the
@b{./configure} script, which is executed as the first step in the
compilation of GNU APL, determines if the prerequisite for an optional
driver are satisfied. IOW an optional driver is only available for @b{⎕PLOT}
if the libraries and C/C++ header files needed to compile the driver are
installed at the time where GNU APL is being compiled. The @b{ASCII} driver
merely generates ESC squences for VT100 terminals and does therefore not
depend on additional libraries.

For the monadic @b{⎕PLOT B}, the driver used by @b{⎕PLOT} is chosen as follows:

@itemize
@item GTK if available, otherwise
@item XCB if available, otherwise
@item ASCII
@end itemize

For the dyadic @b{A ⎕PLOT B}, the driver used is the same, but the choice made
by @b{./configure} can be overridden by the plot window attribute
@b{gui_driver}. The default for this plot window attribute is empty, which
means that the decision of @b{./configure} shall be used (like in the monadic
case where no plot attributes can be specified).

The plot driver also affects the type of result returned by @b{⎕PLOT}:

@itemize
@item drivers @b{GTK} and @b{XCB} return an integer scalar @b{H} (a plot 
      window handle) that can later be used from APL to programmatically
      close the plot window (i.e. with @b{⎕PLOT H}) with that handle.
      (Alternatively a GTK or XCB plot window can be closed by clicking
      the close button of the plot window).

@item driver @b{ASCII} returns a 3-dimensional APL value @b{Z}, where:
    @itemize
    @item @b{Z[1;;]} is a character matrix with the Unicode characters used
          to draw the plot window,
    @item @b{Z[2;;]} is an integer matrix with the the forground colors
          (in terms of @b{VT100} terminal colors), and
    @item @b{Z[3;;]} is an integer matrix with the the backround colors
          (again in terms of @b{VT100} terminal colors).
    @end itemize
@end itemize

The @b{VT100}  colors returnd by the ASCII driver are essentially 4-bit RGB
values with the following integer encoding:

@itemize
@item @b{0 = 0b000}: (dark) black,
@item @b{1 = 0b001}: (dark) red,
@item @b{2 = 0b010}: (dark) green,
@item @b{3 = 0b011}: (dark) yellow,
@item @b{4 = 0b100}: (dark) blue,
@item @b{5 = 0b101}: (dark) magenta,
@item @b{6 = 0b110}: (dark) cyan,
@item @b{7 = 0b111}: (dark) white (kind of grey),
@ifhtml
@verbatim
@end verbatim
@end ifhtml
@item @b{60 = 60 + 0b000}: bright black (another kind of grey),
@item @b{61 = 60 + 0b001}: bright red,
@item @b{62 = 60 + 0b010}: bright green,
@item @b{63 = 60 + 0b011}: bright yellow,
@item @b{64 = 60 + 0b100}: bright blue,
@item @b{65 = 60 + 0b101}: bright magenta,
@item @b{66 = 60 + 0b110}: bright cyan, and
@item @b{67 = 60 + 0b111}: bright white
@end itemize

The @b{ASCII} driver returns its result as a committed value (so that the
plot window is not drawn twice if the value is not assigned to a variable).
Naturally the @b{ASCII} driver can not support most of the @b{⎕PLOT} atrributes
like a true graphical driver like @b{GTK} or @b{XCB}. But at least the color
attributes are honored by rounding 3×8-bit RGB values to their nearest 4-bit
@b{VT100} color.

Some terminal emulators have issues when characters are printed on the last
column of the terminal. This is addressed by a user preference named
@b{plot-ASCII-columns}, which should be set to one less the number of columns
of the terminal (so that the last columns in not written).

Another preferencefor the ASCII driver that might be of interest is
@b{plot-ASCII-background}. It sets the background color for the terminal
if the Plot attribute canvas-color is not provided (e.g. in monadic ⎕PLOT
which has no plot attributes).

The @b{XCB} driver was the first driver for @b{⎕PLOT} but has since become
somewhat obsolete. It has limitation when it comes to the display of texts
(in particular for APL characters). The long-term GNU APL design strategy
is therefore @b{GTK} in favor of the good old @b{XCB}. @b{XCB} will still
be supported for a while, but new features will only be added to the
@b{GTK} driver.

@c ----------------------------------------------------------------------------
@node ⎕PNG - Portable Network Graphics
@section ⎕PNG - Portable Network Graphics

Portable Network Graphics is a file format for images, defined in
@strong{RFC 2083}. To quote the RFC:

@cite{The PNG format provides a portable, legally unencumbered,
well-compressed, well-specified standard for lossless bitmapped image files.}

@c ............................................................................
@node The GNU APL Color Model
@subsection The GNU APL Color Model

An image is a rectangular matrix of @strong{Pixels} which makes APL the
perfect language for manipulating images. In GNU APL (and also in the PNG
standard) a single pixel may have one, two, three, or four color components,
or short colors, as follows:

@itemize
@item a pixel with @strong{one} color defines an uncolored (grayscale)
pixel between black and white,
@item a pixel with @strong{two} colors defines a grayscale pixel between
black and white, and its opacity (see below),
@item a pixel with @strong{three} colors defines a colored pixel whose colors
are red, green and blue (or RGB for short), and
@item a pixel with @strong{four} colors defines an RGB pixel and its opacity
(RGBA for short).
@end itemize

The opacity (sometimes called the "alpha channel" and abbreviated A in the
RGBA format) defines how transparent a pixel is; lower values cause the
background of a pixel to shine through while larger values make the pixel
more opaque. One important use of the alpha channel is images that shall
not appear rectangular (even though their underlying color matrix is).
For example, to display a red circle on top of something else, one can set
the opacity of the pixels inside the circle to their maximum (as to making
them fully opaque and completely hiding the background or some other image
below) and the pixels outside the circle to 0 (making them fully transparent).

Since every pixel of an image may have 1, 2, 3, or 4 color components, the
representation of an image in APL is a three dimensional array
@strong{Image} where:

@itemize
@item @strong{(COLORS HEIGHT WIDTH) ← ⍴ IMAGE},
@item @strong{COLORS} is a number that determines @strong{@emph{general
format}} of the image (and, by chance, the number of its color components):
  @itemize
  @item @strong{COLORS = 1}: a grayscale image,
  @item @strong{COLORS = 2}: a grayscale image with an alpha channel,
  @item @strong{COLORS = 3}: a colored RGB (red/green/blue) image, or
  @item @strong{COLORS = 4}: a colored RGB image with an alpha channel,
   aka. RGBA)
  @end itemize
@item @strong{Height} is the number of pixels of the image in the vertical
direction, and
@item @strong{Width} is the number of pixels of the image in the horizontal
direction.
@end itemize

Above is: @strong{WIDTH} the number of pixels in the horizontal direction,
@strong{HEIGHT} the number of the pixels in the vertical direction, and
@strong{COLORS} the number of color components as above. For simplicity we
can treat the opacity as yet another color although, strictly speaking, it
is not.

For an image I, the horizontal pixel position I[;;x] increases from left to
right and the vertical pixel position I[;y;] increases from top to bottom.

In addition to the 4 general image formats above, the PNG standard allows
each format to have different resolutions (of 1-bit, 2-bit, 4-bit, 8-bit,
and 16-bit, even though not all combinations of image format and resolution
are permitted. For example, the lower resolutions (of 1-bit, 2-bit, or 4-bit)
are only permitted for grayscale images without alpha channel, 16-bit is
allowed for all images except those with a color table, and so on.

In theory there are 20 different combinations of 4 general image formats and
5 resolutions of the color components. However, the PNG standard allows only
15 of them, in particular by permitting 1-bit, 2-bit, 4-bit resolutions only
for grayscale images without alpha channel. The GNU APL color model reduces
these 15 combinations further to the 4 general types with the following rules:

@itemize
@item On input, i.e. when reading a PNG image file and producing the APL
array described above, the different resolutions allowed by the PNG standard
are normalized to a single range of real numbers between 0.0 and 1.0 (this
is by chance also the range used in the GTK standard used in ⎕PNG and ⎕PLOT).
@item Also on input, images that use color tables (for compressing the
image to reduce the file size) are automatically expanded. The resulting
APL value is therefore independent of the use of color tables.
@item On output, i.e. when storing the APL array in a PNG file, the
resolution is 8-bit by default. 8-bit is the only resolution which is valid
for all image formats in the PNG standard.
@item Also on output, one may optionally specify a different resolution.
@end itemize

In GNU APL, integers and floating point numbers take the same space in
memory. The advantage of smaller PNG files with lower resolutions would
therefore disappear as soon as converted to APL. On the other hand, the
disadvantage (primarily different max. values for different resolutions)
would remain. In the GNU APL color model a purely red RGB pixel has the same
value @strong{I[;y[x] ← 1.0 0.0 0.0} regardless of resolution, which
simplifies matters considerably. In a PNG file that same red RGB pixel
would instead have either:

@itemize
@item the value $01 0 0 in a 1-bit RGB image, or
@item the value $03 0 0 in a 2-bit RGB image, or
@item the value $07 0 0 in a 4-bit RGB image, or
@item the value $FF 0 0 in a 8-bit RGB image, or
@item the value $FFFF 0 0 in a 16-bit RGB image.
@end itemize

It may be useful to note that the human eye can only distinguish around 120
different colors, i.e. about 7 bits (for all color planes together). For this
reason 16-bit color components are quite an overkill and the 8-bit default
therefore should be more than sufficient for most purposes.

@c ............................................................................
@node Monadic ⎕PNG
@subsection Monadic ⎕PNG

Monadic ⎕PNG, i.e. ⎕PNG B is used for 3 purposes:

@enumerate
@item Conversion of PNG files into the GNU APL color model,
@item Display of an APL array (in the GNU APL color model) in a window on
the screen (interactive use of PNG), and
@item Controlling of ⎕PNG itself.
@end enumerate

The desired purpose is dictated by the properties of @strong{B}:

@enumerate
@item Any APL text vector (i.e. a string) @strong{B} is taken as the name of
  a file that supposedly contains an image in the PNG format. ⎕PNG will
  read the file and return the pixels in it as a numeric APL array of rank 3.
  The array contains the color components of the image in the file as
  described above.
@item Any array @strong{B} with rank 3 must be numeric (with values from
  0.0 to 1.0) and is then assumed to be an array in the GNU APL color model
  (such as the result of the case above). This image is displayed in a new
  window on the screen; the window survives even after ⎕PNG has returned.
  The result returned by ⎕PNG is a window handle (a positive integer
  identifying the window) which may later be used (in the case described
  below) to close the window programmatically from APL. Alternatively the
  window can also be closed interactively by the user via its close button.
@item Finally, if @strong{B} is a numeric scalar it fine-tunes the ⎕PNG
behaviour:

@itemize
@item  ⎕PNG N with N ≥ 1 (as returned in the case above) closes the window 
       with handle N programmatically.

@item ⎕PNG 0 disables all debug printouts related to ⎕PNG.
@item ⎕PNG ¯1 turns on a debug printout related to ⎕PNG (X events).
@item ⎕PNG ¯2 turns on a debug printout related to ⎕PNG (image data).
@item ⎕PNG ¯3 closes all open ⎕PNG windows.
@item ⎕PNG ¯4 turns on a debug printout related to ⎕PNG (GTK drawing).
@item ⎕PNG ¯5 turns on a debug printout related to ⎕PNG (function calls).
@item ⎕PNG ¯6 returns a vector of all open ⎕PNG window handles.
@end itemize
@end enumerate

@c ............................................................................
@node Dyadic ⎕PNG
@subsection Dyadic ⎕PNG

Dyadic @strong{A ⎕PNG B} has only one purpose: to store an APL array in
the GNU APL color model into a PNG file.

@itemize
@item If @strong{A} is a text vector then it is taken as the name of a PNG
  file into which the image @strong{B} shall be written. In this case the
  resolution is 8 bits for each color component.
@item If @strong{A} is a 2-element vector, say @strong{(A1 A2) ← A},
  then the nested @strong{A1} is the name of a PNG file as before and 
  @strong{A2} is the resolution (1, 2, 4, 8, or 16) to be used for the
  output file. In this case the combination of the general image format
  (as implied by @strong{↑⍴B}) and resolution @strong{A2} must be a valid
  one in the PNG standard.
@end itemize

@c ............................................................................
@node APL Examples
@subsection APL Examples

@example
      ⍝ load PNG file image.png
      Image ← ⎕PNG 'image.png'

      ⍝ display the image
      Handle ← ⎕PNG Image

      ⍝ close the window that displays the image
      ⎕PNG Handle

      ⍝ display PNG file image.png without storing it in a variable
      ⎕PNG ⎕PNG 'image.png'

      ⍝ Simple color conversions...

      ⍝ convert a monochrome image into the equivalent RGB image
      RGB_Gray ← 3 ⌿ Mono

      ⍝ paint all pixels of a monochrome image red
      RGB_Red ← 1 0 0 ⍀ Mono

      ⍝ paint all pixels of a monochrome image green
      RGB_Green ← 0 1 0 ⍀ Mono

      ⍝ paint all pixels of a monochrome image blue
      RGB_Blue ← 0 0 1 ⍀ Mono
@end example


@c ----------------------------------------------------------------------------
@node ⎕PS - Print Style
@section ⎕PS - Print Style

⎕PS is an integer vector that currently (read: as of SVN 982) contains
two integers which control some details of how APL values are printed.
The default value of ⎕PS is 0 0. ⎕PS is a session variable which survives
the @strong{)LOAD}ing of workspaces.

⎕PS[1] = 0: print rational quotients as normal floating point numbers
(digits, fractional point, possibly an exponent).

⎕PS[1] = 1: print rational quotients as Numerator÷Denominator.

Note: ⎕PS[1] has no effect if rational numbers were not ./configure'd.

⎕PS[2] = 0: no "boxing" of APL values

⎕PS[2] > 0: "boxing" of APL values according to ⎕PS[2].

Setting ⎕PS[2] has the same effect as the debug command ]BOXING and uses
the same values.

Note: For compatibility with older workspaces, assigning a single value to
⎕PS assigns that value to ⎕PS[2] and sets ⎕PS[1] to 0.


@c ----------------------------------------------------------------------------
@node ⎕RE - Regular Expressions
@section ⎕RE - Regular Expressions

⎕RE is a function that provides access to a subset of libpcre2, which is a
powerful regular expression matching library. "pcre" is an acronym for "Perl
compatible regular expressions". libpcre2 is Copyright (c) 1997-2017
University of Cambridge, England.

@c ............................................................................
@node Preconditions
@subsection Preconditions

libpcre2 is NOT shipped with GNU APL but must be installed before the GNU APL
./configure script is executed. The ./configure script tries to locate
libpcre2 and enables ⎕RE if the library (and its header files) are found.

If ./configure does not find libpcre2, then ⎕RE will raise a SYNTAX ERROR when
⎕RE is used in an APL program.

@c ............................................................................
@node ⎕RE Syntax
@subsection ⎕RE Syntax

⎕RE has two forms. The short form (without an axis argument) is intended to
cover the most likely use cases of ⎕RE, while the long form gives more control
over the format of the result produced by ⎕RE:

@example
Z ← A ⎕RE B      (short form, no axis)
Z ← A ⎕RE[X] B   (long form, with axis X)
@end example

The short form is equivalent to the long form with an empty axis:

@example
    A ⎕RE B   ←→   A ⎕RE[''] B
@end example

It therefore suffices to explain the long form. The long form has 3 different
arguments A, B, and X.

@c ............................................................................
@node The Regular Expression A
@subsection The Regular Expression A

A is a simple character vector which is expected to be a regular expression
as defined for the libpcre2 library. ⎕RE passes its left argument A unmodified
to libpcre2. The format of A ranges from very simple REs to very complex REs.
In fact, regular expressions are a language of their own, and it is unnecessary
to describe them here. The user of ⎕RE is referred to the description that
comes with libpcre2.

See in particular: @b{man pcre2pattern}

@c ............................................................................
@node The String(s) B Being Matched
@subsection The String(s) B Being Matched

B is either:

@enumerate
@item a simple character vector (which is then also passed unmodified
to libpcre2), or

@item a (nested) APL array consisting of simple character vectors.
@end enumerate

In the second case above, the result of ⎕RE is conceptually the same as if
the EACH operator had been applied to ⎕RE:

@example
    A ⎕RE B   ←→   A ⎕RE ¨ B
@end example

The libpcre2 first compiles A into an internal representation that improves
the speed of matching B. As a consequence, if several strings shall be matched
with the same regular expression A, then it is (sometimes considerably) faster
to A ⎕RE B once with several items of a nested B, than to A ⎕RE B multiple
times with the same A but different Bs. This is especially true when longer
text files with many lines will be matched.

If multiple (nested) strings B are used then A is compiled once and after that
the different ⊃B[i] in B are passed unmodified to libpcre2. In that case the
result Z is nested and (⍴B) ≡ (⍴Z←A ⎕RE[X] B). We can therefore limit the
description to the case where B is a simple character vector.

Any string B that is passed to libpcre2 is called "Subject" in the
libpcre2 documentation.

@c ............................................................................
@node The Flags X
@subsection The Flags X

By default, i.e. without an axis argument X or - equivalently - with an empty
string X←@b{""} as axis argument, ⎕RE returns the first match of the regular
expression A in the string B as an APL string.

By means of the optional axis argument @b{X ≢ ""} (called flags below) one may
override the default behavior to adapt various aspects of the matching to
satisfy the requirements that the user of ⎕RE may have.

The X argument of @b{A⎕RE[X] B} is a character vector where each character
is a flag. Each flag is either understood by (and then passed unmodified to)
libpcre2 or else understood by ⎕RE itself (and then not passed to libpcre2).

@node The libpcre2 flags: i m s and x
@subsubsection The libpcre2 flags: i, m, s, and x

The libpcre2 flags accepted by ⎕RE are: i, m, s, and x. They correspond to the
following libpcre2 flags:

@itemize
@item i: PCRE2_CASELESS   (perform case insensitive matching)
@item m: PCRE2_MULTILINE  (perform matching across multiple lines)
@item s: PCRE2_DOTALL     (dot (.) in A also matches CR/LF)
@item x: PCRE2_EXTENDED   (see 'man pcre2pattern')
@end itemize

Please consult the libpcre2 documentation for a more detailed description
of these flags.

@node The ⎕RE (APL output format) flags ↓ ⊂ and /
@subsubsection The ⎕RE (APL output format) flags ↓, ⊂, and /

⎕RE provides different output formats, some of which are tailored to different
APL functions, so that these other APL functions can use the result of ⎕RE
without major conversions. There are currently 4 different output formats:

@itemize
@item no ⎕RE flag: ⎕RE returns the matching sub-string(s) of B.

@item ↓ flag: ⎕RE returns the position(s) and the length(s) of the matched
sub-string of B, Like other left arguments of ↑ or ↓, the string position(s)
in the result start at 0 (i.e. not at ⎕IO) and can therefore be used directly
by dyadic ↓ and ↑. For instance to cut out the matched pieces.

@item ⊂ flag: ⎕RE returns an integer vector suitable as the left argument of 
partition (dyadic ⊂).

@item / flag: ⎕RE returns a Boolean vector suitable as the left argument of 
compress (dyadic /).
@end itemize

The following examples may better explain the nature of the output flags:

@example
      ⍝ return (first) matched string
      4 ⎕CR 'f..' ⎕RE[''] '__foo___fun____fox'
┏→━━┓
┃foo┃
┗━━━┛

      ⍝ return (first) pair (position, length)
      4 ⎕CR 'f..' ⎕RE['↓'] '__foo___fun____fox'
┏→━━┓
┃2 3┃
┗━━━┛

      ⍝ return left argument A of A ⊂ B
      4 ⎕CR 'f..' ⎕RE['⊂'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return left argument A of A / B
      4 ⎕CR 'f..' ⎕RE['/'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
@end example

In the absence of a 'g' flag (see below) the results of output formats
⊂ and / are identical; if 'g' is present, however, then they may differ.

@node the ⎕RE global flag g
@subsubsection the ⎕RE global flag g

The default behavior of ⎕RE is to return only the first match of A in B.
The 'g' flag causes ⎕RE to return all matches instead. For example:

@example
      ⍝ return first match as string
      4 ⎕CR 'f..' ⎕RE[''] '__foo___fun____fox'
┏→━━┓
┃foo┃
┗━━━┛

      ⍝ return all matches as strings
      4 ⎕CR 'f..' ⎕RE['g'] '__foo___fun____fox'
┏→━━━━━k━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━┓┃
┃┃foo┃ ┃fun┃ ┃fox┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━━━┛

      ⍝ return first match as pair (position, length)
      4 ⎕CR 'f..' ⎕RE['↓'] '__foo___fun____fox'
┏→━━┓
┃2 3┃
┗━━━┛

      ⍝ return all matches as pair (position, length)
      4 ⎕CR 'f..' ⎕RE['↓g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━━┓┃
┃┃2 3┃ ┃8 3┃ ┃15 3┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━┛

      ⍝ return first match as left argument of ⊂ (aka. partition)
      4 ⎕CR 'f..' ⎕RE['⊂'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return all matches as left argument of ⊂ (aka. partition)
      4 ⎕CR 'f..' ⎕RE['⊂g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 2 2 2 0 0 0 0 3 3 3┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return first match as left argument of / (aka. compress)
      4 ⎕CR 'f..' ⎕RE['/'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return all matches as left argument of / (aka. compress)
      4 ⎕CR 'f..' ⎕RE['/g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
@end example

Please note that the 'g' flag increments the depth of the result of the output
formats 'string' and 'position/length', but not of the result of output formats
'partition' and 'compress'.

The output formats ⊂ and / are related like this:

@example
      A ⌷RE['/'] B   ←→   0 ≠ A  A ⌷RE['⊂'] B
@end example


@node the ⎕RE error flag E
@subsubsection the ⎕RE error flag E

The default behavior of ⎕RE is to return an empty result if no match of
the regular expression A is found in B. It is sometimes more convenient to
raise an error if no match is found. This can be achieved with the E flag:

@example
       4 ⎕CR 'g..' ⎕RE[''] '__foo___fun____fox'
┏⊖┓
┃0┃
┗━┛
       'g..' ⎕RE['E'] '__foo___fun____fox'
DOMAIN ERROR+
      'g..' ⎕RE['E']'__foo___fun____fox'
      ^            ^
      )MORE
No match
@end example

The 'E' flag has no effect if the 'g' flag is also given because in that case
it is natural to check the length of the result:

@example
      4 ⎕CR 'g..' ⎕RE['Eg'] '__foo___fun____fox'
┏⊖┓
┃0┃
┗━┛
@end example

@c ............................................................................
@node Complex Matches
@subsection Complex Matches

The regular expressions considered so far were simple, which means that they
did not have (regular) sub-expressions. In other words, the regular expression
A did not contain non-literal (unquoted) parentheses. However, the libpcre2
allows sub-expressions and ⎕RE follows suit to a certain extent. If A contains
sub-expressions, the result of A ⎕RE B becomes deeper and the structure of the
result is determined by the structure of the parentheses in A.

The result of a general regular expression A is recursively constructed by ⎕RE
as follows:

1. If A contains no sub-expressions then the result is constructed as described
   in the previous sections.

2. If A contains sub-expressions, say A1, ... An, then the result is a (nested)
   vector with N+1 elements where the first element is the entire match (which
   contains the sub-matches) and the following elements are the next-level
   sub-matches.

Since the sub-matches are contained in their parent matches, it makes no sense
to handle sub-matches in the partition and compress output formats. For the
string and pos+len formats, the following examples may give some insight into
how the results look.

The regular expression A ← 'f(.)(.)' used in the example below is a top-level
regular expression which contains two 1-character sub-expressions (.).
The result is therefore the top-level match plus two 1-character sub-matches:

@example
      4 ⎕CR 'f(.)(.)' ⎕RE[''] '__foo___fun____fox'
┏→━━━━━━━━━━━━┓
┃┏→━━┓ ┏→┓ ┏→┓┃
┃┃foo┃ ┃o┃ ┃o┃┃
┃┗━━━┛ ┗━┛ ┗━┛┃
┗∊━━━━━━━━━━━━┛
@end example

Matching all instead of only the first then gives:

@example
      4 ⎕CR 'f(.)(.)' ⎕RE['g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━┓┃
┃┃┏→━━┓ ┏→┓ ┏→┓┃ ┃┏→━━┓ ┏→┓ ┏→┓┃ ┃┏→━━┓ ┏→┓ ┏→┓┃┃
┃┃┃foo┃ ┃o┃ ┃o┃┃ ┃┃fun┃ ┃u┃ ┃n┃┃ ┃┃fox┃ ┃o┃ ┃x┃┃┃
┃┃┗━━━┛ ┗━┛ ┗━┛┃ ┃┗━━━┛ ┗━┛ ┗━┛┃ ┃┗━━━┛ ┗━┛ ┗━┛┃┃
┃┗∊━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━┛┃
┗∊∊━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
@end example

The same for the pos+len output format:

@example
      4 ⎕CR 'f(.)(.)' ⎕RE['↓'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━┓
┃2 3 ┏→━━┓ ┏→━━┓┃
┃    ┃3 1┃ ┃4 1┃┃
┃    ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━┛
      4 ⎕CR 'f(.)(.)' ⎕RE['↓g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━━━━━━┓┃
┃┃2 3 ┏→━━┓ ┏→━━┓┃ ┃8 3 ┏→━━┓ ┏→━━━┓┃ ┃15 3 ┏→━━━┓ ┏→━━━┓┃┃
┃┃    ┃3 1┃ ┃4 1┃┃ ┃    ┃9 1┃ ┃10 1┃┃ ┃     ┃16 1┃ ┃17 1┃┃┃
┃┃    ┗━━━┛ ┗━━━┛┃ ┃    ┗━━━┛ ┗━━━━┛┃ ┃     ┗━━━━┛ ┗━━━━┛┃┃
┃┗∊━━━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━━━━━━┛┃
┗∊∊━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
@end example


@c ----------------------------------------------------------------------------
@node ⎕RVAL - Random APL value
@section ⎕RVAL - Random APL value

The standard way to produce random numbers is the primitive function ?
aka. @b{Roll}. Roll returns a simple random array whose ravel elements are
positive integers chosen in a (pseudo-) random fashion. While this is sufficient
for many applications, one sometimes needs APL values whose randomness not
only affects the ravel elements themselves, but also other aspects like:

@itemize
@item the rank of the value,
@item the shape of the value,
@item the data type of the ravel elements, and
@item the depth (nesting) of the values.
@end itemize

Such values can be produced with system function ⎕RVAL. The main use case for
⎕RVAL is the production of test data for the interpreter, but ⎕RVAL might be
useful for other purposes as well.

@c ............................................................................
@node General
@subsection General

In general, monadic ⎕RVAL B, returns a single random APL value Z,
while dyadic A RVAL B is used to control the properties of
subsequently generated random APL values returned by monadic ⎕RVAL.

The typical use of ⎕RVAL is to first call A ⎕RVAL B zero or more times to
override individual default properties of the subsequently produced values,
followed by one or more calls of monadic ⎕RVAL B.

The default properties, i.e. the properties used unless they were overridden
by dyadic A ⎕RVAL B, are:

@example
Rank:  0
Shape: 1 1 1 1 1 1 1 1 (initially irrelevant since Rank = 0)
Type:  0 1 0 0 0       (only integer random values)
Max. Depth: 4          (initially irrelevant since type = 0 1 0 0 0)
@end example

With these default properties, every call of monadic ⎕RVAL B returns an
integer scalar.

@c ............................................................................
@node Dyadic ⎕RVAL
@subsection Dyadic ⎕RVAL

The left argument A of A ⎕RVAL B determines which aspect of the
subsequent random APL values shall be controlled. A is an integer scalar:

@itemize
@item
A=0 controls the state of the underlying random number generator,

@item
A=1 controls the ranks of the subsequent random APL values,

@item
A=2 controls the shapes of the subsequent random APL values,

@item
A=3 controls the data types in the ravels of the subsequent random APL values,
and

@item
A=4 controls a maximum depth of the subsequent random APL values.

@end itemize

@node Control of the underlying random number generator: 0 ⎕RVAL B
@subsubsection Control of the underlying random number generator: 0 ⎕RVAL B

Every random number generator has a state (similar to @b{⎕RL} for @b{'?'}).
While the primitive @b{?} keeps its state in @b{⎕RL}, i.e. in the workspace,
⎕RVAL has no associated system variable like i@b{⎕RL} which keeps its state.
Instead, the user needs to store the state of i@b{⎕RVAL} in a workspace
variable if the workspace shall be stored and continued at a later point.
This rarely happens, therefore no effort was spent on implementing another
system variable. For the same reasonxi does, the state of @b{⎕RVAL}
survive @b{)CLEAR}.

The current state of @b{⎕RVAL}'s random number generator can be retrieved
with:

@example
      STATE ← 0 ⎕RVAL ''
@end example

The state changes with every call of monadic ⎕RVAL. The random number generator
can later be restored to the same state like this:

@example
      0 ⎕RVAL STATE
@end example

Restoring the state is needed when the same sequence of random values shall
be generated again. The random number generators of ? and ⎕RVAL are
independent; generating a random number or random value with one does not
change the state of the other.

@node Control of the ranks: 1 ⎕RVAL B
@subsubsection Control of the ranks: 1 ⎕RVAL B

The rank of subsequently generated random values can be controlled with
1 ⎕RVAL B as follows:

@itemize
@item
If B is a non-negative scalar with 0 ≤ B < MAX_RANK then the subsequently
generated random values will have the fixed rank B. MAX_RANK is a 
./configurable constant with a default of 8.

@item
If B is a negative scalar with 0 > B > -MAX_RANK then the ranks of the
subsequently generated random values will be chosen at random between
0 and -B (inclusive). All possible ranks 0 ... B will have the same
probability P = 100÷1-B %.

@item
Otherwise B must be a vector of positive integers. The integers are the
relative frequencies of the ranks 0, 1, ... MAX_RANK respectively. Too many
integers (i.e. MAX_RANK > ⍴B) will raise a LENGTH ERROR while fewer (than
MAX_RANK) are set to 0.

@end itemize

1 ⎕RVAL always returns its previous setting (i.e. before applying B).
An empty B, e.g. 1 ⎕RVAL ⍬ will not change anything and can be used to
query the current setting for 1 ⎕RVAL.

Examples:

@itemize
@item
1 ⎕RVAL 2 will produce only matrices,

@item
1 ⎕RVAL ¯2 will produce scalars, vectors or matrices, each with a
probability of 100÷3 %, and

@item
1 ⎕RVAL 50 30 20 will produce scalars with probability 50%, vectors
with probability 30%, and matrices with probability 20%. The numbers
need not add up to 100, but will be scaled accordingly.
@end itemize

@node Control of the shapes: 2 ⎕RVAL B
@subsubsection Control of the shapes: 2 ⎕RVAL B

The shapes of subsequently generated random values can be controlled with
2 ⎕RVAL B as follows.

B must be a vector of positive or negative integers. Vectors longer than
MAX_RANK + 1 will give a LENGTH ERROR while vectors shorter than MAX_RANK + 1
will be filled with 1 until the length of B becomes MAX_RANK + 1.

If an element B@sub{i} of the resulting shape B = B@sub{1} ... B@sub{MAXRANK}
is ≥ 0 then the corresponding axis i will have a fixed length of B@sub{i}.
If  B@sub{i} < 0 then the length of that axis will be chosen at random with
a probability of 1÷B@sub{i}. Note that the same B can still produce different
shapes even if all  B@sub{i} are ≥ 0 (i.e. if the rank is also chosen at
random).

2 ⎕RVAL always returns its previous setting (i.e. before applying B).
An empty B, e.g. 2 ⎕RVAL ⍬, will not change anything and can be used to
query the current setting for 2 ⎕RVAL.

Example:

@example
1 ⎕RVAL 2        ⍝ produce matrices
2 ⎕RVAL 3 ¯10    ⍝ the first axis of every matrix will have length 3, and
                 ⍝ the last axis of every matrix will vary between 0 and 10
@end example

This will produce scalars, vectors with a fixed length of 3 and matrices
with 3 rows and between 0 and 10 columns.

@node Control of the data types: 3 ⎕RVAL B
@subsubsection Control of the data types: 3 ⎕RVAL B

The types (i.e. CHARACTER, INTEGER, REAL, COMPLEX, or NESTED) of subsequently
generated random values can be controlled with 3 ⎕RVAL B as follows.

B must be a vector with up to 5 positive numbers, corresponding to the types.
Shorter vectors are filled with 0 (the corresponding types will not occur) and
longer vectors will raise a LENGTH error.

Once the type was chosen, the ravel item will be:

@itemize
@item
for type CHARACTER: a random character with ⎕UCS between 0 and $3000
(exclusive). Larger Unicodes typically cause display problems. If you need
larger Unicodes, use ⎕UCS with random integers.

@item
for type INTEGER: a random integer with ⎕UCS between -$80000000 and
$7FFFFFFF.

@item
for type REAL: a floating point number between 0.0 and 1.0 (exclusive)

@item
for type COMPLEX: a complex point number with real and imaginaty part
between 0.0 and 1.0 (exclusive), and

@item
for type NESTED: a nested APL value with the same properties (rank, shape,
and type) as their parent.
@end itemize

3 ⎕RVAL always returns its previous setting (i.e. before applying B).
An empty B, e.g. 3 ⎕RVAL ⍬ will not change anything and can be used to
query the current setting for 3 ⎕RVAL.

Example:

@example
      1 ⎕RVAL 2         ⍝ produce matrices
      2 ⎕RVAL 3 3       ⍝ produce 3×3 matrices
      3 ⎕RVAL 0 50 50   ⍝ produce a mix of integer and real values
      ⎕RVAL 0
¯975954163190248487                   0.5892132425
7677327503669503253 2152001972871424768
@end example

@node Control of a maximum depth: 4 ⎕RVAL B
@subsubsection Control of a maximum depth: 4 ⎕RVAL B

If the probability of generating a nested value is > 0 then the depth of
random values can become arbitrarily large and could therefore exceed the
system limit for the depth of the random value being produced. This can be
prevented by setting a maximum depth for every value generated. ⎕RVAL sets
the probability of the deepest allowed depth to 0.

4 ⎕RVAL B expects B to be a positive integer which is deepest allowed depth;

An empty B, e.g. 4 ⎕RVAL ⍬, will not change anything and can be used to
query the current setting for 4 ⎕RVAL.

@node monadic ⎕RVAL B
@subsubsection monadic ⎕RVAL B

Calling ⎕RVAL monadically either produces one new random APL value or displays
a list of arguments for dyadic @b{⎕RVAL}:

@itemize
@item @b{⎕RVAL ⍬} shows the valid left arguments of @b{A ⎕RVAL B} with a brief
  description,
@item @b{⎕RVAL ''} shows the valid syntaces of @b{A ⎕RVAL B}
  (see also @xref{Function Groups}.
@item @b{⎕RVAL 0} returns the next random value according to the properties set with dyadic @b{⎕RVAL}.
@end itemize

The properties of that next value can be controlled by the dyadic calls
of ⎕RVAL as described above. If no prior dyadic call of ⎕RVAL was made then
the properties have the following defaults:

@example
      1 ⎕RVAL ⍬    ⍝ rank (scalar)
0

      2 ⎕RVAL ⍬    ⍝ shape (not used since rank = 0)
1 1 1 1 1 1 1 1

      3 ⎕RVAL ⍬    ⍝ types (integer)
0 1 0 0 0

      4 ⎕RVAL ⍬    ⍝ max. depth (not used since probability of NESTED = 0)
4
@end example

That is, by default, ⎕RVAL ⍬ produces random integer scalars.

It is sometimes convenient to be able to change the properties of the next
random value without modifying (and later restoring) the properties with
dyadic ⎕RVAL. This can be achieved with monadic ⎕RVAL with non-empty B.

Let 4=⍴B. Then

@example
   Z←⎕RVAL B does essentially:

   B[1]←1 ⎕RVAL B[1]
   B[2]←2 ⎕RVAL B[2]
   B[3]←3 ⎕RVAL B[3]
   B[4]←4 ⎕RVAL B[4]

   Z←⎕RVAL 0

   B[1]←1 ⎕RVAL B[1]
   B[2]←2 ⎕RVAL B[2]
   B[3]←3 ⎕RVAL B[3]
   B[4]←4 ⎕RVAL B[4]
@end example

If 4 > ⍴B then the missing properties are not changed.

Examples:

@example
⎕RVAL 1 (,¯4) (0 1)    ⍝ return a random 0-4 element integer vector
⎕RVAL 2 (2 2) (1 0)    ⍝ return a random 2×2 character array
@end example

@node Syntax Alternatives
@subsubsection Syntax Alternatives

For an improved readability, ⎕RVAL also accepts string arguments for its left
argument A and also the sub-function notation:

@example
      0 ⎕RVAL B  ←→  'state' ⎕RVAL B  ←→  ⎕RVAL.state B
      1 ⎕RVAL B  ←→   'rank' ⎕RVAL B  ←→  ⎕RVAL.rank  B
      2 ⎕RVAL B  ←→  'shape' ⎕RVAL B  ←→  ⎕RVAL.shape B
      3 ⎕RVAL B  ←→   'type' ⎕RVAL B  ←→  ⎕RVAL.type  B
      4 ⎕RVAL B  ←→  'depth' ⎕RVAL B  ←→  ⎕RVAL.depth B
@end example

@c ----------------------------------------------------------------------------
@node ⎕SI - State Indicator
@section ⎕SI - State Indicator

⎕SI returns aspects of the current State Indicator, similar to the standard
command )SI. This can be used, for example, to create debug functions similar
to the assert() macro in C/C++:

@example
∇Assert B;COND;LOC;VAR
 →(1≡B)⍴0
 ' '
 COND←7↓,¯2 ⎕SI 4
 LOC←,¯2 ⎕SI 3
 '************************************************'
 ' '
 '*** Assertion (', COND, ') failed at ',LOC
 ''

 ⍝ show stack
 ⍝
 ' '
 'Stack:'
 7 ⎕CR ⊃¯1↓⎕SI 3
 ' '
 '************************************************</pre>'
 →
∇
@end example

The right argument of ⎕SI specifies which aspect of the State Indicator
shall be returned:

@itemize

@item
⎕SI 1: The name of the context. That name is either:
@itemize
@item the name of a defined function, or
@item  ◊ for an immediate execution context, or
@item  ⍎ for an execute context.
@end itemize

@item
⎕SI 2: The line number (of a defined function) or 0 for immediate execution
and execute contexts.

@item
⎕SI 3: The function name and line number in square brackets, for example:
@b{"FOO[3]"}

@item
⎕SI 4: Either the statement text of the function line or the error text of
 an error that has occurred on the line

@item
⎕SI 5: The program counter (= token number counted from the start of the
function text)

@item
⎕SI 6: the parse mode of the context (immediate execution, execute, or
defined function).

@end itemize

If no left argument is provided then the result of ⎕SI is a vector with
one entry per State Indicator level (and hence ⍴⎕SI B is the depth of the SI
stack).

If the optional left argument A is provided then it specifies a particular
level of the SI instead of the entire SI. A should be an integer scalar.
If A is positive then the level is counted from the oldest entry to the latest,
while negative A counts from the latest to the oldest level.

For example, ¯1 ⎕SI refers to the currently executing context, ¯2 ⎕SI is
the caller, and so on.


@c ----------------------------------------------------------------------------
@node ⎕SQL - SQL Database Interface
@section ⎕SQL - SQL Database Interface

⎕SQL was kindly contributed by Elias Mårtenson.

As of GNU APL 1.6, the native function SQL has been replaced by the
system function ⎕SQL, described below. ⎕SQL has an axis argument that
selects a sub-function of ⎕SQL.

@c ............................................................................
@node ⎕SQL[0] ⍬ : display sub-function numbers
@subsection ⎕SQL[0] ⍬ : display sub-function numbers

The system function ⎕SQL is divided into a number of sub-functions. The
desired sub-function of ⎕SQL is selected by means of a numeric axis argument:

@example
    Valid (sub-)function numbers for ⎕SQL:

    Legend: Fs - database file name (path)
            Ty - database type ('sqlite' or 'postgres')
            Db - database handle (small integer)
            Qs - SQL query string
            Pv - query parameters (APL values, to be bound to Qs)
            Ts - name of a table in the database (string)
            Vi - DB provider (library) version (integer)
            Vs - DB provider (library) version (string)

            ⎕SQL[0] ''        list the ⎕SQL (sub-)function names
            ⎕SQL[0] ⍬         list the ⎕SQL (sub-)function numbers
    Db ← Ty ⎕SQL[1] Fs        open database Fs & return a handle for it
            ⎕SQL[2] Db        close database handle Db
         Qs ⎕SQL[3, Db] Pv    perform SQL query Qs
         Qs ⎕SQL[4, Db] Pv    perform SQL update Qs
            ⎕SQL[5] Db        begin a transaction
            ⎕SQL[6] Db        commit the current transaction
            ⎕SQL[7] Db        roll the current transaction back
            ⎕SQL[8] Db        list the tables in database Db
         Db ⎕SQL[9] Ts        list the column names and types of table Tn
    Vi ←    ⎕SQL[10] Ty       the provider version number for type Ty
    Vs ←    ⎕SQL[11] Ty       the provider version string for type Ty
@end example

The axis argument [0] is optional; one can also use @b{⎕SQL ⍬}.

@node ⎕SQL[0] @code{''} : display sub-function names
@subsection ⎕SQL[0] @code{''} : display sub-function names

As of @b{SVN version 1901}, ⎕SQL has been promoted to be a function group.
Instead of the function nunbers above, one can therefore also use function
names with the following syntax:

@example
    With a small performance penalty, ⎕SQL also accepts the following
    sub-function names instead of sub-function numbers as axis argument:

    Legend: Fs - database file name (path)
            Ty - database type ('sqlite' or 'postgres')
            Db - database handle (small integer)
            Qs - SQL query string
            Pv - query parameters (APL values, to be bound to Qs)
            Ts - name of a table in the database (string)
            Vi - DB provider (library) version (integer)
            Vs - DB provider (library) version (string)

       ⎕SQL[5]         ←→  ⎕SQL.begin         ⍝ begin a transaction
       ⎕SQL[9]         ←→  ⎕SQL.columns       ⍝ show the columns of a table
       ⎕SQL[6]         ←→  ⎕SQL.commit        ⍝ end a transaction
       ⎕SQL[2]         ←→  ⎕SQL.close         ⍝ close a database handle
       ⎕SQL[0]         ←→  ⎕SQL.list ''       ⍝ ⎕SQL function names/numbers
       ⎕SQL[1] Fs      ←→  ⎕SQL.open          ⍝ open the database file
       ⎕SQL[3, Db] Pv  ←→  ⎕SQL.query Db Pv   ⍝ SQL database query
       ⎕SQL[7] Db      ←→  ⎕SQL.rollback      ⍝ roll a transaction back
       ⎕SQL[8] Db      ←→  ⎕SQL.tables        ⍝ show all tables
    Qs ⎕SQL[4, Db] Pv  ←→  ⎕SQL.update Db Pv  ⍝ SQL database update
       ⎕SQL[10]        ←→  ⎕SQL.version       ⍝ SQL provider version number
       ⎕SQL[11]        ←→  ⎕SQL.vstring       ⍝ SQL provider version string
@end example

@b{NOTE:} The mapping from function numbers to function names of a function
group does not allow for non-scalar axis arguments (such as @b{[3, Db]})
for @b{query} or @b{[4, Db]} for @b{update}. For that reason, the @b{Db}
argument of functions @b{query} and @b{update} had to be moved from the
second item in the axis argument to the first item of the right value argument,
and the right argument @b{Pv} (aka. query parameters) of the function axis
is now the (nested) second argument of the right argument in the named
function symtax.

Like for @b{⎕SQL[0] ⍬} above is the axis argument [0] optional; one can
therefore also use @b{⎕SQL ''} instead of @b{⎕SQL[0] ''}.

@c ............................................................................
@node ref ← A ⎕SQL[1] B : open database
@subsection ref ← A ⎕SQL[1] B : open database

@example
      type  ⎕SQL[1] file
@end example

Connect to database of type A using connection arguments B. The result is a
database handle @b{ref} to be used in the other ⎕SQL functions.

@b{A} must be a string indicating the database type. Currently supported
values are 'postgresql' and 'sqlite'.

@b{B} is the connection parameters which depends on the type of
database:

  - For type≡'sqlite': the argument is string pointing to the
    database file.

  - For type≡'postgresql', the argument is a standard connect
    string as described in the PostgreSQL documentation.

This function returns a database handle that should be used when
using other SQL functions. This value should be seen as an opaque
handle. It is, however, guaranteed that the handle is a scalar
value.

@c ............................................................................
@node ⎕SQL[2] B : close database
@subsection ⎕SQL[2] B : close database

@example
      ⎕SQL[2] ref
@end example

Disconnect from database @b{ref}.

B is the database handle that should be disconnected. After this
function has been called, no further operations are to be performed
on this handle. Future calls to ⎕SQL[1] may reuse previously
disconnected handles.

@c ............................................................................
@node Z ← A ⎕SQL[3 ref] B : database query (with result)
@subsection Z ← A ⎕SQL[3, ref] B : database query (with result)

@example
      query ⎕SQL[3, ref] params
@end example

Execute a select statement and return the result table.

The axis parameter indicates the database handle.

A is a select statement to be executed. Positional parameters can
be supplied by specifying a question mark "?" in the statement.

B is an array containing the values for the positional parameters.
If the array is of rank 2, the statement will be executed multiple
times with each row being the values for each call.

The return value Z is a rank-2 array representing the result of the
select statement. Null values are returned as ⍬ and empty strings
are returned as ''.

@c ............................................................................
@node Z ← A ⎕SQL[4 ref] B : database query (w/o result)
@subsection Z ← A ⎕SQL[4, ref] B : database query (w/o result)

@example
      query ⎕SQL[4, ref] params
@end example

Execute an SQL statement that does not return a result.

This function is identical to SQL∆Select with the exception that it
is used on statements which do not return a result table.

@c ............................................................................
@node ⎕SQL[5] B : start transaction
@subsection ⎕SQL[5] B : start transaction

@example
      ⎕SQL[5] ref
@end example

Begin a transaction.

@c ............................................................................
@node ⎕SQL[6] B : commit transaction
@subsection ⎕SQL[6] B : commit transaction

@example
      ⎕SQL[6] ref
@end example

Commit a transaction.

@c ............................................................................
@node ⎕SQL[7] B : rollback transaction
@subsection ⎕SQL[7] B : rollback transaction

@example
      ⎕SQL[7] ref
@end example

Rolls back the current transaction.

@c ............................................................................
@node Z←⎕SQL[8] B : table names
@subsection Z←⎕SQL[8] B : table names

@example
      ⎕SQL[8] ref
@end example

Return an array containing the name of all tables.

@c ............................................................................
@node Z←⎕SQL[9] B : column names
@subsection Z←⎕SQL[9] B : column names

@example
      ref  ⎕SQL[9] table
@end example

Return an array containing information about the columns in the
given table. Currently, the column layout is as follows:

  Name
  Type

More columns containing additional information may be added in a future
release.


@c ............................................................................
@node Z ← ⎕SQL[10] B : library version number
@subsection Z ← ⎕SQL[10] B : library version number

Return the version number (integer scalar) of the library that was used
to implement the database type @b{B}. Depending on the SQL libraries
detected by ./configure, ⎕SQL may support no, one, or more than one
kind of SQL database type (the database type is sometimes refered to as
"SQL provider).

@c ............................................................................
@node Z ← ⎕SQL[11] B : library version string
@subsection Z ← ⎕SQL[11] B : library version string

Return the version string of the library that was used to implement the
database type @b{B}. Depending on the SQL libraries
detected by ./configure, ⎕SQL may support no, one, or more than one
kind of SQL database type (the database type is sometimes refered to as
"SQL provider).

@c ----------------------------------------------------------------------------
@node SQLite Quickstart
@subsection SQLite Quickstart

@b{SQL} is a powerful query language with many functions for manipulating
the data that is stored in the tables of databases. Similarly, APL is
a powerful language with many functions for manipulating the data stored
in matrices (the APL conterpart of SQL database tables). The overlap in
functionality between SQL and APL is considerable. That means that an APL
programmer always has the follwing options:

@itemize

@item Manipulate SQL database tables directly. This requires a detailed
      understanding of the SQL query lannguage.

@item Manipulate the data in APL. That means:
  @itemize
  @item Reading SQL database tables (or parts of them) into APL matrices,
  @item manipulate the matrices in APL, and
  @item write the result back into the SQL tables.
  @end itemize
  This approach requires only rather basic knowledge of the SQL query
  lannguage. It might have a small performance panalty, but that performance
  penalty will only be noticable if the databases involved are huge.
@item A mix of the options above, e.g if performance matters.
@end itemize

This SQLite Quickstart follows the second option. The basic SQL operations
explained in this Quickstart are:

@itemize
@item deleting a database,
@item creating a new database,
@item creating a database table,
@item writing an APL matrix into a database table,
@item reading a database table into an APL matrix,
@item updating a database table, and
@item deleting rows from a database table.
@end itemize

@node Deleting a Database
@subsubsection Deleting a Database

In order to have a well-defined starting point we delete any
already existing database. In SQLite every database is contained
in a single file, frequently named @b{db.sqlite}.

@b{CAUTION:} The @b{⎕FIO.unlink} command below will erase any existing
database without a warning.

@example
      )CLEAR
CLEAR WS

      ⍝ delete any existing database
      ⍝
      Filename ← "db.sqlite"                ⍝ name of the database to be deleted
      ⎕FIO.strerror ⎕FIO.unlink Filename
Success

      ⍝ verify that it was deleted
      ⍝
      ⎕FIO.strerror ⎕FIO.unlink Filename
No such file or directory
@end example

@b{NOTES:} An SQLite database can be closed in two ways: using @b{⎕FIO.unlink}
as shown above, or with an SQL query such as
@b{"DROP DATABASE IF EXISTS db.sqlite"}. However, to issue an SQL query we
would need to create an SQL connection to a database that we do not yet have
at this point.

For databases that are more complex than @b{SQLite}, an SQL query or a utility
program shipped with the database might be required. For example, postgreSQL
comes with a program named @b{dropdb} that deletes a database. Such programs
can be called from APL with the @b{)HOST} command or with @b{⎕FIO.fork_daemon}. 

@node Creating a new Database
@subsubsection Creating a new Database

To read from or write into a database we need a database handle (sometimes
called a database connection) for it:

@example
      )CLEAR
CLEAR WS

      ⍝ create a new (and empty) database
      ⍝
      Filename ← 'db.sqlite'            ⍝ name of the database to be created
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ create the database

      ⎕SQL.close DB                     ⍝ and close it
@end example

@b{NOTES:} If a file with the given @b{Filename} already exists, then
@b{⎕SQL.open} simply connects to it and returns a handle (a small integer)
for the connection. The handle is used like a file descri[tor for an open
file in the operating system. The handle is later used for subsequent database
operations of ⎕SQL (until the connection is closed).

If no file with the given @b{Filename} exists, then @b{⎕SQL.open}
creates a file with than name and then connects to it. It is possible to have
several such connections to different databases at the same time. It is also
possible to have several connections to the same database at the same time
(and then one should use database transactions to avoid conflicts with the
data written to the same database via different connections).

@b{postgreSQL} is shipped with programs @b{createdb} for creating databases,
@b{createuser} for creating database users, and @b{psql} for various other
database administration tasks. These programs can be called from APL with
the @b{)HOST} command or with @b{⎕FIO.fork_daemon}.

@node Creating a Database Table
@subsubsection Creating a Database Table

An SQL database is a set of tables. The next step after a new database was
created and a connection to it was opened is to define one or more tables
in the database. Every table has one or more named columns and zero or more
rows. Every column has attributes, some of which are mandatory (e.g. the
column name and in some SQL dialects the data type) while others are optional
(e.g. the data type in other SQL dialects or the size of the items in the
column). Creating a database table means to specify a name for a table and
to specify the attributes of the table columns.

In this quickstart we create a table named @b{EMPLOYEES} with 5 columns named
@b{ID}, @b{NAME}, @b{ADDRESS}, @b{AGE}, and @b{SALARY}.

@example
      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      ⍝ define an SQL Query string Qs that creates a table
      ⍝
      Qs ← ,⊃«««
      CREATE TABLE EMPLOYEES(
             ID      INT  NOT NULL PRIMARY KEY,
             NAME    TEXT NOT NULL,
             AGE     INT  NOT NULL,
             ADDRESS CHAR(50),
             SALARY  REAL);
             »»»

      ⍝ perform the SQL query according to query string Qs.
      ⍝
      Qs ⎕SQL.query DB ⍬

      ⍝ show the tables in the database
      ⍝
      ⎕SQL.tables DB
 EMPLOYEES 

      DB ⎕SQL.columns 'EMPLOYEES'
 ID      INT      
 NAME    TEXT     
 AGE     INT      
 ADDRESS CHAR(50) 
 SALARY  REAL     

      ⎕SQL.close DB                     ⍝ and close DB
@end example

@b{NOTES:} As we see above, SQL query strings tend to be quite long. We could
have removed most of the blanks and all of the linefeeds in the query string
above, but readability of the SQL query would have suffered a lot by doing
that. Instead we used a multi-line GNU APL script ««« ... »»» for the query
text, disclose it (which turns the APL vector of strings into an APL matrix,
and finally ravel it (to make it a string).

Many SQL query strings contain strings themselves, e.g. for column names that
contain blanks or for string values in columns. IOW, the query string is a
string that contains other strings. To avoid the need to escape the SQL quotes
inside (quoted) APL strings we have quoted the multi-line query string with
««« ... »»» instead of """ ... """ or ''' ... '''.

The right argument of @b{⎕SQL.query} is @b{DB ⍬} (and not simply @b{DB} as
in other ⎕SQL functions). This is because an SQL query string can have
parameters as we will see below. The query string above has no parameters and
therefore the parameter list for the query string is @b{⍬} aka. @b{⍳0}.

Finally, SQL table columns have names, but SQL table rows do not. It is
therefore common practice (though not required) that the first column of a
table is a unique integer ID that identifies the row and can later be used
to quickly locate the row in a large database.

@node Excursus: Dynamic Query Strings
@subsubsection Excursus: Dynamic Query Strings

The SQL query strings @b{(Qs)} used so far were @b{static}, which means that
they are sent verbatim to the database machinery. In the following we will
also use SQL query strings that are @c{dynamic}. Such dynamic query string
are a mix of static text and @c{Placeholders}. In SQLite placeholders are
the character @b{%} in the query string. In other SQL dialects the placeholders
may be different. For example in @b{postgreSQL} the placeholders in a query
string are @b{$1}, @b{$2}, @b{$3}, and so forth.

The placeholders in a dynamic query string are replaced by
@c{query string parameters} before the resulting static query
string is to the database machinery. For example:

@example
      ⍝ dynamic query string Qs
      ⍝
      Qs ← ,⊃«««
      INSERT INTO EMPLOYEES(ID, NAME, AGE, ADDRESS, SALARY
               VALUES(?,  ?,    ?,   ?,       ?);
           »»»

      ⍝ query string parameters
      ⍝
      PARAMS ← 1 "Paul" 32 "California" 20000

      ⍝ number of placeholders in Qs
      ⍝
      +/Qs = '?'
5

      ⍝ number of parameters (shall be equal to the number of placeholders)
      ⍝
      ⍴PARAMS
5

      ⍝ the (intermediate) result of replacing the placeholders in the dynamic
      ⍝ query string Qs would then be this static query string:
      ⍝
      RESULT ← ,⊃«««
      INSERT INTO EMPLOYEES(ID, NAME, AGE, ADDRESS, SALARY
               VALUES(1, "Paul", 32, "California", 20000);
                 »»»
@end example

The example above shows the binding of query string parameters to query
strings for the case where the rank of the parameters is ≤ 1. @b{⎕SQL} also
supports parameters with rank 2; in that case the query is repeated for
every row of the parameter matrix. I.e. the placeholders in same query
string @b{Qs} are being replaced with every row of the parameter matrix
before the result of each replacement is sent as a (static) query to the
database machinery. This feature makes it rather simple to write an APL
matrix (= query parameter) to the database.

Dynamic query strings are allowed in two ⎕SQL sub-functions: in function 3
(⎕SQL.query) and in function 4 (⎕SQL.update). The syntax differs slightly
beween ⎕SQL function numbers and ⎕SQL function names:

@example
      Qs ⎕SQL.query DB PARAMS       ⍝ ⎕SQL sub-function name
      Qs ⎕SQL[3, DB] PARAMS         ⍝ ⎕SQL sub-function number

      Qs ⎕SQL.update DB PARAMS      ⍝ ⎕SQL sub-function name
      Qs ⎕SQL[4, DB] PARAMS         ⍝ ⎕SQL sub-function number
@end example

For rank 2 parameters shall the number of placeholders be equal to the
number of colums of the APL matrix. If a query string contains no placeholders
then the parameters shall be ⍬ (aka. ⍳0).

@node Writing an APL matrix into a Database Table
@subsubsection Writing an APL matrix into a Database Table

Now suppose that we have an APL matrix with the same columns as our
SQL table created above.

@example
      )CLEAR
CLEAR WS

      MATRIX ← 4 5⍴0
      MATRIX[1;] ← 1 'Paul'   32 'California' 20000.00
      MATRIX[2;] ← 2 'Allan'  25 'Texas'      15000.00
      MATRIX[3;] ← 3 'Teddy'  23 'Norway'     20000.00
      MATRIX[4;] ← 4 'Gunnar' 26 'Sweden'     18000.00

      ⍝ show MATRXIX
      ⍝
      8 ⎕CR MATRIX
┌→─────────────────────────────┐
↓┌→───┐   32 ┌→─────────┐ 20000│
││Paul│      │California│      │
│└────┘      └──────────┘      │
│┌→────┐  25 ┌→────┐      15000│
││Allan│     │Texas│           │
│└─────┘     └─────┘           │
│┌→────┐  23 ┌→─────┐     20000│
││Teddy│     │Norway│          │
│└─────┘     └──────┘          │
│┌→─────┐ 26 ┌→─────┐     18000│
││Gunnar│    │Sweden│          │
│└──────┘    └──────┘          │
└ϵ─────────────────────────────┘
@end example

This @b{MATRIX} can then be written (more precisely: appended) to the
database table @b{EMPLOYEES} like this:

@example
      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      Qs ← ,⊃«««
      INSERT INTO EMPLOYEES(ID, NAME, AGE, ADDRESS, SALARY)
                     VALUES(?,  ?,    ?,   ?,       ?);
           »»»
     Qs ⎕SQL.query DB MATRIX

      ⎕SQL.close DB                     ⍝ and close DB
@end example

@node Reading a Database Table into an APL Matrix
@subsubsection Reading a Database Table into an APL Matrix

Reading a matrix back from a database table uses a @b{SELECT} query. @b{SELECT}
is the counterpart of indexed reference or indexed assignment in APL and works,
like in APL, in both directions (i.e. reading from tables and writing to
tables). In a @b{SELECT} query one can select specific rows and columns of the
table, which is left as an exercise for the reader. In this quickstart we want
to keep things simple and read an entire table back:

Due to the absence of a binding mechanism is reading back of an SQL table
into an APL matrix even simpler:

@example
      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      Qs ← ,⊃«««
      SELECT * FROM EMPLOYEES;
             »»»

      RESULT ← Qs ⎕SQL.query DB ⍬
      RESULT
 1 Paul   32 California 20000 
 2 Allan  25 Texas      15000 
 3 Teddy  23 Norway     20000 
 4 Gunnar 26 Sweden     18000 

      ⍴RESULT
4 5

      ⎕SQL.close DB                     ⍝ and close DB
@end example

@b{NOTES:} The @b{*} in the query string stands for @b{all columns} of the
table. @b{*} is NOT a placeholder because it is not replaced with a parameter.
One can also retrieve a only subset of columns by using a comma separated list
of column names instead of @b{*} (similar to indexed reference in APL).


@node Updating a Database Table
@subsubsection Updating a Database Table

The ⎕SQL sub-function introduced so far suffice, at least in principle. The
APL programmer can read a table from the database into APL, analyse and/or
modify it in APL, and optionally write it back into the database.

There is, however, one case where this approach becomes arbitrarily
 inefficient: a large database with only a small change. In that case it
is far more efficient to make a change directly in the database, i.e. without
transferring the entire database back and forth between APL and SQL.
This is achieved with an SQL UPDATE query, and @b{⎕SQL.update} does
exactly that.

Suppose we want to increase Teddy's salary from 20000 to 22000. Then:

@example
      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      Qs ← ,⊃«««
      UPDATE EMPLOYEES SET SALARY=? WHERE ID=?;
             »»»

      Qs ⎕SQL.query DB (22000 3)        ⍝ update column SALARY in row 3

      ⍝ check result
      Qs ← ,⊃«««
      SELECT * FROM EMPLOYEES;
             »»»

      RESULT ← Qs ⎕SQL.query DB ⍬
      RESULT
 1 Paul   32 California 20000 
 2 Allan  25 Texas      15000 
 3 Teddy  23 Norway     22000 
 4 Gunnar 26 Sweden     18000 

      ⍴RESULT
4 5

      ⎕SQL.close DB                     ⍝ and close DB
@end example

@b{NOTES:}  an @b{UPDATE} query in SQL is pretty much what an indexed
assignment in APL. In the example above we changed a single column @b{SALARY}
in a single row @b{ID} in the database table @b{COMPANY}. The placeholder
@b{%} in @b{SET SALARY=%} was set to parameter 22000 and the place holder in 
@b{where ID=?} was set to Terry's row with @b{ID}. Like in APL one can,
however, also change multiple rows and/or multiple columns in a database table.
To change multiple columns one can set a comma separated list of column names
in the @b{SET} clause (or @b{*} for all columns). For multiple rows one can
set multiple rows in the @b{WHERE} clause. Omitting the @b{WHERE} clause
updates all rows (similar to @b{*} for columns) in the database.
For example:

@example
      UPDATE COMPANY SET AGE=?, SALARY=? WHERE ID IN (?, ?, ?);
@end example

The shape of a parameter matrix for an SQL @b{UPDATE} must, of course,
match the number of selected rows and columns.

@c ----------------------------------------------------------------------------

@node Deleting Rows From a Database Table
@subsection Deleting Rows From a Database Table

Like @b{UPDATE} queries are efficient for changing large tables of a database,
there is also an SQL query for deleting one or more rows of a database:

@example
      Qs←«DELETE FROM COMPANY WHERE ID=?;»                ⍝ delete one row
      Qs←«DELETE FROM COMPANY WHERE ID IN (?, ?, ?);»     ⍝ delete 3 rows
      Qs←«DELETE FROM COMPANY;»                           ⍝ delete all (!) rows

      Qs ⎕SQL.query DB B
@end example

As usual, the number of rows must match the number of placeholders in the
query parameter @b{B} (an APL vector).

@c ----------------------------------------------------------------------------
@node ⎕SYL - System limits
@section ⎕SYL - System limits

System variable ⎕SYL shows a number of system limits when referenced:

@example
      ⎕SYL
 SI depth limit          (0 = no limit)                    0 
 number of values limit  (0 = no limit)                    0 
 total ravel bytes limit (0 = no limit)                    0 
 current SI depth                                          1 
 current number of values                                 54 
 current total ravel bytes                              9072 
 max. rank for APL values                                  8 
 min. ⎕PW                                                 30 
 max. ⎕PW                                               1000 
 min. ⎕PP                                                  1 
 max. ⎕PP                                                 16 
 max. input line length                                 2000 
 hash table size (obsolete)                            65536 
 max. shared variable name length                         64 
 max. length of filenames (paths)                       4096 
 max. # of shared variables (obsolete)                    64 
 max. number of APs                                       16 
 max. operators per statement                             16 
 largest integer                         9000000000000000000 
 smallest integer                       ¯9000000000000000000 
 largest numeric exponent                                308 
 max. shared variable size (bytes)                     65000 
 max. cores (per ./configure)                              0 
 max. cores (per max_cores())                              1 
 current cores (per core_count())                          1 
 print length limit (0 = no limit)                         0 
@end example

Most system limits are read-only; attempting to assign ⎕SYL will result
in a SYNTAX ERROR. Indexed assignment to ⎕SYL will result in an
INDEX ERROR unless the limit is writable. Some read-only limits can
be changed at compile-time via ./configure

@c ............................................................................
@node ⎕SYL[1 2 3 4 5 6;]
@subsection ⎕SYL[1 2 3 4 5 6;]

The first three limits are writable; their purpose is to stop execution
if:

@enumerate
@item The SI stack of the interpreter reaches a limit, or
@item The number of APL values created reaches a limit, or
@item The memory size of APL values created reaches a limit.
@end enumerate

Whenever a writable limit is reached, a corresponding LIMIT ERROR
is reported and the limit is reset to 0 (meaning no limit) in order
to prevent the interpreter from locking up.

The main use of ⎕SYL is to troubleshoot infinite recursion in user-defined
function that would otherwise cause the machine to start swapping or to
become non-responding.

Example (stop execution if function call stack exceeds 20):

@example
      ⎕SYL[1;2]←20
@end example

Note that small APL values (up to a configurable number of ravel elements)
count as values (in the number of values limit), but not in the total ravel
bytes limit (the ravel does not allocate memory in addition to the
space already in the number of values limit). It is therefore typical to
set the number of values limit and the total ravel bytes limit together.

An attempt to set a too small limit, for example one that is close to the
current value, is rejected with a DOMAIN ERROR.

@c ............................................................................
@node ⎕SYL[7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23;]
@subsection ⎕SYL[7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23;]

These system limits are read-only. Some of them can be changed at
compile time via ./configure. Most others can be changed in the source code
of GNU APL, but doing so is not recommended.

@c ............................................................................
@node ⎕SYL[24 25 26;]
@subsection ⎕SYL[24 25 26;]

These system limits have been added in preparation for multi-core GNU APL.
They are currently not used.

@c ............................................................................
@node ⎕SYL[27;]
@subsection ⎕SYL[27;]

This is not a limit that raises a limit error. Instead it limits the
size of APL values when printed as statement results. It is fairly easy
in APL to create huge APL values by mistake. During development it can
be useful to limit the amount of data printed. If the ravel length of,
say, VALUE exceeds ⎕SYL[27;2] (i.e. if ⎕SYL[27;2] ≤ ⍴,VALUE) then SH↑VALUE
is printed instead of VALUE. The shape SH is computed like this:

@example
      SH←⍴VALUE
      while (ravel-length(SH) ≥ ⎕SYL[27;2]) @{ divide longest axis in SH by 2 @}
@end example

@c ----------------------------------------------------------------------------
@node ⎕XML - XML Parsing
@section ⎕XML  - XML Parsing

@c ............................................................................
@node Purpose
@subsection Purpose

GNU APL provides ⎕XML as a means of performing @b{elementary}
manipulations of APL character strings that are formatted according
to the XML standard. The anticipated procedure for such modifications
is something like:

@example
      XML_string ← "<A><B><C>Hello</C></B></A>"   ⍝ input data (usually from an .xml file)
      APL ← ⎕XML XML_string       ⍝ convert XML_string to associative array APL
      ABC ← APL.a.b.c             ⍝ retrieve node a.b.c in APL
      APL.a.b.c ← ABC, " World"   ⍝ modify node a.b.c in APL
      Z ← ⎕XML APL                ⍝ convert associative array APL to XML string Z

      ABC   ⍝ the original value of XML.a.b.c
Hello
      Z     ⍝ the modified XML string
<A><B><C>Hello World</C></B></A>"
@end example

The scope (functionality) of ⎕XML is then:

@itemize
@item Conversion of (valid) XML strings to associative APL arrays that can be
  more easily processed in APL than the XML strings,
@item Conversion of associative APL arrays back to valid XML strings,
@item Simple (at APL level) access to the data contained in XML strings, and
@item Transparent access to those (mostly XML specific) portions of the XML
  string that are not (yet) handled by ⎕XML; this means that the user of ⎕XML
  shall be able to handle advanced XML aspects (like document type
  declarations (DTDs), document validation, etc.) in APL without much overhead.
@end itemize

@c ............................................................................
@node Some XML Terminology
@subsection Some XML Terminology

An XML document is a character string with a standardized format.
The characters must be valid XML characters as defined in the XML
standard (almost all Unicode characters except most of the ASCII control
characters). GNU APL raises a DOMAIN ERROR when an XML document contains
invalid XML characters.

The XML document is a sequence of adjacent sub-strings, where each sub-string
is:

@itemize
@item either markup (-string),
@item or non-markup (-string).
@end itemize

A markup (sub-)string:

@itemize
@item starts with character '<', and
@item ends with character '>', and
@item all other characters in the markup string differ from '<' and '>'.
@end itemize

A non-markup (sub-)string:

@itemize
@item starts either at the beginning of the XML document or else directly
      after a markup string,
@item end at the end of the document or else directly before a markup
      string, and therefore
@item cannot be adjacent to another non-markup string.
@end itemize

Non-markup is sometimes referred to as "unstructured text". The producer of
an XML document has a principal choice about where to locate the information
that the document contains: in the unstructured text or in the attribute
values inside the tags. For that reason the XML standard is somewhat picky
about keeping the unstructured text as well as attribute values intact
and requires that whitespace in non-markup (the only allowed whitespace
characters in XML being TAB, CR, LF, and SPACE) is preserved in non-markup
strings.

As a consequence, many non-markup strings begin with LF because the
preceeding markup was on the previous line and therefore the LF after the
preceeding markup is whitespace of the non-markup and therefore needs to be
preserved.

Every markup belongs to one of 4 categories:

@itemize
@item Processing instructions: "<? ... ?>",
@item Comments: "<!--- ... --->",
@item Declarations "<! ... !>", and
@item Tags, which are either:
  @itemize
  @item Empty tags (leaf tags): "<tagname attributes... />
  @item Start tags: "<tagname attributes... >, or
  @item End tags: "</tagname>
  @end itemize
@end itemize

The XML tags recursively define a tree structure where:

@itemize
@item the nodes of the trees are either Empty Tags or else a start tag,
   followed by a (possibly empty) sequence of non-markup, comments, and
   sub-nodes (of the tree), followed by an end tag (with the same name
   as the start tag),
@item the leafs of the trees are either Empty Tags or else a start tag,
   followed by a (possibly empty) sequence of non-markup and comments,
   followed by an end tag (with the same name as the start tag). The difference
   between a leaf node and a non-leaf is therefore the absence of sub-nodes
   between a start and an end tag.
@item optionally non-markup and comments before, after, and between the nodes.
@end itemize

Processing instructions and declarations are typically located at the
beginning of the XML document and at the same level as the root of the tree
defined by the document. Document type declarations are preserved (so that
they can be processed in APL), but are currently ignored by ⎕XML. As a
consequence, entity declarations have no effect and only the predefined XML
entities (i.e. &, <, >, ', and ") are recognized (and translated) by ⎕XML.

@c ............................................................................
@node How ⎕XML maps XML documents to APL Values
@subsection How ⎕XML maps XML documents to APL Values

The principal mapping between XML strings and APL values is:

@itemize
@item @b{XML document} ←→ structured (top-level) APL value or variable
  (= an associative array)
@item @b{XML node} ←→ structured APL (sub-) value (= an associative (sub-)
  array of either the top-level value or of another XML node)
@item @b{XML name} ←→ name of a member in a structured APL value
@item @b{Other} ←→ member value
@end itemize

Every valid name in XML is also a valid name for an APL variable or structure
member. Unfortunately, mapping XML names directly to APL member names cannot
work because:

@itemize
@item In any associative array the names of the members are necessarily
 distinct. Assigning a value to an already existing member of a structured
 value overwrites the current value of that member instead of creating a
 second member with that value. On the other hand, in XML nodes tags may (and
 frequently do) occur repeatedly. If an XML document contains several nodes
 with the same tag, then the applications usually knows how to interpret the
 order in which the tags occur, but this order may not be detectable at the XML
 level (i.e. the nodes may not only have the same tag names, but also the same
 attributes and attribute values).

 The applications that use the XML document frequently require that the order
 of XML nodes with the same name remain intact when the document is modified.
 For example, the entries of a telephone book may be sorted alphabetically
 without XML being aware of that. Reading, modifying, and writing back the
 data in the telephone book must therefore not destroy that order.

@item Likewise, at least for readability reasons, the order of node
 attributes should not be changed. XML documents are frequently processed
 with text tools like @b{diff} which would incorrectly show differences
 between XML documents that are equivalent from an XML perspective.

@item Conflicts between tag names and attribute names. In XML, the same name
  can designate an attribute in one node and a tag in another node. ⎕XML keeps
  all information from the < of a start tag to the > of an end tag in a single
  associative array, therefore the attribute names must somehow differ from
  the (sub-) tag names.

@item Every XML node has a tag name that can be used to derive a
  corresponding APL member name. However, the text segments between an XML
  tag and its corresponding end tag, are a mixture of unstructured text, XML
  subnodes and comments of which only the XML subnodes have tags of their own.
  These untagged segments may, however, contain relevant information and,
  from an XML standard perspective, both their content and their order must
  be preserved.
@end itemize

⎕XML solves these issues with a simple naming convention for all member names
in the (recursive) associative array that is produced by monadic ⎕XML. Every
member name consists of 3 components that together yield a valid APL variable
name (and therefore a valid member name that is understood by the APL parser):

@itemize
@item A single character ⍙, ∆, or _, that indicates the role of the name in
  the XML document as follows:
  @itemize
  @item the character ⍙ characterizes an attribute name, i.e. a name that
  was used @b{inside} (i.e. between < and >) of an XML start or leaf tag.
  The character ⍙ alone is the name of a special member which is present
  in every associative array below the top-level and whose value is the
  original XML tag name of a tag, or

  @item the character ∆ characterizes a name that was synthesized by ⎕XML
  itself to tag a text (sub-) segment that has no tag of its own. The
  corresponding XML entity was not a tag (and therefore had no tag name), or

  @item the character _ characterizes the name of an XML entity, i.e. a CML
  name that occurred @b{between} the start tag and the corresponding end tag
  (if any: leaf tags have no such entities)
  @end itemize

@item a position prefix of the form @b{NN}, where NN is the position
 of the member in relation to the tag to which it belongs. The numbering
 starts at ⎕IO and restarts at every XML start or leaf tag.

@item the attribute name or tag name. For those parts of an XML entity that
  do not have a tag name, the member name is synthesized by ⎕XML and the
  member value will be the text in the XML document with predefined XML
  entities replaced by their single characters, i.e.:
  @itemize
  @item & for &amp,
  @item < for &lt,
  @item > for &gt,
  @item ' for &apos,
  @item " for &quot,
  @item ⎕UCS nnn for &#nnn, and
  @item ⎕UCS $xxx for &#xxx
  @end itemize
@end itemize

Let NN be a sequence of decimal digits. Then only the following 4
synthesized names are produced by ⎕XML:

@itemize
@item @b{∆NNcomment} for XML comments, i.e. <!--- ... ---> in XML,
@item @b{∆NNdeclaration} for XML declarations, i.e. <? ... ?> in XML,
@item @b{∆NNdoctype} for document type declarations, i.e.
    <!DOCTYPE ... > in XML
@item @b{∆NNtext} for unstructured (and hence untagged) text segments in XML.
@end itemize

Every XML name (of a tag or attribute) is also a valid APL name, but
valid APL names may also contain the characters _, ∆, and ⍙ which are not
valid in XML and used in the naming convention above to clearly separate
the member names used in APL from the names used in the XML document.

The essence of all this is that @b{everything} that belongs to an XML
node or document, i.e. the attributes inside a tag, the sub-nodes between a
start tag and its corresponding end tag (if any), declarations, etc. will
be contained in a single associative array.

Examples of APL member names (assuming ⎕IO←1):

@example
      ⍙1              ⍝ the first member in every XML node. Its value is the node tag.
      ∆1declaration   ⍝ the first member in most XML documents. E.g. <?xml version= ...>
      ∆2text          ⍝ whitespace (LF) between _1∆declaration and _3∆doctype
      ∆3doctype       ⍝ DTD in XML documents that have one. E.g. <!DOCTYPE  ...
      _6Workspace     ⍝ Sub-array for <Workspace ... at position 6.
      ⍙1wsid          ⍝ first attribute wsid from e.g. <Workspace wsid=...
      ⍙2year          ⍝ second attribute year
@end example

@c ............................................................................
@node Monadic ⎕XML
@subsection Monadic ⎕XML

@b{Z ← ⎕XML B} converts APL string B that contains a valid XML document
to an associative APL array Z and vice versa.

@b{Note}: If you convert an associative APL array back to a an XML document,
the result is a possibly very long string containing several linefeed
characters. That is OK if the string is written to a file, but causes problems
when the string is displayed and longer than ⎕PW. The standard APL
folding of output near ⎕PW make the string then look somewhat garbled.
This effect can be suppressed in 2 ways:

@enumerate
@item set ⎕PW large enough, or
@item prefix the string with ⊃35⎕CR as in the examples below.
@end enumerate

@c ............................................................................
@node Dyadic ⎕XML
@subsection Dyadic ⎕XML

@b{Z ← A ⎕XML B} provides a number of convenience functions. Its left
argument A is an integer function number, possibly followed by arguments
of the function number ↑A.

In the following examples, let

@example
      XML←36 ⎕CR """
<?xml version='1.0' encoding='UTF-8'?>
  <Document>
    <Tag1 name='tag1'>   <!-- first tag -->
      TEXT1
      <Subtag name='subag1.1'>   <!-- first subtag -->
        SUBTEXT1_1
      </Subtag>
    </Tag1>
  </Document>
"""

APL←⎕XML XML
@end example

Then

@itemize
@item @b{Z←0 ⎕XML B} is the same as monadic Z←⎕XML B.

@item @b{Z←1 ⎕XML B} is like 0 ⎕XML B, except that B is not the XML
document (string) itself but the name of a file from which an XML document
is read.

@item @b{Z←2 ⎕XML B} splits an APL member path (a sequence of member names)
into a vector P of (nested) member names. This vector P is understood by
the APL primitive PICK (i.e. Z←P⊃APL or (P⊃APL)←value and can be used to
address the data in an XML node.

@item @b{Z←3 ⎕XML B} splits an APL member name (as produced by e.g.
monadic ⎕XML into a 3-item vector Z. Z[1] is a single character indicating
the category (i.e. a character scalar ∆, ⍙, or _). Z[2] is the position (an
integer scalar ≥ ⎕IO). Z[3] is a (nested) APL string. 3 ⎕XML has an
inverse function ¯3 ⎕XML which converts a 3-item vector as produced by 3 ⎕XML
back into an APL member name indicating the category (i.e. _, ∆, or ⍙)
and the name from the XML tag:

@example
            8 ⎕CR (C P N)←3 ⎕XML "∆2comment"
┌→────────────┐
│∆ 2 ┌→──────┐│
│    │comment││
│    └───────┘│
└∊────────────┘

      C   ⍝ category
∆
      P   ⍝ position
2
      N   ⍝ (XML-) name
comment

      ¯3 ⎕XML C P N   ⍝ inverse of 3 ⎕XML
∆2comment
@end example

@item @b{Z←4 ⎕XML B} is a string that shows the member names of the
 associative array B in a tree-like fashion. For example:

@example
      ⊃35⎕CR 4 ⎕XML APL
XML                     
 │                      
 └─── Document          
       │                
       └─── Tag1        
             │          
             └─── Subtag
@end example

@item @b{Z←5 ⎕XML B} is like 4 ⎕XML but displays not only the XML tags but
also other XML nodes (declarations, attributes, etc.):

@example
      ⊃35⎕CR 5 ⎕XML APL
XML                               
 │                                
 ├─── _1∆declaration              
 ├─── _2∆text                     
 └─── _3_Document                 
       │                          
       ├─── _1∆text               
       ├─── _2_Tag1               
       │     │                    
       │     ├─── _1∆text         
       │     ├─── _2∆comment      
       │     ├─── _3∆text         
       │     ├─── _4_Subtag       
       │     │     │              
       │     │     ├─── _1∆text   
       │     │     ├─── _2∆comment
       │     │     └─── _3∆text   
       │     └─── _5∆text         
       └─── _3∆text               
@end example

@item @b{Z←6 ⎕XML B} is like 4 ⎕XML but displays not only the member
name but the entire path to the member:

@example
      ⊃35⎕CR 6 ⎕XML APL
XML                                             
 │                                              
 └─── ._3_Document                              
       │                                        
       └─── ._3_Document._2_Tag1                
             │                                  
             └─── ._3_Document._2_Tag1._4_Subtag
@end example

@item @b{Z←7 ⎕XML B} returns a list of all members in a depth-first order.
This list can be used as an iterator to traverse the entire tree (and possibly
filter out items along the way).

@example
      ⊃7 ⎕XML APL
._1∆declaration                          
._2∆text                                 
._18∆text                                
._3_Document                             
._3_Document._1⍙                         
._3_Document._1∆text                     
._3_Document._3∆text                     
._3_Document._2_Tag1                     
._3_Document._2_Tag1._1⍙                 
._3_Document._2_Tag1._2⍙name             
._3_Document._2_Tag1._1∆text             
._3_Document._2_Tag1._2∆comment          
._3_Document._2_Tag1._3∆text             
._3_Document._2_Tag1._5∆text             
._3_Document._2_Tag1._4_Subtag           
._3_Document._2_Tag1._4_Subtag._1⍙       
._3_Document._2_Tag1._4_Subtag._2⍙name   
._3_Document._2_Tag1._4_Subtag._1∆text   
._3_Document._2_Tag1._4_Subtag._2∆comment
._3_Document._2_Tag1._4_Subtag._3∆text   
@end example

@item @b{Z←8 ⎕XML B} is like Z←7 ⎕XML B, except that only the subset of
those members that are related to tags (the tag name ⍙ and tag attributes)
are returned:

@example
      ⊃8 ⎕XML APL   ⍝ all tag related member names (tag names and attributes)
._3Document.⍙1····················
._3Document._2Tag1.⍙1·············
._3Document._2Tag1.⍙2name·········
._3Document._2Tag1._4Subtag.⍙1····
._3Document._2Tag1._4Subtag.⍙2name
@end example

@item @b{Z←9 ⎕XML B} is like Z←7 ⎕XML B, except that only the subset of
synthesized members other than ∆text are returned:

@example
      ⊃9 ⎕XML APL   ⍝ all synthetic member names (except unstructured text)
.∆1declaration·······················
._3Document._2Tag1.∆2comment·········
._3Document._2Tag1._4Subtag.∆2comment
@end example

@item @b{Z←10 ⎕XML B} is like Z←7 ⎕XML B, except that only the subset of
synthesized ∆text members are returned:

@example
.∆2text···························
._3Document.∆1text················
._3Document._2Tag1.∆1text·········
._3Document._2Tag1.∆3text·········
._3Document._2Tag1._4Subtag.∆1text
._3Document._2Tag1._4Subtag.∆3text
._3Document._2Tag1.∆5text·········
._3Document.∆3text················
.∆18text··························
@end example

@item @b{Z←11 ⎕XML B} is like Z←7 ⎕XML B, except that only the tag
names are returned:

@example
      ⊃11 ⎕XML APL   ⍝ all tag member names
._3Document················
._3Document._2Tag1·········
._3Document._2Tag1._4Subtag
@end example

@item @b{Z←12 ⎕XML B} is like Z←7 ⎕XML B, except that all member
names of B are returned without descending into the sub-tree:

@example
      ⊃12 ⎕XML APL   ⍝ all member names (7 ⎕XML, but non-recursive)
.∆1declaration
.∆2text·······
._3Document···
.∆18text······
@end example

This variant may also be useful for structured variables that were not
created by ⎕XML.

@item @b{Z←13 ⎕XML B} shows the (flat) tag names in B, but but does
not descend into any sub-tree of B.

@example
     ⊃13 ⎕XML APL   ⍝ all tag members (8 ⎕XML, but non-recursive)
 ._3Document 
@end example

@item @b{Z←(14 "member-name") ⎕XML B} shows the next member in B
after "member-name". It can be used as an iterator that walks through
the tags. An empty member name returns the first member:

@example
      APL ← ⎕XML XML←"<ROOT><A/><B/><C/></ROOT>"

      (14 "") ⎕XML APL._1ROOT   ⍝ get first member in APL._1ROOT
⍙1
      (14 "⍙1") ⎕XML APL._1ROOT   ⍝ next member after APL._1ROOT.⍙1
_1A
      (14 "_1A") ⎕XML APL._1ROOT   ⍝ next member after APL._1ROOT._1A
_2B
      (14 "_2B") ⎕XML APL._1ROOT   ⍝ next member after APL._1ROOT._2B
_3C
      (14 "_3C") ⎕XML APL._1ROOT   ⍝ last member: will be empty
@end example
@end itemize

@c ............................................................................
@node XML Queries
@subsection XML Queries

The different functions in dyadic ⎕XML are intended to construct XML queries.
An XML query extracts information from, or modifies information in, an XML
document. XML queries may differ considerably in their details, but usually
follow these top-level steps:

@enumerate
@item Read an XML document into, say, APL (APL←0 ⎕XML B or APL←1 ⎕XML B)
@item Optionally, display the structure of the document (3, 4, or 5 ⎕XML APL)
@item Select a subset of XML nodes (= GNU APL member names, 6 ⎕XML APL)
@item retrieve or modify the data in the selected XML nodes
 (with something like MEMBERS←(⊂APL) FUN¨6⎕XML APL←⎕XML followed by MEMBER⊃ )
@item Optionally, convert the modified XML document back to XML (0 ⎕XML APL)
@end enumerate

@c ----------------------------------------------------------------------------
@node Conditionals
@section Conditionals

@b{WARNING}: GNU APL conditionals are experimental. Use them with care and
avoid them, wherever possible, in favor of portability. Neither the ISO
APL standard, nor IBM APL2 provide conditionals.

@c ............................................................................
@node Design considerations.
@subsection Design considerations.

The author of GNU APL has, for a long time, hesitated to implement
conditionals in GNU APL. Primarily for the following reasons:

@itemize
@item the lack of standardization of conditionals,
@item the lack of an elegant syntax that fits well into the typical
      (brief) APL programming style, and in particular
@item the large number of keywords for flow control used in some other APL
      interpreters. See below.
@end itemize

The C language has six keywords for flow control: @b{if} @b{else}, @b{for},
@b{while}, @b{do}, @b{switch} and @b{goto}. Two of them (@b{do} and @b{while})
are redundant and rarely used, leaving only 4 base cases. A quick @b{grep -c}
over the 130,000 top-level C++ source code lines of GNU APL reveals the
following statistics:

@itemize
  @item 7000 @b{if} statements, 5400 of them @i{one-sided} (i.e.
  if-then-endif), and the other 1600 of them @i{two-sided} (i.e.
   if-then-else-endif,
  @item 2000 @b{for} statements, 1200 of them trivial iterations over some
  array (aka. @i{for (int i = 0; i < size; ++i) ... },
  @item 200 @b{switch} statements with an averge of 8 case clauses per switch,
  @item 250 @b{while} statements, and
  @item 70 @b{do while} statements
@end itemize

In contrast, a popular non-free APL interpreter came up with a zoo of 30
or so keywords for the purpose of flow control in one way or another:

@b{:If} @b{:While} @b{:Repeat} @b{:For} @b{:In} @b{:InEach} @b{:Select}
 @b{:With} @b{:Trap} @b{:Hold} @b{:Disposable}
@b{:Else} @b{:ElseIf} @b{:AndIf} @b{:OrIf} @b{:Until} @b{:Case} @b{:CaseList}
@b{:EndIf} @b{:EndWhile} @b{:EndRepeat} @b{:EndFor} @b{:EndSelect} @b{:EndWith}
@b{:EndTrap} @b{:EndHold} @b{:EndDisposable} @b{:GoTo}, @b{:Return},
@b{:Leave} @b{:Continue}

This approach, its implementation, and its constraints raise a number of
serious problems. To name a few:

@itemize
  @item A rather common APL design practice is to first debug non-trivial
  APL statements in immediate execution mode and only transfer them into
  the body of defined functions when they work properly. However, the keywords 
  above seem to work only in the bodies of defined functions but @b{not} in
  immediate execution mode. The reason for this limitation might be that
  their implementation depends, under the hood, on APL labels (which are not
   available in immediate execution mode).

  @item Another rather common design practice is to limit the length of
  lines to less than 80 characters (e.g. in the GNU coding conventions,
  @b{https://gcc.gnu.org/codingconventions.html#Line}).
  Even if it were possible to squeeze an :If :Then :Else :Endif into an 80
  character line, and using a single whitespace around the keywords for
  readability, wastes almost 1/3 of the precious 80 characters for keywords.
  In this example, each of the COND, THEN, and ELSE parts must then be < 20
  characters on average. Not to talk about indentation to clarify the code.

  @item Each of the keywords has rather specific constraints as to how and
  where it can be used. For example (to quote a simple one from a reference
  manual):

  @i{Control words, including qualifiers such as :Else and :ElseIf, may occur
  only at the beginning of a line or expression in a diamond-separated
  statement. The only exceptions are :In and :InEach which must appear on
  the same line within a :For expression.}

  These constraints suggest that the keywords mentioned are merely syntactic
  sugar that covers the underlying APL structure, i.e. labels and computed
  branches. GNU APL does not claim to be fundamentally different in that
  regard, but the syntax of the syntactic sugar is somewhat simpler.
@end itemize

Learning from the above, the conclusions for the syntax of GNU APL condionals
were these:

@enumerate
  @item the syntax shall be easy to understand and consistent,
  @item the syntax shall not only work in defined functions, but also in
       immediate execution mode (provided, of course, that the entire
       conditional fits into a single input line),
  @item in order to support 2., lengthy keywords shall be avoided in
        favor of the original APL style, and
  @item the syntax needs to only support those cases where the standard APL
        flow control (i.e. @b{→N}) becomes cumbersome. There is no obvious
        benefit in introducing a new (and redundant) syntax when the
        standard APL syntax is already completely adequate. IOW: in which
        universe would @b{@i{:Goto Label}} be any better than the good old
        @b{@i{→Label}}?
@end enumerate

As to the last point, almost every flow control case (in a standards compliant
APL program) falls into one of the following categories (sorted by
increasing complexity):

@enumerate
@item @b{one-sided if}: if COND then THEN endif
@item @b{two-sided if/else}: if COND then THEN else ELSE endif
@item @b{loop}: while COND do BODY (or its less frequently used twin do BODY
while COND)
@item @b{switch/case}: switch ARG case CASE1 case CASE2 case ... CASEn endswitch
@end enumerate

where THEN, ELSE, CASE1, CASE2, CASEn, and BODY are sequences of one or more
statements, while COND or ARG determine which of these sequences (if any)
shall be executed.

Interestingly, the complexity of the above cases, when coded in APL,
increases in amost the reverse order (compared to other languages).
For example,

@enumerate
@item @b{switch/case}:
@example
    →(ARG) / L1  L2 ...  Ln   ⍝ switch(ARG)
L1: CASE1 ◊ →L_endcase        ⍝ case 1:
L2: CASE2 ◊ →L_endcase        ⍝ case 2:
...
LN: CASEn                     ⍝ case N:
L_endcase:
@end example

@item @b{loop}:
@example
 Loop: BODY ◊ →(COND) / Loop   ⍝ repeat BODY while COND
@end example

@item @b{one-sided if}:
@example
  →(!COND) / L_endif   ⍝ jump over the THEN clause
  THEN
L_endif:
@end example

or, not uncommon and shorter, but less portable and less readable:

@example
  →(!COND) / ↑1+⎕LC ◊ THEN  ⍝ jump over the THEN clause to the next line
@end example

@item @b{two-sided if/else}:
@example
  →COND ↓ L_else L_then
L_then: THEN ◊ →L_endif
L_else: ELSE
L_endif:
@end example
@end enumerate

A closer look at these cases reveals that there is not much room for
syntax improvements in cases 1. (switch) and 2. (loop). The introduction of
a new syntax like :Select / :Case for switches or :For / :Endfor for loops
would therefore only make matters different and incompatible, but not better.

In case 3. (one-sided if) there is also not much to gain. However, case 3. 
can be seen as a special case of case 4. (with an empty ELSE clause). For
the sake of consistency, a new syntax for case 4. should also cover case 3.

The only cases where a new syntax could improve matters are therefore
cases 3. and  4.

Another consideration relates to the end marker @b{L_endif} (and would
likewise relate to @b{L_endcase} if we hadn't already ruled that case out).
There are two reasonable syntax alternatives to handle the branch to the
end of the conditional:

@enumerate
@item allow multiple statements in the THEN and ELSE clauses and mark the end
  of the @b{ELSE} clause (or, in the one-sided case 3. of the @b{THEN} clause
  above) with some kind of @b{ENDIF} marker, or
@item allow only single statement clauses (which implies the @b{ENDIF} and
 makes it obsolete.
@end enumerate

The design decision made in GNU APL was to go for multiple statements because
the advantage in 2. (shorter code) looks much smaller than the advantages
of 1. (e.g. insertion of debug printouts into a clause).

@c ............................................................................
@node Syntax
@subsection Syntax

With the above considerations in mind, GNU APL decided in favor of the
following syntax for APL conditionals:

@itemize
@item @b{one-sided if/else}:
@example
      COND →→ THEN-statement(s) ←←
@end example

@item @b{two-sided if/else}:
@example
      COND →→ THEN-statement(s) ←→ ELSE-statement(s) ←←
@end example

@end itemize

The following rules govern this syntax:

@itemize
  @item The @b{→→}, @b{←→}, and @b{←←} markers must be single tokens (2
        adjacent Unicode characters). No whitespace (and in particular
        no line feed) is permitted between the first and the second arrow.
        A single @b{→} is a valid APL statement (named Escape). A linefeed
        between two @b{→} turns the single token @b{→→} into two valid (though
        most of the time useless) Escape statements.

  @item The @b{COND}, @b{THEN}, and @b{ELSE} clauses shall be sequences
         of valid APL statements. In particular:
        @itemize
        @item The condition @b{COND} shall be a @i{single statement} that
              evaluates to a 1-item Boolean array (of any rank).
              The statement result of @b{COND} is a committed value and is
              therefore not being displayed.
       
        @item The @b{THEN} clause shall be @i{one or more valid statements},
              possibly separated by @b{◊} for more than one statement.
       
        @item Likewise, the @b{ELSE} clause (if present) shall be
              @i{one or more valid statements}, separated by @b{◊} for
              more than one statement.
        @end itemize
@end itemize

Examples:
@example
      ⍝ short format. Valid in immediate execution and in defined functions
      ⍝
      X←5 ◊ X≥0 →→ X 'is positive' ←→ X 'is negative' ←←
 5 is positive 


      ⍝ long format. Only possible in defined functions because immediate
      ⍝ execution works line by line (and parsing the first line would fail).

      ⍝ valid split into several lines (in a defined function)
      ⍝
      ∇FOO X
       X≥0             →→
       X 'is positive' ←→
       X 'is negative' ←←
      ∇
      FOO 5
 5 is positive 
      FOO ¯5
 ¯5 is negative 


      ⍝ invalid split into several lines: ←→ spread over 2 lines.
      ⍝ FOO can be properly parsed, but calling FOO fails.
      ⍝
      ∇FOO X
      X≥0 →→ X 'is positive' ←
           → X 'is negative' ←←
      ∇
      FOO 5
SYNTAX ERROR+
FOO[1]  X 'is positive'←
        *              *
@end example

The recommended usage of the corresponding @b{→→}, @b{←→}, and @b{←←}
token of the same conditional is to:

@itemize
  @item either put all of them on the same line (short format), or
  @item else align all of them on the same column (long form), but
  @item never mix the short and long forms
@end itemize

In theory one could place (and align) the token @i{before} or @i{after}
their statement(s). However, placing them @i{after} their statement(s),
like shown in the examples above, makes them arguably more visible and
is therefore the recommended way. In particular when conditionals are
being nested.

@c ----------------------------------------------------------------------------
@node A Possible Pitfall
@subsection A Possible Pitfall

In APL, as well as in C/C++ (and most other languages), an expression is also
a statement, but not vice versa. Expressions yield a value, while statements
may not.

The @b{C/C++} language knows @i{conditional statements} and
@i{conditional expressions}:

@example
   if (COND) @{ IF-statements @}       // conditional C/C++ statement
   else      @{ ELSE-statements @}

   (COND) ? IF-value : ELSE-value    // conditional C/C++ expression
@end example

Likewise, GNU APL also has @i{conditional statements} and
@i{conditional expressions}:

@example
      COND →→ IF-statements ←→ ELSE-statements ←←    ⍝ conditional APL statement

      IF-value ⊢[COND] ELSE-value                    ⍝ conditional APL expression
@end example

If a value is needed, e.g. for assigning it to a variable, but the expression
becomes too long to fit in a single line, then one can replicate the
assignment in both of the @b{IF} and the @b{ELSE} clauses:

@example
      // C/C++
      x = -5;
      Result = (x < 0) ? "negative" : "positive";                          // OK
      Result = if (x < 0) @{ "negative"; @} else @{ "positive"; @}         // syntax error
      if (x < 0) @{ Result = "negative"; @} else @{ Result = "positive"; @}   // OK

      ⍝ GNU APL conditional expression, aka. ⊢[]
      X←¯5
      Result ← "positive" ⊢[X < 0] "negative"                     ⍝ OK
      Result ← (X < 0) →→ ⊣"negative" ←→ ⊣"positive" ←←           ⍝ ???
      Result
1

      (X < 0) →→ Result ← "negative" ←→ Result ← "positive" ←←    ⍝ OK
      Result
negative
@end example

The correct, although possibly unexpected, result @b{1} of the APL conditional
in the line with comment @b{⍝ ???} above was evaluated as follows:

@enumerate 1
@item APL computes different statements from right to left.
@item The conditional tokens @b{→→}, @b{←→}, and @b{←←} are statement
      separators like @b{◊}, but with the additional capability to jump
      over their @b{THEN} and/or @b{ELSE} clauses according to the value
      of their @b{COND} clause.
@item The first statement in the example above is @code{Result ← (X < 0)}
      i.e. @code{Result←1}.
@item The statement result of @b{COND} is always a committed value, regardless
      of whether it is assigned to a variable (like in the example above)
      or not. Therefore the @b{COND} clause does not produce any output.
@item The next statement that is being executed it either @code{⊣"negative"}
      or else @code{⊣"positive"}, depending on the value of variable @b{X}.
      Normally it would be displayed, but due to @b{⊣} the display is
      suppressed. The entire conditional does therefore produce no output.
@item As a consequence, @code{Result ← 1} (and not, for example,
      @code{"negative"} !). 
@item The @code{⊣"negative"} and @code{⊣"positive"} are separate statements
      which do not assign anything to @b{Result} in the previous @b{COND}
      clause. The pitfall in the example is that the conditional looks like
      it would assign either @b{"negative"} or @b{"positive"} to @b{Result},
      but in fact assigns @b{1}.
@end enumerate

Needless to say that, as in the example above, the assignment of a APL
conditional statement to a variable is, although possible, a dubious
programming style that should be avoided for the sake of readability. The
key takeaway here is that @i{the clauses of a GNU APL conditional are (two or
three) different statements, and that an assignment of the conditional
to a variable assigns only the first of these statements to the variable}.

In C/C++, the condition of a conditional statement MUST be put in parentheses,
while the condition of a conditional expression MAY be put in parentheses.

In GNU APL the condition of a conditional statement MAY be put in parentheses,
while the condition of conditional expression MUST be put in brackets (since it
is the axis argument @b{X} of @b{A ⊢[X] B.}. A good programming practice
in C/C++ and in GNU APL is to always put the condition into parentheses even if
that if not syntactically required. Some people even use double parentheses to
emphasize the condition of a conditional statement or expression.

@c ----------------------------------------------------------------------------
@node Matrix Product
@section Matrix Product

In standard APL, the inner product Z of two matrices A and B as known from
linear algebra, is:

@example
      Z←A +.× B      ⍝ the most frequent special case of A f.g B
@end example

However, the inner product in APL is more general than the inner product in
linear algebra. In the general case @b{A f.g B} may the functions @b{f}
and @b{g} also be non-scalar APL primitives or even defined functions.
This generality may occasionally be helpful, but it comes with a performance
penalty since many argument checks are duplicated. At the same time
@b{A +.× B} is by far the most frequent use of the inner product operator
@b{'.'}.

For this reason GNU APL also provides a slightly more efficient dyadic
function @b{∘} (Matrix Product) for the special case @b{+.×} of the dyadic
operator '.'. This function ∘ has been optimized for numeric arguments
and computes the matrix product as known from linear algebra.

For two-dimensional matrices @b{A} and @b{B} is function @b{∘} conceptually
the same as @b{+.×}:

@b{A∘B ←→ A +.× B}

However, there are some subtle differences between the function @b{∘} and
the operator @b{+.×}:

@itemize
@item @b{A +.× B} allows @b{A} and @b{B} to be of any rank, while @b{A∘B}
raises a RANK ERROR if the rank of @b{A} or @b{B} is more than 2.

@item The product @b{A +.× B} can be a vector (e.g. if @b{A} is a vector
and @b{B} is a matrix), usually understood as row-vector.
 In contrast (and more similar to linear algebra) @b{A∘B} is always a matrix.
@item In order to preserve the distinction between row vectors and column
  vectors the arguments @b{A} and @b{B} of @b{∘} are possibly reshaped
  according to the following rules:

@itemize
@item if @b{A} and/or @b{B} is a scalar then the normal (element-wise)
  product of @b{A} and @b{B} is computed. That is, @b{A∘B ←→ A×B}.
@item if @b{A} is a APL vector of length @b{N} then it is reshaped to a
row-vector before @b{A∘B} is computed. In this context,
the term  @i{row-vector} shall refer to a matrix of shape @b{(1 N)}.
@item if @b{B} is a APL vector of length @b{N} then it is reshaped to a
column-vector before @b{A∘B} is computed. In this context, the term
@i{column-vector} shall refer to a matrix of shape @b{(N 1)}.
@end itemize

Examples:

@example
      M←2 2⍴1 2 3 4   ⍝ a matrix
      V←10 11         ⍝ a vector

      M +.× V
32 74

      M∘V
32
74

      V +.× M
43 64

      V∘M
43 64

      ⍴M +.× V   ⍝ the result is an (APL-) vector of length 2
2
      ⍴V +.× M   ⍝ the result is an (APL-) vector of length 2
2
      ⍴V∘M       ⍝ the result is row vector
1 2
      ⍴M∘V       ⍝ the result is column vector
2 1
@end example

@item @b{A +.× B} requires that the number of columns in @b{A} is equal to
the number of rows in @b{B}. In contrast, @b{A∘B} has no such requirement
and will add 0-columns to @b{A} resp. 0-rows to @b{B} to match the shapes
of @b{A} and @b{B}. Naturally, the multiplications of these 0-columns or
0-rows is not in fact performed, but optimized away.

@end itemize

@c ----------------------------------------------------------------------------
@node Monadic ≠ (Nub Sieve)
@section Monadic ≠ (Nub Sieve)

The nub sieve @b{Z←≠B} of a value @b{B} is a Boolean vector @b{Z} with the
same shape as @b{B} and:

@b{Z[i]} is 1 if @b{B[i]} is the first occurrence of @b{B[i]} in @b{B} and
0 otherwise. For vectors @b{B} this can also be expressed as:

@example
      ≠ B ←→ (B⍳B)=(⍳⍴B)
@end example

Examples:

@example
      ≠'Hello, World.'
1 1 1 0 1 1 1 1 0 1 0 1 1
      ≠ 1 (2 3) 4 (2 3) 5 6
1 1 1 0 1 1
@end example




@c ============================================================================
@node Limitations and Missing Features
@chapter Limitations and Missing Features

GNU APL is an almost full implementation of the ISO standard 13751.
However, a few limitations exist. See file README-4-compliance for
an up-to-date description of limitations.

@node System Limits
@section System Limits

APL floating point values are 64-bit wide, thus ranging from
-8.98E307 to 8.98E307.

GNU APL integers have a guaranteed range from -9200000000000000000 to
9200000000000000000, which is a slighly smaller range.

A 64-bit integer represents a value between -9223372036854775808 and
9223372036854775807.

If the result of a computation is integer by nature, for example +, -, or ×
with integer arguments, @b{and} falls into the guaranteed range, then
the result will be an APL integer. If the result is too large for a 64-bit
signed integer then the result will be automatically converted to an APL
floating point value. However, if the result is outside the guaranteed range
but still within the 64-bit integer range, e.g. between 9200000000000000000
and 9223372036854775807 (exclusive), then GNU APL gives no guarantees as to
whether the result will be a 64-bit integer or a 64-bit floating point (with
the resulting loss of precision). This decision is usually function-specific
and driven by performance considerations.

APL values have a maximum rank of 8. However, the maximum range can be set
by means of ./configure (see README-2-configure).

@node Shared Variables
@section Shared Variables

The system functions and variables related to shared variables, i.e. ⎕SVO,
⎕SVR, ⎕SVC, ⎕SVS, and ⎕SVQ, are implemented.

Communication between two workspaces is supported, but with some
limitations. Two workspaces can only communicate via shared variables if they
(i.e. their processes) run on the same machine.

However, only two auxiliary processors, AP100 and AP210, are provided as
examples of how to implement auxiliary processors in GNU APL.

Shared variables for auxiliary processors are provided for some backward
compatibility. In the past - without access to the source code of the APL
interpreter - shared variables were the only method available to extend the
functionality of the interpreter.

With this interpreter - and access to its source code - it is often more
convenient to add your own commands or your own system variables to the
APL interpreter rather than adding auxiliary processors.

@c ============================================================================
@node Internationalization
@chapter Internationalization

GNU APL used to be internationalized by means of GNU gettext, but is not
anymore. 

For APL characters to work properly with keyboards and terminal emulators
(see also README-3-keyboard) it is essential that your LANG environment
variable is correctly set. For German, that would be "de_DE.utf8", like:

$export LANG=de_DE.utf8

A misspelling of the language can cause APL characters to become unavailable,
even if your xmodmap is correct. You can check your setting by executing the
program 'locale'. If it shows warnings like these:

@example
locale: Cannot set LC_CTYPE to default locale: No such file or directory
locale: Cannot set LC_MESSAGES to default locale: No such file or directory
locale: Cannot set LC_ALL to default locale: No such file or directory
@end example

then something is wrong and you should check your LANG setting and installed
locales.

As far as xterm is concerned, the xterm command line options --u8, --lc, and
--en might solve problems related to the terminal.

@c ============================================================================
@node Vector Notation
@chapter Vector Notation

This chapter describes details of the GNU APL vector notation. Unfortunately
there exists misleading information concerning the GNU APL vector notation
on the public internet and the attempts of the author of this manual to have
the misleading information corrected were not successful.

@c ----------------------------------------------------------------------------
@node History
@section History

Vector Notation is as old as APL. The famous Book @i{APL \360: An interactive
approach} by @b{Leonard Gilman} and @b{Allen J. Rose} introduces vector
notation already on page 2, but without calling it vector notation.

Quote:

@example
Or we can assign a string of numbers to a variable called X, and ask the
computer to execute the command shown, + / X , with the response 17.1 :
      X+-3 4 1.1 3 6
      +/X
17.1
@end example

The terms @i{vector notation} (used by IBM APL and by GNU APL) was
apparently coined by @b{Kenneth Iverson} (@i{A Programming Language},
p. 163) and the equivalent term @i{strand notation} is used by other
APL vendors.


@c ----------------------------------------------------------------------------
@node Standardization
@section Standardization

The current ISO standard ISO/IEC 13751 @i{Programming Language APL, extended},
in the following abbreviated as @b{iso},  does not formally define vector
notation, but uses it all over the place in its examples.

The IBM APL2 @i{Language Reference Manual}, in the following abbreviated as
@b{lrm}, was the blueprint for GNU APL. The @b{lrm} describes vector notation
briefly. 


@c ----------------------------------------------------------------------------
@node Quotes from the lrm
@section Quotes from the lrm

Different aspects of vector notation are distributed over the @b{lrm}. To
get a compressed view of the matter we quote the relevant pieces:

@enumerate
@item @b{lrm page 14}: @i{The juxtaposition of two or more arrays in an
      expression results in a vector whose items are the arrays. Representing
      a vector in this manner is called @b{vector notation}.}

@item @b{lrm page 14}: @i{Characters in a vector consisting only of
characters can be listed within one set of single quotation marks:
      'FACE' ←→ ('F' 'A' 'C' 'E')}

@item @b{lrm page 33}: @i{The hierarchy of binding strengths is listed
      below in descending order.}
@verbatim
        Binding Strength         What Is Bound
     ──────────────────────      ────────────────────────────────────
     1. Brackets                 Brackets to what is on their left
     2. Specification left       Left arrow to what is on its left
     3. Right operand            Dyadic operator to its right operand
     4. Vector                   Array to an array
     5. Left operand             Operator to its left operand
     6. Left argument            Function to its left argument
     7. Right argument           Function to its right argument
     8. Specification right      Left arrow to what is on its right
     ════════════════════════════════════════════════════════════════
@end verbatim
    The binding strength @b{4.} above is the one for vector notation. It
    defines how the items of a vector notation are bound together (as
    opposed to being bound to other tokens). For example: @b{1 2 3[2]}
    becomes @b{1 2 (3[2])} because @b{[2]} binds stronger to @b{3} left
    of it than @b{3} binds to @b{2} left  of it (which then raises
    a @b{RANK ERROR} because @b{3} is a scalar).
@end enumerate


@c ----------------------------------------------------------------------------
@node Some APL2 Tests
@section Some APL2 Tests

Before explaining the details of vector notation in GNU APL we would like
to share the results of some tests performed with IBM APL2 (again the demo
version for PCs).

@node Test 1: Binding strength of []
@subsection Test 1: Binding strength of []

@example
      ⍝ IBM APL2: Binding strength of []...

      ⍝ Test 1a.
      ⍝
      1 2 3[2]
RANK ERROR
      1 2 3[2]
          ^^

      ⍝ Test 1b.
      ⍝
      'a' 'b' 'c'[2]
RANK ERROR
      'ab' 'c'[2]
           ^  ^

      ⍝ Test 1c.
      ⍝
      'abc'[b][2]
b
@end example

@b{Test 1a}: no surprise. @b{[]} has the strongest binding and raises a
@b{RANK ERROR} when indexing the scalar @b{3}.

@b{Test 1b}: no surprise. since it follows suit.

@b{Test 1c}: no surprise for the binding strength. This test confirms
the second quote on page 14 of the lrm.

@node Test 2: Binding strength of the left operand
@subsection Test 2: Binding strength of the left operand

For this test we create a monadic defined function @b{SHOW_NC}, a monadic
defined operator @b{OPER1}, and a dyadic defined operator @b{OPER2}:

@example
      ⍝ display the name class of B¯B
      ⍝
      ∇Z←SHOW_NC B¯B;N
       N←8↑B,':'
       →(¯1 0 1 2 3 4 = ⎕NC B¯B)/I U L V F O ◊ ++
      I: →0, Z←(8↑B), ': Invalid Name'
      U: →0, Z←(8↑B), ': Unused Name'
      L: →0, Z←(8↑B), ': Label'
      V: →0, Z←(8↑B), ': Value', ⍎B¯B
      F: →0, Z←(8↑B), ': Function'
      O: →0, Z←(8↑B), ': Operator'
      ∇

      ⍝ monadic operator
      ⍝
      ∇Z←A (LO OPER1) B
       SHOW_NC 'A'
       SHOW_NC 'LO'
       SHOW_NC 'OPER1'
       SHOW_NC 'B'
      ∇

      ⍝ dyadic operator
      ⍝
      ∇Z←A (LO OPER2 RO) B
       SHOW_NC 'A'
       SHOW_NC 'LO'
       SHOW_NC 'OPER1'
       SHOW_NC 'RO'
       SHOW_NC 'B'
      ∇
@end example

The somewhat strange looking argument name @b{B¯B} of @b{SHOW_NC} instead of
the rather frequent @b{B} was chosen to avoid name conflicts between the
right value argument of @b{SHOW_NC} and the right value argument the
operators under test (like @b{OPER1} or @b{OPER2}). Then:

@example
      ⍝ Test 2a: binding of the left operand (literal LO)
      ⍝
      11 22 33 OPER1 1 2 3
A       : Unused Name
LO      : Value  11 22 33
OPER1   : Operator
B       : Value  1 2 3

      ⍝ Test 2b: binding of the left operand (parser LO)
      ⍝
      (UU VV WW)←11 22 33
      UU VV WW OPER1 1 2 3
A       : Unused Name
LO      : Value  11 22 33
OPER1   : Operator
B       : Value  1 2 3

      ⍝ Test 2c. binding of the left operand (in parentheses)
      ⍝
      11 22 (33 OPER1) 1 2 3
A       : 11
LO      : Value  22 33
OPER1   : Operator
B       : Value  1 2 3

      ⍝ Test 2d. binding of the right operand
      ⍝
      11 22 33 'lop' OPER2 'rop' 1 2 3
A       : Unused Name
LO      : Value  11 22 33 lop
OPER2   : Operator
RO      : Value  rop
B       : Value  1 2 3
@end example

The results are rather interesting. We will later on discuss a potential
difference between vector notation at tokenization time and vector notation
at execution time. The same outcome of tests @b{2a.} and @b{2b.} demonstrates
that IBM APL2 handles both variants in the same fashion (supposedly because
both cases are resolved at execution time).

Test @b{2a.} proves that vector notation binds stronger than the left operand
to its dyadic operator. Test @b{2c.} demonstrates how the binding strengths
can be undermined with parentheses.

Test @b{2d.} proves that the right operand binds stronger than vector notation.
However, this case is interesting from an @b{iso}. Page 126 of @b{iso} gives
the following example for the dyadic rank operator:

@example
      ⍝ example from ISO/IEC 13751 p.126
      ⍝
      0 1 2⌽∘0 1 'ABC'
ABC
BCA
CAB
@end example

Comparing the results of Test @b{2d.} with GNU APL shows how @b{iso} believes
their example should be parsed:

@example
      ⍝ Test 2e. (GNU APL):
      ⍝
      0 1 2(⌽⍤0 1) 'ABC'
ABC
BCA
CAB

      ⍝ Test 2f. (GNU APL):
      ⍝ 
      0 1 2(⌽⍤0) 1 'ABC'
LENGTH ERROR
      0 1 2(⌽⍤0 0 0)1 'ABC'
      ^            ^
@end example

@b{NOTE}: It is not possible to run tests @b{2e.} and @b{2f.} in IBM APL2
directly (because at least in the PC demo version of IBM APL2 the rank
operator @b{⍤} is not implemented. However using our defined function
@b{OPER2} above instead of @b{⍤} shows that IBM APL2 evaluates the ISO
example as @b{0 1 2(⌽∘0) 1 'ABC'} (in accordance with @b{lrm}) and not
as @b{0 1 2(⌽∘0 1) 'ABC'} as @b{iso} suggests.

@b{Takeaways}: the tests above teach us the following:

@itemize
@item Vector notation differs between:
      @itemize
      @item The ISO standard for APL,
      @item IBM APL2, and
      @item APL implementation of other vendors (including GNU APL).
      @end itemize
@item It is therefore impossible to accommodate both the ISO standard and
      IBM APL2 to 100% at the same time, but
      @item GNU APL tries to be as close as possible to one of them,
@item The issues highlighted above can be easily avoided by always using
      parentheses around operators and their operands (even though they
      may appear to be redundant). These parentheses are the (only) key
      to portable APL programs.
@end itemize


@c ----------------------------------------------------------------------------
@node Definitions
@section Definitions

@b{Definition}: A @i{(complete) strand} consists of two (!) or more adjacent
values in a defined function, or in a string being ⍎'ed, or on a line entered
in immediate execution mode. Such values can be literals (like in @b{1 2 3}),
but also values of variables or results of niladic functions. A strand is
always a vector and we call the values in a strand the @i{items of the strand}.
Strand items can be simple (i.e. numbers or single characters) or nested
(strings or other strands). Strand items are always (possibly nested)
APL scalars.

@b{Definition}: A @i{literal strand} is a strand whose items are only
literals (including nested literals like in 1 2 (3 4) 5). Literal strands are
created when an APL expression is tokenized. At that point in time the
literal strand is incomplete and more strand items may be added to it when
the statement is executed.

@b{Note}: Strands are, in general, subject to what is known
as @i{unexpected stranding}, which occurs when an operator accept values as
operands. This is the case for the power operator @b{⍣} and for the rank
operator @b{⍤}; both can have values as right operands. The clean way of
avoiding unexpected stranding is to put the operator and its operands
into parentheses, but the @b{iso} does not do that and in order to make
the examples in @b{iso} work, GNU APL has dedicated code for the rank and
power operators that prevents unexpected stranding for their right operand
and their right function value. For defined operators this is not possible 
because at tokenization time a defined operator is only a symbol whose
name class at execution time is unknown at tokenization time of its caller.
For the same reason. the binding strengths defined above can not cure this
kind of unexpected stranding.

@b{Definition}: An @i{extended literal strand} is a strand whose items are
either literals (including nested literals) or particular simple primitive
function is with particular literal arguments.

@b{Definition}: A @i{mixed strand} is a strand which is not a literal strand.

@b{Definition}: A strand is @i{open} if more items can be appended to it.

@b{Definition}: A strand is @i{closed} if it is not open.

@b{Definition}: A strand is @i{nested} if at lease one of its items is nested.

Examples:

@example
      1 2 3        ⍝ open literal strand

      1 2 (3⍴4) 5     ⍝ open extended literal strand ←→ 1 2 4 4 4 5
      1 2 (3/4) 5     ⍝ open extended literal strand ←→ 1 2 4 4 4 5
     'ab' (⍳3) 'ef'   ⍝ open extended literal strand ←→ 'ab' (1 2 3) 'ef' 

      VAR←1 2 3       ⍝ mixed strand item
      ∇Z←FOO          ⍝ mixed strand item
       Z←1 2 3
      ∇

      1 VAR 3         ⍝ nested open mixed strand
      1 FOO 3         ⍝ nested open mixed strand

      (1 2 3)       ⍝ closed literal strand
      'abc'         ⍝ closed literal strand
@end example


@c ----------------------------------------------------------------------------
@node Unexpected Stranding
@section Unexpected Stranding

@i{Unexpected stranding} (or better: @i{false stranding}) refers to a situation
where the stranding of adjacent values leads to a wrong result. Unexpected
stranding is not unexpected at all, as we will see in this section.

In non-nested APL interpreters, such as APL \360, the binding strength of
strand items was the highest. That was a wise decision which had two
consequences:

@itemize
@item Expressions like @b{1 2 3[2]} were always valid in APL \360 and friends.
      The current APL2 and, (for the sake of compatibility) also GNU APL cases,
      where @b{1 2 3[2]} raises a RANK ERROR, are merely an unfortunate
      consequence of binding expressions in brackets stronger than strand
      items.
@item Similarly, the IBM APL2 design decision to bind right operand stronger 
      than strand items, together with the decision to allow values as operands,
      create the possibility of false stranding.
@end itemize

@b{Corollary 1:} unexpected stranding was not an issue before binding strength
in APL2 was introduced.

@b{Corollary 2:} The only binding strengths for which unexpected stranding
can occur are those for that bind stronger than vector (i.e. strand item),
i.e. @b{1. bracket}, @b{2. specification left}, and @b{3. right operand}.
Of these 3 binding strengths, only @b{3. right operand} conflicts with
strand notation (because neither @b{[...]} nor @b{←} can be strand items.

To go into a little more detail we consider the following token sequences:

@example
                                                        ⍝ at tokenizatio time...
    PRIMITIVE_FUNCTION         VALUE_A VALUE_B          ⍝  ├── expected
    PRIMITIVE_DYADIC_OPERATOR  VALUE_A VALUE_B          ⍝  ├── unexpected
    VALUE                      VALUE_A VALUE_B          ⍝  ├── expected
    SYMBOL                     VALUE_A VALUE_B          ⍝  └── unknown

                                                        ⍝ at execution time...
    VALUE                      VALUE_A VALUE_B          ⍝  ├── expected
    FUNCTION                   VALUE_A VALUE_B          ⍝  ├── expected
    DYADIC_OPERATOR            VALUE_A VALUE_B          ⍝  └── unexpected
@end example

The fact that @b{VALUE_A} and @b{VALUE_B} are adjacent constitutes what is
known as a @i{shift/reduce conflict} in compiler terminology. Let's add
some parentheses to see how these shift/reduce conflicts are, by means
of the APL2 binding strength, supposed to be resolved:

@example
                                                        ⍝ at tokenization time...
    PRIMITIVE_FUNCTION       ( VALUE_A   VALUE_B )      ⍝  ├── expected
  ( PRIMITIVE_DYADIC_OPERATOR  VALUE_A ) VALUE_B        ⍝  ├── unexpected
    VALUE                    ( VALUE_A   VALUE_B )      ⍝  ├── expected
  ? SYMBOL                   ? VALUE_A ? VALUE_B ?      ⍝  └── unknown

                                                        ⍝ at execution time...
    VALUE                    ( VALUE_A   VALUE_B )      ⍝  ├── expected
    FUNCTION                 ( VALUE_A   VALUE_B )      ⍝  ├── expected
  ( DYADIC_OPERATOR            VALUE_A ) VALUE_B        ⍝  └── unexpected
@end example

The first four sequences occurs at tokenization time, the others at
execution time. The significant difference between primitive functions
and primitive operators on the one hand and defined functions and defined
operators on the other hand is that the parser "knows" in the first three
cases if @b{VALUE_A} and @b{VALUE_B} shall be stranded together or not.
In the fourth case the parser cannot decide whether @b{SYMBOL} refers to a
defined function (including functions derived from monadic operators),
to a variable, or to a dyadic defined operator. Therefore the parser can
resolve the first three cases at tokenization time, i.e. bind @b{VALUE_A}
to @b{PRIMITIVE_DYADIC_OPERATOR} or to @b{VALUE} if so, and to @b{VALUE_B}
otherwise. Only the binding of @b{SYMBOL} is left over and will be decided
at execution time.

At execution time, the first thing that happens is that @b{SYMBOL} is resolved
to its current value (so its token class changes from @b{SYMBOL} to @b{VALUE}
(for variables),  or to @b{FUNCTION} (for defined functions), or to
@b{OPERATOR} for defined operators. At the time when the parser sees the
pattern @b{VALUE_A VALUE_B} it can, with a lookahead of 1 token decide
if the token @b{VALUE_A} shall be bound to it @b{PRIMITIVE_DYADIC_OPERATOR},
or stranded to its adjacent value @b{VALUE_B}. To do this a so-called
@b{LALR-1} parser suffices. In simple terms: if the parser sees a
@b{PRIMITIVE_DYADIC_OPERATOR} left of @b{VALUE_A VALUE_B} then it shifts
@b{VALUE_A} (which will later on become the right operand of that operator)
and otherwise it starts or continues strand building with @b{VALUE_A}.
In @b{iso} terminology: the phrase evaluator for pattern @b{VALUE_A VALUE_B}
stops the strand building (in compiler terminology it @b{shifts} @b{VALUE_A})
when the lookahead token is @b{DYADIC_OPERATOR}), and otherwise it continues
the strand building (in compiler terminology it @b{reduces} the pattern
@b{VALUE_A VALUE_B}). 

As an aside: The ISO standard provides examples with strand notation
that would be illegal in IBM APL2 (rank operator, page 126):

@example
      0 1 2 ⌽⍤0 1 'ABC'
ABC
BCA
CAB
@end example

In IBM APL2, the rightmost @b{0} would bind to @b{⍤} and not to the rightmost
@b{1}, like in (parentheses added according to the APL binding strength:

@example
      (0 1 2) (⌽ ⍤ 0) (1 'ABC')
LENGTH ERROR
      0 1 2(⌽⍤0 0 0)1 'ABC'
      ^            ^
@end example

What saves IBM here is only the fact that the two primitive operators
that accept values as right operands (the rank operator ⍤ and the power 
operator ⍣) are not implemented in IBM APL2. In contrast GNU APL has
implemented both operators and the @b{iso} examples work as shown in
the standard.

The same @b{iso} example in Dyalog APL (who seem to be the inventor of these
operators) does not work either (at least not without parentheses:

@example
      ⍝ Dyalog APL
      ⍝
      0 1 2 ⌽⍤0 1 'ABC'
SYNTAX ERROR: Missing right argument
 0 1 2⌽⍤0 1 'ABC'
      ∧

      ⍝ Dyalog APL
      ⍝
      0 1 2 (⌽⍤0 1) 'ABC'
ABC
BCA
CAB
@end example

and @b{1 2 3[2]} is valid in Dyalog APL, which suggests that Dyalog has not
adapted the idea of binding strength but rather remains backward compatible
with APL \360.

@c @node Performance considerations
@subsection Performance considerations

But why bother? The tokenizer could have made its life easier by only
converting (number-) strings and quoted strings to APL values and postpone
the resolution of shift/reduce conflicts to the execution phase. To understand
why it is important to perform as much of the strand building at tokenization
time, have a look at the following, somewhat crude, APL code (which is not
too uncommon in real life):

@example
      ∇Z←FOO N
       Z←100⍴0 ◊ J←1
       LOOP: Z[J]←J+1 2 3 4 5 6 7 8 9 10 ÷ →(100 < J←J+1)/LOOP
      ∇
@end example

There are smarter ways to do that in APL, but before APL2 an APL programmer
could corectly assume that @b{1 2 3 4 5 6 7 8 9 10} will be tokenized into
a strand created at tokenization time, which would have been more efficient
than e.g. storing that term in an APL variable. That strand would have been
a single APL value.

The tokenization of the LOOP line is pretty fast; the tokenizer runs once
over the line and produces a strand as it goes. Most of the tokenization, such
as converting number strings to integers is the same in both cases compared 
below.

Now, if the strand is built at tokenization time, then the strand is created
once and becomes a single APL value.

If, however, the strand is built at execution time, then:

@itemize
@item the tokenization produces 10 scalar APL values,
@item every iteration of the LOOP line creates 9 more values:
      @itemize
      @item 9 10
      @item 8 (9 10)
      @item 7 (8 (9 10))
      @item ...
      @end itemize
@item and each of the 9 steps in the strand creation requires a pattern
      match for the pattern @b{VALUE_A VALUE_B}
@item
@end itemize

The function FOO would therefore create 900 more APL values and perform
900 more pattern matches compared to the strand creation at tokenization time.
GNU APL has further extended this principle to APL values that may not only
contain literals but also a handful of primitives with literal argument.
For example, the APL expression:

@example
      ⍪3 4⍴"Hello"
@end example

would be tokenized into a single APL value. This kind of optimization
can be disabled at compile time.

@c ----------------------------------------------------------------------------
@node GNU APL Vector Notation
@section GNU APL Vector Notation

With the definitions from the previous section we can explain how GNU APL
has implemented vector notation:

1. GNU APL distinguishes between literal and mixed strands. Literal strands
   are combined into a single APL value at tokenization time. This is a
   very effective optimization (which can be disabled at @b{./configure}
   time). In contrast, mixed strands are created at phrase matching time, i.e.
   when an APL expression is evaluated. The tokenization of APL expressions
   happens only once, while phrase matching occurs every time when a
   statement is being executed and therefore moving major parts of the strand
   building from phrase matching time to tokenization time can bring
   substantial performance benefits.

2. The tokenization expands simple APL primitives with literal arguments and
   short results into their literal equivalent. Currently the APL primitives
   @b{N⍴B}, @b{N/B}, @b{N⌿B}, @b{,B}, @b{⍪B}, and @b{⊂B} are handled this way.

3. The tokenization of literal strands has dedicated code for literal
   arguments of the power (@b{⍣}) and rank operators (@b{⍤}) to make make
   it working like described in the ISO standard. This is, in contrast to
   defined operators, possible because the ISO standard defines in very
   detail how its right operand needs to look like and how deviations shall
   be handled (@b{DOMAIN ERROR}, @b{LENGTH ERROR}) etc.

4. The phrase matching of mixed strands builds the strand step-by-step
   according to the following rules:

@itemize
@item two adjacent APL scalars @b{A} and @b{B} create a new open strand
      (with two strand items). Each of the two scalars @b{A} or @b{B} can
      be simple (like numbers or single characters) or nested (like strings
      or closed strands).
@item An APL scalar @b{A} adjacent to an open strand @b{B} prepends @b{A}
      to @b{B}.  That is, @b{B←A, B}.
@item An open strand @b{A} adjacent to an an APL scalar @b{B} appends @b{B} 
      to @b{A}.  That is, @b{A←A, B}.
@item the previous two rules prevent two adjacent open strands, because the
      strand items of a potential open left strand @b{A} are prepended item
      by item to an open right strand @b{B} (which prevents the creation of a
      new open left strand while an open right strand exists,

@item Like the normal APL evaluation, this procedure is preformed from the
      right to the left.
@item if an open strand hits a closing parenthesis then a new phrase matching
      context is created in which the current procedure is abandoned until
      the corresponding opening parentheses has been processed. The result
      of the expression in parentheses is then taken as the next strand item
      for the abandoned strand (if any).
@item Any symbols encountered during this procedure are being resolved into
      scalars (possibly enclosing them if they are not). Niladic functions are
      called to obtain their result.
@item Likewise, any literal strands that were created at tokenization time
      are considered closed (which cause them to be enclosed when prepended
      or appended to an open mixed strand).
@item If a strand is open then any other token (functions, operators,
      brackets, end-of-statement, etc.) closes it. The result is a single
      value for the strand.
@end itemize

5. The operators @b{⍣} (power operator) and @b{⍤} (rank operator), whose
   right operand can be identified by the rules given in @b{iso}) have
   dedicated code that "unstrands" the right operand from a strand that
   may have been mistakenly created in @b{3.} above.

@c @node Example
@subsection Example

Consider the following APL statement:

@example
      ∇FOO
       "Time is now: " ⎕TS
      ∇
@end example

When function @b{FOO} is created its single function line is being tokenized.
The tokenization of the function line sees the string "Time is now: " which
is a literal token with 13 literal items (= single characters). The result of
the tokenization is therefore 2 tokens: the string and the system function
@b{⎕TS}. Naturally @b{⎕TS} is the time stamp when function @b{FOO} is called
and not the time when function @b{FOO} is created. Therefore the strand
building stops at this point and leaves the string as an (incomplete) strand
because the strand building can not predict when @b{FOO} will be called. At
this point the incomplete strand and the system function @b{⎕TS} will become
adjacent tokens in the function body of @b{FOO}.

Later on (or maybe never) @b{FOO} will be called. The execution of @b{FOO}
(which goes from  right to left) first sees the niladic token @b{⎕TS}, and
evaluates it to an integer vector @b{B}, say @b{2026 1 1 18 5 6 194}. Values
returned from functions, values of variables, and the literal strands from
the tokenization are all closed and incomplete, and so is @b{B} at this point.
The execution then fetches the next token @b{A}, which is the string. The
two adjacent tokens @b{A} and @b{B} now match the pattern @b{A B} in the
phrase table of the interpreter. The corresponding phrase evaluator for
that pattern then prepends the items in @b{A} to @b{B} and replaces the
two tokens @b{A} and @b{B} with the new (single) token for the modified @b{B}.
The next token is then an @b{END} (for end of staement) token which causes
@b{B} to be printed.

@b{A final remark}: It should be noted, that the phrase table defined 
in @b{iso} is somewhat incomplete. In particular the pattern @b{A B}
above (which performs the strand building in the execution phase) is
not present in @b{iso}. Also, @b{iso} defines @b{END} tokens as used
above, but no pattern in which they appear. All this has the unfortunate
consequence that vector notation is provided by every APL interpreter,
but the details of it differ considerably. For example, @b{1 2 3[2]} could
be @b{2} in one interpreter, but raise a @b{RANK ERROR} in another. Even
for different interpreters of the same vendor. The only reliable way to
write portable APL programs is to use (otherwise redundant) parentheses,
in particular around operators and their operands, and around strands that
are being indexed.


@c ============================================================================
@node GNU APL Project Structure
@chapter GNU APL Project Structure

The top-level directory of the GNU APL project has a structure like this:

@example
    apl-2.0/
    ├── build
    ├── debian
    ├── doc
    ├── erlang
    ├── gnu-apl.d
    ├── HOWTOs
    ├── html
    ├── m4
    ├── rpm
    ├── src
    ├── support-files
    ├── tools
    ├── websock
    ├── workspaces
    ├── wslib2
    ├── wslib3
    ├── wslib4
    └── wslib5
@end example

Most of these directories are contained in the project tar file
(e.g. apl-2.0.tar.gz) and some are created during the build of GNU APL. Only
the directories in the top-level are shown above.

The GNU APL project follows the standard layout for autoconf/automake projects.
The top-level itself contains the files needed by autoconf/automake and a
handful of README-XXX files that might be of interest when building GNU APL.

The sub-directories are (in order of decreasing relevance):

@node The sub-directory src
@section The sub-directory src

The sub-directory src contains all C++ source files that are needed to build
either the GNU APL interpreter itself, or libapl (a library that contains the
essential part of GNU APL, but without the interactive REPL loop of the
interpreter). During the build, the object files produced by the compiler are
also placed in this directory.

The sub-directory src contains further sub-directories for components of GNU APL
that are optional and typically require the installation of additional
libraries on the build machine. The ./configure script in the top-level
directory determines, which optional components shall be included in the build.

@example
    apl-2.0/
    ├── src
        ├── APs                     APs for ⎕SVO and friends
        ├── emacs_mode              dito
        ├── Gtk                     ⎕GTK (graphical user interface)
        ├── native                  templates for native functions
        ├── sql                     ⎕SQL
        ├── testcases               dito
@end example

GNU APL comes with an automated test suite, the testcases are
contained in sub-directory src/testcases

@node The sub-directory support-files
@section The sub-directory support-files

The sub-directory support-file contains a number of configuration files that aim
at making a standard keyboard produce APL characters. Details are explained in
top-level file README-3-keyboard.

@node The sub-directory doc
@section The sub-directory doc

The sub-directory doc is used to create @b{this} info file in different file
formats (.info, .html).

@node The sub-directory HOWTOs
@section The sub-directory HOWTOs

The sub-directory HOWTOs contains some documents that try to explain how some
of the non-standard features of GNU APL are intended to be used. These
documents were written for two different audiences: APL programmers and
GNU APL hackers. The documents for APL programmers are, by default, installed
in directory /usr/local/share/doc/apl by @b{make install} while the documents
for GNU APL hackers (i.e. C++ programmers or trouble-shooters) are not
installed by @b{make install}, but rather remain in the source tree.

@node The sub-directories workspaces and wslib2 ... wslib5
@section The sub-directories workspaces and wslib2 ... wslib5

These directories (and their content) are, by default, copied to
/usr/local/lib/apl by @b{make install}. Some sub-directories are empty while
others contain small workspaces or libraries for various purposes (mostly
demonstrating the use of some non-standard APL features).

@node The sub-directory m4
@section The sub-directory m4

The sub-directory m4 contains macros used by autoconf and automake. In most
cases the top-level ./configure script can be used as is. If a platform on
which GNU APL shall be built differs too much from the platform on which the
GNU APL project tar file (e.g. apl-2.0.tar.gz) was created, then it might
be necessary to re-create the ./configure script on the platform.

Re-creating the ./configure script requires:

@itemize
@item the installation of autoconf,
@item the installation of automake,
@item the installation of libtool,
@item maybe running aclocal, libtoolize, and other scripts, and finally
@item running autoreconf
@end itemize

@node The sub-directories debian and debian_tmp
@section The sub-directories debian and debian_tmp

These sub-directories are templates for building Debian
packages that contain GNU APL. Sub-directory debian_tmp is initially empty and
acts as a scratch directory inside which the interpreter is being compiled
when a Debian package is built.

@node The sub-directory tools
@section The sub-directory tools

GNU APL uses generated C++ code. The sub-directory tools contains the source
code for the code generator (phrase_gen) and also some other small tools for
various purposes. These tools are typically used by GNU APL hackers and are
therefore not installed by @b{make install}.

@c ============================================================================
@node Libraries Contributed by GNU APL Users
@chapter Libraries Contributed by GNU APL Users

Since the first release of GNU APL, a number of libraries, but also other
developments, have been made by users of GNU APL. Below are links to these
contributions.

@node The GNU APL Community Webpage
@section The GNU APL Community Webpage

The GNU APL project maintains a web page that lists contributions made by
GNU APL users:

@uref{http://www.gnu.org/software/apl/Community.html}

The contributions listed there were collected from emails sent to
bug-apl@@gnu.org

Since the list was started long after bug-apl@@gnu.org was created. It is
therefore quite possible that your contribution is missing. That does not
mean that your contribution was not considered worthwhile to be listed but
more likely that it was overlooked. Just send an email to bug-apl@@gnu.org
to fix that.

@node Core Libraries
@section Core Libraries

There is a small number of libraries that are considered "core" APL because,
for example, the ISO standard requires their functionality, or because other
APL interpreters normally support them.

These libraries are linked via the GNU APL Community web page. In addition
they are stored in the GNU APL SVN repository and also included in the
GNU APL packages that follows their contribution.

Currently the following core libraries exist:

GNU APL Emacs mode     (for Emacs users)           by Elias Mårtenson

SQL interface          (SQLite and Postgres)       by Elias Mårtenson

Component File System  (as required by ISO 13751)  by David B. Lamkins

Component File System  (as required by ISO 13751)  by Blake McBride

SQL

@c ============================================================================
@node Licenses for this GNU APL Manual and for GNU APL
@chapter Licenses for this GNU APL Manual and for GNU APL

This manual for GNU APL is licensed under the GNU Free Documentation
License, *Note Section 8.1:: below.

GNU APL itself is licensed under the GNU Public License version 3 or later
(aka GPLv3+), *Note Section 8.2:: below.

@node The GNU Free Documentation License
@section The GNU Free Documentation License

@c The GNU Free Documentation License.
@center Version 1.3, 3 November 2008

@display
Copyright @copyright{} 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
@uref{http://fsf.org/}

Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
@end display

@enumerate 0
@item
PREAMBLE

The purpose of this License is to make a manual, textbook, or other
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assure everyone the effective freedom to copy and redistribute it,
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Secondarily, this License preserves for the author and publisher a way
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We have designed this License in order to use it for manuals for free
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@item
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The Document may include Warranty Disclaimers next to the notice which
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@item
VERBATIM COPYING

You may copy and distribute the Document in any medium, either
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You may also lend copies, under the same conditions stated above, and
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@item
COPYING IN QUANTITY

If you publish printed copies (or copies in media that commonly have
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If the required texts for either cover are too voluminous to fit
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If you publish or distribute Opaque copies of the Document numbering
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@item
MODIFICATIONS

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@enumerate A
@item
Use in the Title Page (and on the covers, if any) a title distinct
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@item
List on the Title Page, as authors, one or more persons or entities
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@item
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@item
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@item
Add an appropriate copyright notice for your modifications
adjacent to the other copyright notices.

@item
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@item
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@item
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@item
Preserve the network location, if any, given in the Document for
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@item
For any section Entitled ``Acknowledgements'' or ``Dedications'', Preserve
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@item
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@item
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@item
Do not retitle any existing section to be Entitled ``Endorsements'' or
to conflict in title with any Invariant Section.

@item
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@end enumerate

If the Modified Version includes new front-matter sections or
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@item
COMBINING DOCUMENTS

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Make the same adjustment to the section titles in the list of
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In the combination, you must combine any sections Entitled ``History''
in the various original documents, forming one section Entitled
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and any sections Entitled ``Dedications''. You must delete all
sections Entitled ``Endorsements.''

@item
COLLECTIONS OF DOCUMENTS

You may make a collection consisting of the Document and other documents
released under this License, and replace the individual copies of this
License in the various documents with a single copy that is included in
the collection, provided that you follow the rules of this License for
verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute
it individually under this License, provided you insert a copy of this
License into the extracted document, and follow this License in all
other respects regarding verbatim copying of that document.

@item
AGGREGATION WITH INDEPENDENT WORKS

A compilation of the Document or its derivatives with other separate
and independent documents or works, in or on a volume of a storage or
distribution medium, is called an ``aggregate'' if the copyright
resulting from the compilation is not used to limit the legal rights
of the compilation's users beyond what the individual works permit.
When the Document is included in an aggregate, this License does not
apply to the other works in the aggregate which are not themselves
derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these
copies of the Document, then if the Document is less than one half of
the entire aggregate, the Document's Cover Texts may be placed on
covers that bracket the Document within the aggregate, or the
electronic equivalent of covers if the Document is in electronic form.
Otherwise they must appear on printed covers that bracket the whole
aggregate.

@item
TRANSLATION

Translation is considered a kind of modification, so you may
distribute translations of the Document under the terms of section 4.
Replacing Invariant Sections with translations requires special
permission from their copyright holders, but you may include
translations of some or all Invariant Sections in addition to the
original versions of these Invariant Sections. You may include a
translation of this License, and all the license notices in the
Document, and any Warranty Disclaimers, provided that you also include
the original English version of this License and the original versions
of those notices and disclaimers. In case of a disagreement between
the translation and the original version of this License or a notice
or disclaimer, the original version will prevail.

If a section in the Document is Entitled ``Acknowledgements'',
``Dedications'', or ``History'', the requirement (section 4) to Preserve
its Title (section 1) will typically require changing the actual
title.

@item
TERMINATION

You may not copy, modify, sublicense, or distribute the Document
except as expressly provided under this License. Any attempt
otherwise to copy, modify, sublicense, or distribute it is void, and
will automatically terminate your rights under this License.

However, if you cease all violation of this License, then your license
from a particular copyright holder is reinstated (a) provisionally,
unless and until the copyright holder explicitly and finally
terminates your license, and (b) permanently, if the copyright holder
fails to notify you of the violation by some reasonable means prior to
60 days after the cessation.

Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from that
copyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.

Termination of your rights under this section does not terminate the
licenses of parties who have received copies or rights from you under
this License. If your rights have been terminated and not permanently
reinstated, receipt of a copy of some or all of the same material does
not give you any rights to use it.

@item
FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions
of the GNU Free Documentation License from time to time. Such new
versions will be similar in spirit to the present version, but may
differ in detail to address new problems or concerns. See
@uref{http://www.gnu.org/copyleft/}.

Each version of the License is given a distinguishing version number.
If the Document specifies that a particular numbered version of this
License ``or any later version'' applies to it, you have the option of
following the terms and conditions either of that specified version or
of any later version that has been published (not as a draft) by the
Free Software Foundation. If the Document does not specify a version
number of this License, you may choose any version ever published (not
as a draft) by the Free Software Foundation. If the Document
specifies that a proxy can decide which future versions of this
License can be used, that proxy's public statement of acceptance of a
version permanently authorizes you to choose that version for the
Document.

@item
RELICENSING

``Massive Multiauthor Collaboration Site'' (or ``MMC Site'') means any
World Wide Web server that publishes copyrightable works and also
provides prominent facilities for anybody to edit those works. A
public wiki that anybody can edit is an example of such a server. A
``Massive Multiauthor Collaboration'' (or ``MMC'') contained in the
site means any set of copyrightable works thus published on the MMC
site.

``CC-BY-SA'' means the Creative Commons Attribution-Share Alike 3.0
license published by Creative Commons Corporation, a not-for-profit
corporation with a principal place of business in San Francisco,
California, as well as future copyleft versions of that license
published by that same organization.

``Incorporate'' means to publish or republish a Document, in whole or
in part, as part of another Document.

An MMC is ``eligible for relicensing'' if it is licensed under this
License, and if all works that were first published under this License
somewhere other than this MMC, and subsequently incorporated in whole
or in part into the MMC, (1) had no cover texts or invariant sections,
and (2) were thus incorporated prior to November 1, 2008.

The operator of an MMC Site may republish an MMC contained in the site
under CC-BY-SA on the same site at any time before August 1, 2009,
provided the MMC is eligible for relicensing.

@end enumerate

@page
@heading ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and
license notices just after the title page:

@smallexample
@group
  Copyright (C)  @var{year}  @var{your name}.
  Permission is granted to copy, distribute and/or modify this document
  under the terms of the GNU Free Documentation License, Version 1.3
  or any later version published by the Free Software Foundation;
  with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
  Texts. A copy of the license is included in the section entitled ``GNU
  Free Documentation License''.
@end group
@end smallexample

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
replace the ``with@dots{}Texts.''@: line with this:

@smallexample
@group
    with the Invariant Sections being @var{list their titles}, with
    the Front-Cover Texts being @var{list}, and with the Back-Cover Texts
    being @var{list}.
@end group
@end smallexample

If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.

If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License,
to permit their use in free software.

@node The GNU GENERAL PUBLIC LICENSE
@section The GNU GENERAL PUBLIC LICENSE

                    GNU GENERAL PUBLIC LICENSE
                       Version 3, 29 June 2007

 Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
 Everyone is permitted to copy and distribute verbatim copies
 of this license document, but changing it is not allowed.

                            Preamble

  The GNU General Public License is a free, copyleft license for
software and other kinds of works.

  The licenses for most software and other practical works are designed
to take away your freedom to share and change the works. By contrast,
the GNU General Public License is intended to guarantee your freedom to
share and change all versions of a program--to make sure it remains free
software for all its users. We, the Free Software Foundation, use the
GNU General Public License for most of our software; it applies also to
any other work released this way by its authors. You can apply it to
your programs, too.

  When we speak of free software, we are referring to freedom, not
price. Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
them if you wish), that you receive source code or can get it if you
want it, that you can change the software or use pieces of it in new
free programs, and that you know you can do these things.

  To protect your rights, we need to prevent others from denying you
these rights or asking you to surrender the rights. Therefore, you have
certain responsibilities if you distribute copies of the software, or if
you modify it: responsibilities to respect the freedom of others.

  For example, if you distribute copies of such a program, whether
gratis or for a fee, you must pass on to the recipients the same
freedoms that you received. You must make sure that they, too, receive
or can get the source code. And you must show them these terms so they
know their rights.

  Developers that use the GNU GPL protect your rights with two steps:
(1) assert copyright on the software, and (2) offer you this License
giving you legal permission to copy, distribute and/or modify it.

  For the developers' and authors' protection, the GPL clearly explains
that there is no warranty for this free software. For both users' and
authors' sake, the GPL requires that modified versions be marked as
changed, so that their problems will not be attributed erroneously to
authors of previous versions.

  Some devices are designed to deny users access to install or run
modified versions of the software inside them, although the manufacturer
can do so. This is fundamentally incompatible with the aim of
protecting users' freedom to change the software. The systematic
pattern of such abuse occurs in the area of products for individuals to
use, which is precisely where it is most unacceptable. Therefore, we
have designed this version of the GPL to prohibit the practice for those
products. If such problems arise substantially in other domains, we
stand ready to extend this provision to those domains in future versions
of the GPL, as needed to protect the freedom of users.

  Finally, every program is threatened constantly by software patents.
States should not allow patents to restrict development and use of
software on general-purpose computers, but in those that do, we wish to
avoid the special danger that patents applied to a free program could
make it effectively proprietary. To prevent this, the GPL assures that
patents cannot be used to render the program non-free.

  The precise terms and conditions for copying, distribution and
modification follow.

                       TERMS AND CONDITIONS

  0. Definitions.

  "This License" refers to version 3 of the GNU General Public License.

  "Copyright" also means copyright-like laws that apply to other kinds of
works, such as semiconductor masks.

  "The Program" refers to any copyrightable work licensed under this
License. Each licensee is addressed as "you". "Licensees" and
"recipients" may be individuals or organizations.

  To "modify" a work means to copy from or adapt all or part of the work
in a fashion requiring copyright permission, other than the making of an
exact copy. The resulting work is called a "modified version" of the
earlier work or a work "based on" the earlier work.

  A "covered work" means either the unmodified Program or a work based
on the Program.

  To "propagate" a work means to do anything with it that, without
permission, would make you directly or secondarily liable for
infringement under applicable copyright law, except executing it on a
computer or modifying a private copy. Propagation includes copying,
distribution (with or without modification), making available to the
public, and in some countries other activities as well.

  To "convey" a work means any kind of propagation that enables other
parties to make or receive copies. Mere interaction with a user through
a computer network, with no transfer of a copy, is not conveying.

  An interactive user interface displays "Appropriate Legal Notices"
to the extent that it includes a convenient and prominently visible
feature that (1) displays an appropriate copyright notice, and (2)
tells the user that there is no warranty for the work (except to the
extent that warranties are provided), that licensees may convey the
work under this License, and how to view a copy of this License. If
the interface presents a list of user commands or options, such as a
menu, a prominent item in the list meets this criterion.

  1. Source Code.

  The "source code" for a work means the preferred form of the work
for making modifications to it. "Object code" means any non-source
form of a work.

  A "Standard Interface" means an interface that either is an official
standard defined by a recognized standards body, or, in the case of
interfaces specified for a particular programming language, one that
is widely used among developers working in that language.

  The "System Libraries" of an executable work include anything, other
than the work as a whole, that (a) is included in the normal form of
packaging a Major Component, but which is not part of that Major
Component, and (b) serves only to enable use of the work with that
Major Component, or to implement a Standard Interface for which an
implementation is available to the public in source code form. A
"Major Component", in this context, means a major essential component
(kernel, window system, and so on) of the specific operating system
(if any) on which the executable work runs, or a compiler used to
produce the work, or an object code interpreter used to run it.

  The "Corresponding Source" for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts to
control those activities. However, it does not include the work's
System Libraries, or general-purpose tools or generally available free
programs which are used unmodified in performing those activities but
which are not part of the work. For example, Corresponding Source
includes interface definition files associated with source files for
the work, and the source code for shared libraries and dynamically
linked subprograms that the work is specifically designed to require,
such as by intimate data communication or control flow between those
subprograms and other parts of the work.

  The Corresponding Source need not include anything that users
can regenerate automatically from other parts of the Corresponding
Source.

  The Corresponding Source for a work in source code form is that
same work.

  2. Basic Permissions.

  All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
conditions are met. This License explicitly affirms your unlimited
permission to run the unmodified Program. The output from running a
covered work is covered by this License only if the output, given its
content, constitutes a covered work. This License acknowledges your
rights of fair use or other equivalent, as provided by copyright law.

  You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise remains
in force. You may convey covered works to others for the sole purpose
of having them make modifications exclusively for you, or provide you
with facilities for running those works, provided that you comply with
the terms of this License in conveying all material for which you do
not control copyright. Those thus making or running the covered works
for you must do so exclusively on your behalf, under your direction
and control, on terms that prohibit them from making any copies of
your copyrighted material outside their relationship with you.

  Conveying under any other circumstances is permitted solely under
the conditions stated below. Sublicensing is not allowed; section 10
makes it unnecessary.

  3. Protecting Users' Legal Rights From Anti-Circumvention Law.

  No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under article
11 of the WIPO copyright treaty adopted on 20 December 1996, or
similar laws prohibiting or restricting circumvention of such
measures.

  When you convey a covered work, you waive any legal power to forbid
circumvention of technological measures to the extent such circumvention
is effected by exercising rights under this License with respect to
the covered work, and you disclaim any intention to limit operation or
modification of the work as a means of enforcing, against the work's
users, your or third parties' legal rights to forbid circumvention of
technological measures.

  4. Conveying Verbatim Copies.

  You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
appropriately publish on each copy an appropriate copyright notice;
keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the code;
keep intact all notices of the absence of any warranty; and give all
recipients a copy of this License along with the Program.

  You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.

  5. Conveying Modified Source Versions.

  You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4, provided that you also meet all of these conditions:

    a) The work must carry prominent notices stating that you modified
    it, and giving a relevant date.

    b) The work must carry prominent notices stating that it is
    released under this License and any conditions added under section
    7. This requirement modifies the requirement in section 4 to
    "keep intact all notices".

    c) You must license the entire work, as a whole, under this
    License to anyone who comes into possession of a copy. This
    License will therefore apply, along with any applicable section 7
    additional terms, to the whole of the work, and all its parts,
    regardless of how they are packaged. This License gives no
    permission to license the work in any other way, but it does not
    invalidate such permission if you have separately received it.

    d) If the work has interactive user interfaces, each must display
    Appropriate Legal Notices; however, if the Program has interactive
    interfaces that do not display Appropriate Legal Notices, your
    work need not make them do so.

  A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work,
and which are not combined with it such as to form a larger program,
in or on a volume of a storage or distribution medium, is called an
"aggregate" if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit. Inclusion of a covered work
in an aggregate does not cause this License to apply to the other
parts of the aggregate.

  6. Conveying Non-Source Forms.

  You may convey a covered work in object code form under the terms
of sections 4 and 5, provided that you also convey the
machine-readable Corresponding Source under the terms of this License,
in one of these ways:

    a) Convey the object code in, or embodied in, a physical product
    (including a physical distribution medium), accompanied by the
    Corresponding Source fixed on a durable physical medium
    customarily used for software interchange.

    b) Convey the object code in, or embodied in, a physical product
    (including a physical distribution medium), accompanied by a
    written offer, valid for at least three years and valid for as
    long as you offer spare parts or customer support for that product
    model, to give anyone who possesses the object code either (1) a
    copy of the Corresponding Source for all the software in the
    product that is covered by this License, on a durable physical
    medium customarily used for software interchange, for a price no
    more than your reasonable cost of physically performing this
    conveying of source, or (2) access to copy the
    Corresponding Source from a network server at no charge.

    c) Convey individual copies of the object code with a copy of the
    written offer to provide the Corresponding Source. This
    alternative is allowed only occasionally and noncommercially, and
    only if you received the object code with such an offer, in accord
    with subsection 6b.

    d) Convey the object code by offering access from a designated
    place (gratis or for a charge), and offer equivalent access to the
    Corresponding Source in the same way through the same place at no
    further charge. You need not require recipients to copy the
    Corresponding Source along with the object code. If the place to
    copy the object code is a network server, the Corresponding Source
    may be on a different server (operated by you or a third party)
    that supports equivalent copying facilities, provided you maintain
    clear directions next to the object code saying where to find the
    Corresponding Source. Regardless of what server hosts the
    Corresponding Source, you remain obligated to ensure that it is
    available for as long as needed to satisfy these requirements.

    e) Convey the object code using peer-to-peer transmission, provided
    you inform other peers where the object code and Corresponding
    Source of the work are being offered to the general public at no
    charge under subsection 6d.

  A separable portion of the object code, whose source code is excluded
from the Corresponding Source as a System Library, need not be
included in conveying the object code work.

  A "User Product" is either (1) a "consumer product", which means any
tangible personal property which is normally used for personal, family,
or household purposes, or (2) anything designed or sold for incorporation
into a dwelling. In determining whether a product is a consumer product,
doubtful cases shall be resolved in favor of coverage. For a particular
product received by a particular user, "normally used" refers to a
typical or common use of that class of product, regardless of the status
of the particular user or of the way in which the particular user
actually uses, or expects or is expected to use, the product. A product
is a consumer product regardless of whether the product has substantial
commercial, industrial or non-consumer uses, unless such uses represent
the only significant mode of use of the product.

  "Installation Information" for a User Product means any methods,
procedures, authorization keys, or other information required to install
and execute modified versions of a covered work in that User Product from
a modified version of its Corresponding Source. The information must
suffice to ensure that the continued functioning of the modified object
code is in no case prevented or interfered with solely because
modification has been made.

  If you convey an object code work under this section in, or with, or
specifically for use in, a User Product, and the conveying occurs as
part of a transaction in which the right of possession and use of the
User Product is transferred to the recipient in perpetuity or for a
fixed term (regardless of how the transaction is characterized), the
Corresponding Source conveyed under this section must be accompanied
by the Installation Information. But this requirement does not apply
if neither you nor any third party retains the ability to install
modified object code on the User Product (for example, the work has
been installed in ROM).

  The requirement to provide Installation Information does not include a
requirement to continue to provide support service, warranty, or updates
for a work that has been modified or installed by the recipient, or for
the User Product in which it has been modified or installed. Access to a
network may be denied when the modification itself materially and
adversely affects the operation of the network or violates the rules and
protocols for communication across the network.

  Corresponding Source conveyed, and Installation Information provided,
in accord with this section must be in a format that is publicly
documented (and with an implementation available to the public in
source code form), and must require no special password or key for
unpacking, reading or copying.

  7. Additional Terms.

  "Additional permissions" are terms that supplement the terms of this
License by making exceptions from one or more of its conditions.
Additional permissions that are applicable to the entire Program shall
be treated as though they were included in this License, to the extent
that they are valid under applicable law. If additional permissions
apply only to part of the Program, that part may be used separately
under those permissions, but the entire Program remains governed by
this License without regard to the additional permissions.

  When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part of
it. (Additional permissions may be written to require their own
removal in certain cases when you modify the work.)  You may place
additional permissions on material, added by you to a covered work,
for which you have or can give appropriate copyright permission.

  Notwithstanding any other provision of this License, for material you
add to a covered work, you may (if authorized by the copyright holders of
that material) supplement the terms of this License with terms:

    a) Disclaiming warranty or limiting liability differently from the
    terms of sections 15 and 16 of this License; or

    b) Requiring preservation of specified reasonable legal notices or
    author attributions in that material or in the Appropriate Legal
    Notices displayed by works containing it; or

    c) Prohibiting misrepresentation of the origin of that material, or
    requiring that modified versions of such material be marked in
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    d) Limiting the use for publicity purposes of names of licensors or
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    e) Declining to grant rights under trademark law for use of some
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    f) Requiring indemnification of licensors and authors of that
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  All other non-permissive additional terms are considered "further
restrictions" within the meaning of section 10. If the Program as you
received it, or any part of it, contains a notice stating that it is
governed by this License along with a term that is a further
restriction, you may remove that term. If a license document contains
a further restriction but permits relicensing or conveying under this
License, you may add to a covered work material governed by the terms
of that license document, provided that the further restriction does
not survive such relicensing or conveying.

  If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the
additional terms that apply to those files, or a notice indicating
where to find the applicable terms.

  Additional terms, permissive or non-permissive, may be stated in the
form of a separately written license, or stated as exceptions;
the above requirements apply either way.

  8. Termination.

  You may not propagate or modify a covered work except as expressly
provided under this License. Any attempt otherwise to propagate or
modify it is void, and will automatically terminate your rights under
this License (including any patent licenses granted under the third
paragraph of section 11).

  However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly and
finally terminates your license, and (b) permanently, if the copyright
holder fails to notify you of the violation by some reasonable means
prior to 60 days after the cessation.

  Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from that
copyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.

  Termination of your rights under this section does not terminate the
licenses of parties who have received copies or rights from you under
this License. If your rights have been terminated and not permanently
reinstated, you do not qualify to receive new licenses for the same
material under section 10.

  9. Acceptance Not Required for Having Copies.

  You are not required to accept this License in order to receive or
run a copy of the Program. Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer transmission
to receive a copy likewise does not require acceptance. However,
nothing other than this License grants you permission to propagate or
modify any covered work. These actions infringe copyright if you do
not accept this License. Therefore, by modifying or propagating a
covered work, you indicate your acceptance of this License to do so.

  10. Automatic Licensing of Downstream Recipients.

  Each time you convey a covered work, the recipient automatically
receives a license from the original licensors, to run, modify and
propagate that work, subject to this License. You are not responsible
for enforcing compliance by third parties with this License.

  An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a covered
work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or could
give under the previous paragraph, plus a right to possession of the
Corresponding Source of the work from the predecessor in interest, if
the predecessor has it or can get it with reasonable efforts.

  You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you may
not impose a license fee, royalty, or other charge for exercise of
rights granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that
any patent claim is infringed by making, using, selling, offering for
sale, or importing the Program or any portion of it.

  11. Patents.

  A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based. The
work thus licensed is called the contributor's "contributor version".

  A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a
consequence of further modification of the contributor version. For
purposes of this definition, "control" includes the right to grant
patent sublicenses in a manner consistent with the requirements of
this License.

  Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.

  In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement). To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.

  If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) arrange to deprive yourself of the benefit of the
patent license for this particular work, or (3) arrange, in a manner
consistent with the requirements of this License, to extend the patent
license to downstream recipients. "Knowingly relying" means you have
actual knowledge that, but for the patent license, your conveying the
covered work in a country, or your recipient's use of the covered work
in a country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.

  If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate, modify
or convey a specific copy of the covered work, then the patent license
you grant is automatically extended to all recipients of the covered
work and works based on it.

  A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License. You may not convey a covered
work if you are a party to an arrangement with a third party that is
in the business of distributing software, under which you make payment
to the third party based on the extent of your activity of conveying
the work, and under which the third party grants, to any of the
parties who would receive the covered work from you, a discriminatory
patent license (a) in connection with copies of the covered work
conveyed by you (or copies made from those copies), or (b) primarily
for and in connection with specific products or compilations that
contain the covered work, unless you entered into that arrangement,
or that patent license was granted, prior to 28 March 2007.

  Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.

  12. No Surrender of Others' Freedom.

  If conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot convey a
covered work so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you may
not convey it at all. For example, if you agree to terms that obligate you
to collect a royalty for further conveying from those to whom you convey
the Program, the only way you could satisfy both those terms and this
License would be to refrain entirely from conveying the Program.

  13. Use with the GNU Affero General Public License.

  Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a single
combined work, and to convey the resulting work. The terms of this
License will continue to apply to the part which is the covered work,
but the special requirements of the GNU Affero General Public License,
section 13, concerning interaction through a network will apply to the
combination as such.

  14. Revised Versions of this License.

  The Free Software Foundation may publish revised and/or new versions of
the GNU General Public License from time to time. Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.

  Each version is given a distinguishing version number. If the
Program specifies that a certain numbered version of the GNU General
Public License "or any later version" applies to it, you have the
option of following the terms and conditions either of that numbered
version or of any later version published by the Free Software
Foundation. If the Program does not specify a version number of the
GNU General Public License, you may choose any version ever published
by the Free Software Foundation.

  If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that proxy's
public statement of acceptance of a version permanently authorizes you
to choose that version for the Program.

  Later license versions may give you additional or different
permissions. However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
later version.

  15. Disclaimer of Warranty.

  THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

  16. Limitation of Liability.

  IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
SUCH DAMAGES.

  17. Interpretation of Sections 15 and 16.

  If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely approximates
an absolute waiver of all civil liability in connection with the
Program, unless a warranty or assumption of liability accompanies a
copy of the Program in return for a fee.

                     END OF TERMS AND CONDITIONS

            How to Apply These Terms to Your New Programs

  If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.

  To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.

    <one line to give the program's name and a brief idea of what it does.>
    Copyright (C) <year>  <name of author>

    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program. If not, see <http://www.gnu.org/licenses/>.

Also add information on how to contact you by electronic and paper mail.

  If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:

    <program>  Copyright (C) <year>  <name of author>
    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
    This is free software, and you are welcome to redistribute it
    under certain conditions; type `show c' for details.

The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".

  You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<http://www.gnu.org/licenses/>.

  The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<http://www.gnu.org/philosophy/why-not-lgpl.html>.

@c ispell-local-pdict: "ispell-dict"
@c End:

@c ============================================================================
@c ====== Part 6: The End of the Document =====================================
@c ============================================================================

@c node Index
@c @unnumbered Index
@c @printindex cp

@bye

