/usr/share/genius/help/ru/genius.xml is in genius-common 1.0.23-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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<!DOCTYPE book PUBLIC "-//OASIS//DTD DocBook XML V4.1.2//EN" "http://www.oasis-open.org/docbook/xml/4.1.2/docbookx.dtd" [
<!ENTITY app "<application>Genius Mathematics Tool</application>">
<!ENTITY appname "Genius">
<!ENTITY appversion "1.0.22">
<!ENTITY date "September 2016">
<!ENTITY legal SYSTEM "legal.xml">
<!ENTITY manrevision "0.2">
<!ENTITY lt "<">
<!ENTITY gt ">">
<!ENTITY le "≤">
<!ENTITY ge "≥">
<!ENTITY lsquo "‘">
<!ENTITY rsquo "’">
<!--<!ENTITY gel-function-list SYSTEM "gel-function-list.xml">-->]>
<!--
(Do not remove this comment block.)
Maintained by the GNOME Documentation Project
http://developer.gnome.org/projects/gdp
Template version: 2.0 beta
Template last modified Apr 11, 2002
-->
<!-- =============Document Header ============================= -->
<book id="index" lang="ru">
<!-- please do not change the id; for translations, change lang to -->
<!-- appropriate code -->
<bookinfo>
<abstract role="description"><para>Руководство по Математическому инструменту Genius.</para></abstract>
<title>Руководство пользователя Genius</title>
<copyright>
<year>1997-2016</year>
<holder>Jiří (George) Lebl</holder>
</copyright>
<copyright>
<year>2004</year>
<holder>Kai Willadsen</holder>
</copyright><copyright><year>2012</year><holder>Алексей Кабанов (ak099@mail.ru)</holder></copyright>
<!-- translators: uncomment this:
<copyright>
<year>2002</year>
<holder>ME-THE-TRANSLATOR (Latin translation)</holder>
</copyright>
-->
<publisher>
<publishername/>
</publisher>
<legalnotice id="legalnotice">
<para>
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation
License (GFDL), Version 1.1 or any later version published
by the Free Software Foundation with no Invariant Sections,
no Front-Cover Texts, and no Back-Cover Texts. You can find
a copy of the GFDL at this <ulink type="help" url="ghelp:fdl">link</ulink> or in the file COPYING-DOCS
distributed with this manual.
</para>
<para> This manual is part of a collection of GNOME manuals
distributed under the GFDL. If you want to distribute this
manual separately from the collection, you can do so by
adding a copy of the license to the manual, as described in
section 6 of the license.
</para>
<para>
Many of the names used by companies to distinguish their
products and services are claimed as trademarks. Where those
names appear in any GNOME documentation, and the members of
the GNOME Documentation Project are made aware of those
trademarks, then the names are in capital letters or initial
capital letters.
</para>
<para>
DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED
UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE
WITH THE FURTHER UNDERSTANDING THAT:
<orderedlist>
<listitem>
<para>DOCUMENT IS PROVIDED ON AN "AS IS" BASIS,
WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR
IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES
THAT THE DOCUMENT OR MODIFIED VERSION OF THE
DOCUMENT IS FREE OF DEFECTS MERCHANTABLE, FIT FOR
A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE
RISK AS TO THE QUALITY, ACCURACY, AND PERFORMANCE
OF THE DOCUMENT OR MODIFIED VERSION OF THE
DOCUMENT IS WITH YOU. SHOULD ANY DOCUMENT OR
MODIFIED VERSION PROVE DEFECTIVE IN ANY RESPECT,
YOU (NOT THE INITIAL WRITER, AUTHOR OR ANY
CONTRIBUTOR) ASSUME THE COST OF ANY NECESSARY
SERVICING, REPAIR OR CORRECTION. THIS DISCLAIMER
OF WARRANTY CONSTITUTES AN ESSENTIAL PART OF THIS
LICENSE. NO USE OF ANY DOCUMENT OR MODIFIED
VERSION OF THE DOCUMENT IS AUTHORIZED HEREUNDER
EXCEPT UNDER THIS DISCLAIMER; AND
</para>
</listitem>
<listitem>
<para>UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL
THEORY, WHETHER IN TORT (INCLUDING NEGLIGENCE),
CONTRACT, OR OTHERWISE, SHALL THE AUTHOR,
INITIAL WRITER, ANY CONTRIBUTOR, OR ANY
DISTRIBUTOR OF THE DOCUMENT OR MODIFIED VERSION
OF THE DOCUMENT, OR ANY SUPPLIER OF ANY OF SUCH
PARTIES, BE LIABLE TO ANY PERSON FOR ANY
DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR
CONSEQUENTIAL DAMAGES OF ANY CHARACTER
INCLUDING, WITHOUT LIMITATION, DAMAGES FOR LOSS
OF GOODWILL, WORK STOPPAGE, COMPUTER FAILURE OR
MALFUNCTION, OR ANY AND ALL OTHER DAMAGES OR
LOSSES ARISING OUT OF OR RELATING TO USE OF THE
DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT,
EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED OF
THE POSSIBILITY OF SUCH DAMAGES.
</para>
</listitem>
</orderedlist>
</para>
</legalnotice>
<!-- This file contains link to license for the documentation (GNU FDL), and
other legal stuff such as "NO WARRANTY" statement. Please do not change
any of this. -->
<authorgroup>
<author role="maintainer">
<firstname>Jiří</firstname>
<surname>Lebl</surname>
<affiliation>
<orgname>Oklahoma State University</orgname>
<address> <email>jirka@5z.com</email> </address>
</affiliation>
</author>
<author>
<firstname>Kai</firstname>
<surname>Willadsen</surname>
<affiliation>
<orgname>Университет Квинслэнда, Австралия</orgname>
<address> <email>kaiw@itee.uq.edu.au</email> </address>
</affiliation>
</author>
<!-- This is appropriate place for other contributors: translators,
maintainers, etc. Commented out by default.
<othercredit role="translator">
<firstname>Latin</firstname>
<surname>Translator 1</surname>
<affiliation>
<orgname>Latin Translation Team</orgname>
<address> <email>translator@gnome.org</email> </address>
</affiliation>
<contrib>Latin translation</contrib>
</othercredit>
-->
</authorgroup>
<!-- According to GNU FDL, revision history is mandatory if you are -->
<!-- modifying/reusing someone else's document. If not, you can omit it. -->
<!-- Remember to remove the &manrevision; entity from the revision entries other
-->
<!-- than the current revision. -->
<!-- The revision numbering system for GNOME manuals is as follows: -->
<!-- * the revision number consists of two components -->
<!-- * the first component of the revision number reflects the release version of the GNOME desktop. -->
<!-- * the second component of the revision number is a decimal unit that is incremented with each revision of the manual. -->
<!-- For example, if the GNOME desktop release is V2.x, the first version of the manual that -->
<!-- is written in that desktop timeframe is V2.0, the second version of the manual is V2.1, etc. -->
<!-- When the desktop release version changes to V3.x, the revision number of the manual changes -->
<!-- to V3.0, and so on. -->
<!-- This is unmaintained quite a bit so screw this it just makes things
ugly and we don't update the manrevision stuff anyway
<revhistory>
<revision>
<revnumber>&manrevision;</revnumber>
<date>&date;</date>
<revdescription>
<para role="author">Jiri (George) Lebl
<email>jirka@5z.com</email>
</para>
</revdescription>
</revision>
<revision>
<revnumber>0.1</revnumber>
<date>September 2004</date>
<revdescription>
<para role="author">Kai Willadsen
<email>kaiw@itee.uq.edu.au</email>
</para>
</revdescription>
</revision>
<revision>
<revnumber>Genius Calculator Manual</revnumber>
<date>August 2004</date>
<revdescription>
<para role="author">Jiri (George) Lebl
<email>jirka@5z.com</email>
</para>
</revdescription>
</revision>
</revhistory>
-->
<revhistory>
<revision>
<revnumber>0.2</revnumber>
<date>September 2016</date>
<revdescription>
<para role="author">Jiri (George) Lebl <email>jirka@5z.com</email></para>
</revdescription>
</revision>
</revhistory>
<releaseinfo>This manual describes version 1.0.22 of Genius.
</releaseinfo>
<legalnotice>
<title>Обратная связь</title>
<para>
To report a bug or make a suggestion regarding the <application>Genius Mathematics Tool</application>
application or this manual, please visit the
<ulink url="http://www.jirka.org/genius.html" type="http">Genius
Web page</ulink>
or email me at <email>jirka@5z.com</email>.
</para>
<!-- Translators may also add here feedback address for translations -->
</legalnotice>
</bookinfo>
<!-- ============= Introduction =============================== -->
<chapter id="genius-introduction">
<title>Введение</title>
<para>
The <application>Genius Mathematics Tool</application> application is a general calculator for use as a desktop
calculator, an educational tool in mathematics, and is useful even for
research. The language used in <application>Genius Mathematics Tool</application> is designed to be
‘mathematical’ in the sense that it should be ‘what
you mean is what you get’. Of course that is not an
entirely attainable goal. <application>Genius Mathematics Tool</application> features rationals, arbitrary
precision integers and multiple precision floats using the GMP library.
It handles complex numbers using cartesian notation. It has good
vector and matrix manipulation and can handle basic linear algebra.
The programming language allows user defined functions, variables and
modification of parameters.
</para>
<para><application>Математический инструмент Genius</application> имеет две версии. Одна из них — это версия с графическим интерфейсом для GNOME, имеющая интерфейс в стиле интегрированных сред разработки (IDE) и способная строить графики функций одной или двух переменных. Версия для командной строки не требует наличия GNOME, но и не реализует тех возможностей, которые требуют наличия графического интерфейса.</para>
<para>
Parts of this manual describe the graphical version of the calculator,
but the language is of course the same. The command line only version
lacks the graphing capabilities and all other capabilities that require
the graphical user interface.
</para>
<para>
Generally, when some feature of the language (function, operator, etc...)
is new in some version past 1.0.5, it is mentioned, but
below 1.0.5 you would have to look at the NEWS file.
</para>
</chapter>
<!-- =========== Getting Started ============================== -->
<chapter id="genius-getting-started">
<title>Приступая к работе</title>
<sect1 id="genius-to-start">
<title>Запуск <application>Математического инструмента Genius</application></title>
<para>You can start <application>Genius Mathematics Tool</application> in the following ways:
</para>
<variablelist>
<varlistentry>
<term>Меню <guimenu>Приложения</guimenu></term>
<listitem>
<para>В зависимости от установленной операционной системы и её версии, команда меню для запуска <application>Математического инструмента Genius</application> может находиться в различных местах. Она может располагаться в подменю <guisubmenu>Образование</guisubmenu>, <guisubmenu>Стандартные</guisubmenu>, <guisubmenu>Офис</guisubmenu>, <guisubmenu>Наука</guisubmenu> или других подобных подменю. Нужный вам элемент меню называется <guimenuitem>Математический инструмент Genius</guimenuitem>. Когда вы найдёте этот элемент меню, щёлкните на нём, чтобы запустить <application>Математический инструмент Genius</application>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Диалог запуска</term>
<listitem>
<para>В некоторых операционных системах упомянутая выше команда меню может быть недоступна. В этом случае можно открыть диалог запуска и выполнить команду <command>gnome-genius</command>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Командная строка</term>
<listitem>
<para>Чтобы запустить версию <application>Математического инструмента Genius</application> для GNOME, выполните в командной строке <command>gnome-genius</command>.</para>
<para>Чтобы запустить версию для командной строки, выполните следующую команду: <command>genius</command>. Эта версия не содержит графического интерфейса и некоторые возможности, такие как построение графиков, будут в ней недоступны.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-when-start">
<title>После запуска Genius</title>
<para>После запуска версии <application>Математического инструмента Genius</application> для GNOME появится окно, изображённое на иллюстрации <xref linkend="mainwindow-fig"/>.</para>
<figure id="mainwindow-fig">
<title>Окно <application>Математического инструмента Genius</application></title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/genius_window.png" format="PNG" scalefit="1"/>
</imageobject>
<textobject>
<phrase>Shows <application>Математический инструмент Genius</application> main window. Contains titlebar, menubar,
toolbar and working area. Menubar contains <guilabel>Файл</guilabel>,
<guilabel>Правка</guilabel>, <guilabel>Калькулятор</guilabel>,
<guilabel>Examples</guilabel>,
<guilabel>Programs</guilabel>,
<guilabel>Настройки</guilabel>, and <guilabel>Справка</guilabel> menus.</phrase>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>Окно <application>Математического инструмента Genius</application> содержит следующие элементы:</para>
<variablelist>
<varlistentry>
<term>Строка меню.</term>
<listitem>
<para>Строка меню содержит все команды, необходимые для работы с файлами в <application>Математическом инструменте Genius</application>. Меню <guilabel>Файл</guilabel> содержит команды для загрузки и сохранения объектов и создания новых программ. Команда <guilabel>Загрузить и выполнить...</guilabel> не открывает новое окно для программы, а просто сразу выполняет программу. Её действие эквивалентно действию команды <command>load</command>.</para>
<para>
The <guilabel>Calculator</guilabel> menu controls the
calculator engine. It allows you to run the currently selected program or to
interrupt the current calculation. You can also look at the full expression of
the last answer (useful if the last answer was too large to fit onto the
console), or you can view a listing of the values of all user defined
variables. You can also monitor user variables, which is especially useful
while a long calculation is running, or to debug a certain program.
Finally the <guilabel>Calculator</guilabel> allows plotting functions using a user friendly dialog box.
</para>
<para>
The <guilabel>Examples</guilabel> menu is a list of example
programs or demos. If you open the menu, it will load the
example into a new program, which you can run, edit, modify,
and save. These programs should be well documented
and generally demonstrate either some feature of <application>Genius Mathematics Tool</application>
or some mathematical concept.
</para>
<para>
The <guilabel>Programs</guilabel> menu lists
the currently open programs and allows you to switch
between them.
</para>
<para>Остальные меню выполняют такие же действия, как в других приложениях.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Панель инструментов.</term>
<listitem>
<para>Панель инструментов содержит некоторые из команд, к которым можно получить доступ через строку меню.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Рабочая область</term>
<listitem>
<para>Рабочая область — основной способ взаимодействия с приложением.</para>
<para>
The working area initially has just the <guilabel>Console</guilabel> tab, which is
the main way of interacting with the calculator. Here you
type expressions and the results are immediately returned
after you hit the Enter key.
</para>
<para>
Alternatively you can write longer programs and those can
appear in separate tabs. The programs are a set of commands or
functions that can be run all at once rather than entering them
at the command line. The programs can be saved in files for later
retrieval.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
</chapter>
<!-- ================ Usage =================================== -->
<chapter id="genius-usage">
<title>Основы работы с программой</title>
<sect1 id="genius-usage-workarea">
<title>Использование рабочей области</title>
<para>
Normally you interact with the calculator in the <guilabel>Console</guilabel> tab of the
work area. If you are running the text only version then the console
will be the only thing that is available to you. If you want to use
<application>Genius Mathematics Tool</application> as a calculator only, just type in your expression in the console, it
will be evaluated, and the returned value will be printed.
</para>
<para>
To evaluate an expression, type it into the <guilabel>Console</guilabel> work area and press enter.
Expressions are written in a
language called GEL. The most simple GEL expressions just looks like
mathematics. For example
<screen><prompt>genius> </prompt><userinput>30*70 + 67^3.0 + ln(7) * (88.8/100)</userinput>
</screen>
or
<screen><prompt>genius> </prompt><userinput>62734 + 812634 + 77^4 mod 5</userinput>
</screen>
or
<screen><prompt>genius> </prompt><userinput>| sin(37) - e^7 |</userinput>
</screen>
or
<screen><prompt>genius> </prompt><userinput>sum n=1 to 70 do 1/n</userinput>
</screen>
(Last is the harmonic sum from 1 to 70)
</para>
<para>Чтобы получить список функций и команд, наберите: <screen><prompt>genius> </prompt><userinput>help</userinput>
</screen> Для получения дополнительной справки по отдельной функции наберите : <screen><prompt>genius> </prompt><userinput>help ИмяФункции</userinput>
</screen> Чтобы увидеть это руководство, наберите: <screen><prompt>genius> </prompt><userinput>manual</userinput>
</screen></para>
<para>
Suppose you have previously saved some GEL commands as a program to a file and
you now want to execute them.
To load this program from the file <filename>path/to/program.gel</filename>,
type
<screen><prompt>genius> </prompt><userinput>load path/to/program.gel</userinput>
</screen>
<application>Genius Mathematics Tool</application> keeps track of the current directory.
To list files in the current directory type <command>ls</command>, to change directory
do <userinput>cd directory</userinput> as in the UNIX command shell.
</para>
</sect1>
<sect1 id="genius-usage-create-program">
<title>Создание новой программы</title>
<para>
If you wish to enter several more complicated commands, or perhaps write a complicated
function using the <link linkend="genius-gel">GEL</link> language, you can create a new
program.
</para>
<para>
To start writing a new program, choose
<menuchoice><guimenu>File</guimenu><guimenuitem>New
Program</guimenuitem></menuchoice>. A new tab will appear in the work area. You
can write a <link linkend="genius-gel">GEL</link> program in this work area.
Once you have written your program you can run it by
<menuchoice><guimenu>Calculator</guimenu><guimenuitem>Run</guimenuitem></menuchoice> (or
the <guilabel>Run</guilabel> toolbar button).
This will execute your program and will display any output on the <guilabel>Console</guilabel> tab.
Executing a program is equivalent of taking the text of the program and
typing it into the console. The only difference is that this input is done
independent of the console and just the output goes onto the console.
<menuchoice><guimenu>Calculator</guimenu><guimenuitem>Run</guimenuitem></menuchoice>
will always run the currently selected program even if you are on the <guilabel>Console</guilabel>
tab. The currently selected program has its tab in bold type. To select a
program, just click on its tab.
</para>
<para>
To save the program you've just written, choose <menuchoice><guimenu>File</guimenu><guimenuitem>Save As...</guimenuitem></menuchoice>.
Similarly as in other programs you can choose
<menuchoice><guimenu>File</guimenu><guimenuitem>Save</guimenuitem></menuchoice> to save a program that already has
a filename attached to it. If you have many opened programs you have edited and wish to save you can also choose
<menuchoice><guimenu>File</guimenu><guimenuitem>Save All Unsaved</guimenuitem></menuchoice>.
</para>
<para>
Programs that have unsaved changes will have a "[+]" next to their filename. This way you can see if the file
on disk and the currently opened tab differ in content. Programs which have not yet had a filename associated
with them are always considered unsaved and no "[+]" is printed.
</para>
</sect1>
<sect1 id="genius-usage-open-program">
<title>Открытие и запуск программы</title>
<para>Чтобы открыть файл, выберите <menuchoice><guimenu>Файл</guimenu><guimenuitem>Открыть</guimenuitem></menuchoice>. В рабочей области появится новая вкладка с содержимым файла. Вы можете использовать её для редактирования файла.</para>
<para>Чтобы запустить программу из файла, выберите <menuchoice><guimenu>Файл</guimenu><guimenuitem>Загрузить и выполнить...</guimenuitem></menuchoice>. Это действие запустит программу, не открывая её в отдельной вкладке. Это эквивалентно команде <command>load</command>.</para>
<para>
If you have made edits to a file you wish to throw away and want to reload to the version that's on disk,
you can choose the
<menuchoice><guimenu>File</guimenu><guimenuitem>Reload from Disk</guimenuitem></menuchoice> menuitem. This is useful for experimenting
with a program and making temporary edits, to run a program, but that you do not intend to keep.
</para>
</sect1>
</chapter>
<!-- ============= Plotting ============================== -->
<chapter id="genius-gel-plotting">
<title>Построение графиков</title>
<para>Построение графиков поддерживается только в версии с графическим интерфейсом для GNOME. Все варианты графиков, имеющиеся в графическом интерфейсе, доступны из окна <guilabel>Создать график</guilabel>. Чтобы открыть это окно, нажмите кнопку <guilabel>График</guilabel> на панели инструментов или выберите <guilabel>Нарисовать график</guilabel> из меню <guilabel>Калькулятор</guilabel>. Можно также использовать команды <link linkend="genius-gel-function-list-plotting">построения графиков</link> языка GEL. Смотрите <xref linkend="genius-gel"/>, чтобы узнать, как вводить выражения, которые понимает Genius.</para>
<sect1 id="genius-line-plots">
<title>Линейные графики</title>
<para>
To graph real valued functions of one variable open the <guilabel>Create Plot</guilabel>
window. You can also use the
<link linkend="gel-function-LinePlot"><function>LinePlot</function></link> function
on the command line (see its documentation).
</para>
<para>После нажатия кнопки <guilabel>График</guilabel> откроется окно с несколькими вкладками. Вам нужна вкладка <guilabel>Линейный график функции</guilabel>, внутри которой следует выбрать вкладку <guilabel>Функции / Выражения</guilabel>. Смотрите <xref linkend="lineplot-fig"/>.</para>
<figure id="lineplot-fig">
<title>Create Plot Window</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/line_plot.png" format="PNG" scalefit="1"/>
</imageobject>
<textobject>
<phrase>Shows the line plotting window.</phrase>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>
Type expressions with <userinput>x</userinput> as
the independent variable into the textboxes. Alternatively you can give names of functions such as
<userinput>cos</userinput> rather then having to type <userinput>cos(x)</userinput>.
You can graph up to ten functions. If you make a mistake and Genius cannot
parse the input it will signify this with a warning icon on the right of the text
input box where the error occurred, as well as giving you an error dialog.
You can change the ranges of the dependent and independent variables in the bottom
part of the dialog.
The <varname>y</varname> (dependent) range can be set automatically by turning on the <guilabel>Fit dependent axis</guilabel>
checkbox.
The names of the variables can also be changed.
Pressing the <guilabel>Plot</guilabel> button produces the graph shown in <xref linkend="lineplot2-fig"/>.
</para>
<para>
The variables can be renamed by clicking the <guilabel>Change variable names...</guilabel> button, which is useful if you wish to print or save the figure and don't want to use the standard
names. Finally you can also avoid printing the legend and the axis labels completely,
which is also useful if printing or
saving, when the legend might simply be clutter.
</para>
<figure id="lineplot2-fig">
<title>Plot Window</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/line_plot_graph.png" format="PNG" scalefit="1"/>
</imageobject>
<textobject>
<phrase>The graph produced.</phrase>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>
From here you can print out the plot, create encapsulated postscript
or a PNG version of the plot or change the zoom. If the dependent axis was
not set correctly you can have Genius fit it by finding out the extrema of
the graphed functions.
</para>
<para>
For plotting using the command line see the documentation of the
<link linkend="gel-function-LinePlot"><function>LinePlot</function></link> function.
</para>
</sect1>
<sect1 id="genius-parametric-plots">
<title>Parametric Plots</title>
<para>
In the create plot window, you can also choose the <guilabel>Parametric</guilabel> notebook
tab to create two dimensional parametric plots. This way you can
plot a single parametric function. You can either specify the
points as <varname>x</varname> and <varname>y</varname>, or giving a single complex number
as a function of the variable <varname>t</varname>.
The range of the variable <varname>t</varname> is given explicitly, and the function is sampled
according to the given increment.
The <varname>x</varname> and <varname>y</varname> range can be set
automatically by turning on the <guilabel>Fit dependent axis</guilabel>
checkbox, or it can be specified explicitly.
See <xref linkend="paramplot-fig"/>.
</para>
<figure id="paramplot-fig">
<title>Parametric Plot Tab</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/parametric.png" format="PNG" scalefit="1"/>
</imageobject>
<textobject>
<phrase>Parametric plotting tab in the <guilabel>Create Plot</guilabel> window.</phrase>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>
An example of a parametric plot is given in
<xref linkend="paramplot2-fig"/>.
Similar operations can be
done on such graphs as can be done on the other line plots.
For plotting using the command line see the documentation of the
<link linkend="gel-function-LinePlotParametric"><function>LinePlotParametric</function></link> or
<link linkend="gel-function-LinePlotCParametric"><function>LinePlotCParametric</function></link> function.
</para>
<figure id="paramplot2-fig">
<title>Parametric Plot</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/parametric_graph.png" format="PNG" scalefit="1"/>
</imageobject>
<textobject>
<phrase>Parametric plot produced</phrase>
</textobject>
</mediaobject>
</screenshot>
</figure>
</sect1>
<sect1 id="genius-slopefield-plots">
<title>Slopefield Plots</title>
<para>
In the create plot window, you can also choose the <guilabel>Slope field</guilabel> notebook
tab to create a two dimensional slope field plot.
Similar operations can be
done on such graphs as can be done on the other line plots.
For plotting using the command line see the documentation of the
<link linkend="gel-function-SlopefieldPlot"><function>SlopefieldPlot</function></link> function.
</para>
<para>
When a slope field is active, there is an extra <guilabel>Solver</guilabel> menu available,
through which you can bring up the solver dialog. Here you can have Genius plot specific
solutions for the given initial conditions. You can either specify initial conditions in the dialog,
or you can click on the plot directly to specify the initial point. While the solver dialog
is active, the zooming by clicking and dragging does not work. You have to close the dialog first
if you want to zoom using the mouse.
</para>
<para>
The solver uses the standard Runge-Kutta method.
The plots will stay on the screen until cleared. The solver will stop whenever it reaches the boundary
of the plot window. Zooming does not change the limits or parameters of the solutions,
you will have to clear and redraw them with appropriate parameters.
You can also use the
<link linkend="gel-function-SlopefieldDrawSolution"><function>SlopefieldDrawSolution</function></link>
function to draw solutions from the command line or programs.
</para>
</sect1>
<sect1 id="genius-vectorfield-plots">
<title>Vectorfield Plots</title>
<para>
In the create plot window, you can also choose the <guilabel>Vector field</guilabel> notebook
tab to create a two dimensional vector field plot.
Similar operations can be
done on such graphs as can be done on the other line plots.
For plotting using the command line see the documentation of the
<link linkend="gel-function-VectorfieldPlot"><function>VectorfieldPlot</function></link> function.
</para>
<para>
By default the direction and magnitude of the vector field is shown.
To only show direction and not the magnitude, check the appropriate
checkbox to normalize the arrow lengths.
</para>
<para>
When a vector field is active, there is an extra <guilabel>Solver</guilabel> menu available,
through which you can bring up the solver dialog. Here you can have Genius plot specific
solutions for the given initial conditions. You can either specify initial conditions in the dialog,
or you can click on the plot directly to specify the initial point. While the solver dialog
is active, the zooming by clicking and dragging does not work. You have to close the dialog first
if you want to zoom using the mouse.
</para>
<para>
The solver uses the standard Runge-Kutta method.
The plots will stay on the screen until cleared.
Zooming does not change the limits or parameters of the solutions,
you will have to clear and redraw them with appropriate parameters.
You can also use the
<link linkend="gel-function-VectorfieldDrawSolution"><function>VectorfieldDrawSolution</function></link>
function to draw solutions from the command line or programs.
</para>
</sect1>
<sect1 id="genius-surface-plots">
<title>Графики поверхностей</title>
<para>
Genius can also plot surfaces. Select the <guilabel>Surface plot</guilabel> tab in the
main notebook of the <guilabel>Create Plot</guilabel> window. Here you can specify a single
expression that should use either <varname>x</varname> and <varname>y</varname> as real independent variables
or <varname>z</varname> as a complex variable (where <varname>x</varname> is the real part of <varname>z</varname> and <varname>y</varname> is the
imaginary part). For example to plot the modulus of the cosine
function for complex parameters,
you could enter <userinput>|cos(z)|</userinput>. This would be
equivalent to <userinput>|cos(x+1i*y)|</userinput>.
See <xref linkend="surfaceplot-fig"/>.
For plotting using the command line see the documentation of the
<link linkend="gel-function-SurfacePlot"><function>SurfacePlot</function></link> function.
</para>
<para>
The <varname>z</varname> range can be set automatically by turning on the <guilabel>Fit dependent axis</guilabel>
checkbox. The variables can be renamed by clicking the <guilabel>Change variable names...</guilabel> button, which is useful if you wish to print or save the figure and don't want to use the standard
names. Finally you can also avoid printing the legend, which is also useful if printing or
saving, when the legend might simply be clutter.
</para>
<figure id="surfaceplot-fig">
<title>Surface Plot</title>
<screenshot>
<mediaobject>
<imageobject>
<imagedata fileref="figures/surface_graph.png" format="PNG" scalefit="1"/>
</imageobject>
<textobject>
<phrase>Modulus of the complex cosine function.</phrase>
</textobject>
</mediaobject>
</screenshot>
</figure>
<para>
In surface mode, left and right arrow keys on your keyboard will rotate the
view along the z axis. Alternatively you can rotate along any axis by
selecting <guilabel>Rotate axis...</guilabel> in the <guilabel>View</guilabel>
menu. The <guilabel>View</guilabel> menu also has a top view mode which rotates the
graph so that the z axis is facing straight out, that is, we view the graph from the top
and get essentially just the colors that define the values of the function getting a
temperature plot of the function. Finally you should
try <guilabel>Start rotate animation</guilabel>, to start a continuous slow rotation.
This is especially good if using <application>Genius Mathematics Tool</application> to present to an audience.
</para>
</sect1>
</chapter>
<!-- ============= GEL ====================================== -->
<chapter id="genius-gel">
<title>Основы GEL</title>
<para>
GEL stands for Genius Extension Language. It is the language you use
to write programs in Genius. A program in GEL is simply an
expression that evaluates to a number, a matrix, or another object
in GEL.
<application>Genius Mathematics Tool</application> can be used as a simple calculator, or as a
powerful theoretical research tool. The syntax is meant to
have as shallow of a learning curve as possible, especially for use
as a calculator.
</para>
<sect1 id="genius-gel-values">
<title>Значения</title>
<para>
Values in GEL can be <link linkend="genius-gel-values-numbers">numbers</link>, <link linkend="genius-gel-values-booleans">Booleans</link>, or <link linkend="genius-gel-values-strings">strings</link>. GEL also treats
<link linkend="genius-gel-matrices">matrices</link> as values.
Values can be used in calculations, assigned to variables and returned from functions, among other uses.
</para>
<sect2 id="genius-gel-values-numbers">
<title>Числа</title>
<para>Целые числа — первый тип чисел в GEL. Целые числа записываются общепринятым способом. <programlisting>1234
</programlisting> Шестнадцатиричные и восьмиричные числа можно записать, используя нотацию языка C. Например: <programlisting>0x123ABC
01234
</programlisting> Можно также набрать числа в произвольной системе счисления, используя запись <literal><основание>\<число></literal>. Для цифр больше 10 используются буквы, как и в шестнадцатиричном счислении. Например, число по основанию 23 может быть записано в виде: <programlisting>23\1234ABCD
</programlisting></para>
<para>Второй тип чисел в GEL — это рациональные числа. Они получаются делением двух целых чисел. Поэтому можно написать: <programlisting>3/4
</programlisting> чтобы обозначить три четвёртых. Рациональные числа также можно записывать в виде смешанных дробей. Чтобы указать одну целую три десятых, можно написать: <programlisting>1 3/10
</programlisting></para>
<para>
The next type of number is floating point. These are entered in a similar fashion to C notation. You can use <literal>E</literal>, <literal>e</literal> or <literal>@</literal> as the exponent delimiter. Note that using the exponent delimiter gives a float even if there is no decimal point in the number. Examples:
<programlisting>1.315
7.887e77
7.887e-77
.3
0.3
77e5
</programlisting>
When Genius prints a floating point number it will always append a
<computeroutput>.0</computeroutput> even if the number is whole. This is to indicate that
floating point numbers are taken as imprecise quantities. When a number is written in the
scientific notation, it is always a floating point number and thus Genius does not
print the <computeroutput>.0</computeroutput>.
</para>
<para>
The final type of number in GEL is the complex numbers. You can enter a complex number as a sum of real and imaginary parts. To add an imaginary part, append an <literal>i</literal>. Here are examples of entering complex numbers:
<programlisting>1+2i
8.01i
77*e^(1.3i)
</programlisting>
</para>
<important>
<para>При вводе мнимых чисел перед символом <literal>i</literal> должно стоять число. Если использовать символ <literal>i</literal> сам по себе, Genius интерпретирует его как ссылку на переменную <varname>i</varname>. Если нужно указать саму мнимую единицу <literal>i</literal>, используйте вместо неё <literal>1i</literal>.</para>
<para>Чтобы использовать смешанные дроби в мнимых числах, нужно взять смешанную дробь в круглые скобки: (например, <userinput>(1 2/5)i</userinput>)</para>
</important>
</sect2>
<sect2 id="genius-gel-values-booleans">
<title>Логические значения</title>
<para>Genius также поддерживает логические значения. Определены две логические константы: <constant>true</constant> и <constant>false</constant>; их можно использовать, как и любую переменную. В качестве псевдонимов к ним можно также использовать <constant>True</constant>, <constant>TRUE</constant>, <constant>False</constant> и <constant>FALSE</constant>.</para>
<para>Там, где требуется логическое выражение, можно использовать логическое значение или любое выражение, дающее в результате число или логическое значение. Если Genius нужно использовать число как логическое значение, он будет интерпретировать 0 как <constant>false</constant> и любое другое число как <constant>true</constant>.</para>
<para>Кроме того, с логическими значениями можно выполнять арифметические операции. Например: <programlisting>( (1 + true) - false ) * true
</programlisting> это то же самое, что и: <programlisting>( (true or true) or not false ) and true
</programlisting> Поддерживаются только сложение, вычитание и умножение. Если вы используете в выражении смесь чисел с логическими значениями, то числа преобразовываются в логические значения, как описано выше. То есть, результатом выражения: <programlisting>1 == true
</programlisting> всегда будет <constant>true</constant>, так как 1 преобразовывается в <constant>true</constant> перед сравнением с <constant>true</constant>.</para>
</sect2>
<sect2 id="genius-gel-values-strings">
<title>Строки</title>
<para>
Like numbers and Booleans, strings in GEL can be stored as values inside variables and passed to functions. You can also concatenate a string with another value using the plus operator. For example:
<programlisting>a=2+3;"Результат равен: "+a
</programlisting>
will create the string:
<programlisting>Результат равен: 5
</programlisting>
You can also use C-like escape sequences such as <literal>\n</literal>,<literal>\t</literal>,<literal>\b</literal>,<literal>\a</literal> and <literal>\r</literal>. To get a <literal>\</literal> or <literal>"</literal> into the string you can quote it with a <literal>\</literal>. For example:
<programlisting>"Косая черта: \\ Кавычки: \" Табуляция: \t1\t2\t3"
</programlisting>
will make a string:
<programlisting>Косая черта: \ Кавычки: " Табуляция: 1 2 3
</programlisting>
Do note however that when a string is returned from a function, escapes are
quoted, so that the output can be used as input. If you wish to print the
string as it is (without escapes), use the
<link linkend="gel-function-print"><function>print</function></link>
or
<link linkend="gel-function-printn"><function>printn</function></link> functions.
</para>
<para>
In addition, you can use the library function <link linkend="gel-function-string"><function>string</function></link> to convert anything to a string. For example:
<programlisting>string(22)
</programlisting>
will return
<programlisting>"22"
</programlisting>
Strings can also be compared with <literal>==</literal> (equal), <literal>!=</literal> (not equal) and <literal><=></literal> (comparison) operators
</para>
</sect2>
<sect2 id="genius-gel-values-null">
<title>Null</title>
<para>
There is a special value called
<constant>null</constant>. No operations can be performed on
it, and nothing is printed when it is returned. Therefore,
<constant>null</constant> is useful when you do not want output from an
expression. The value <constant>null</constant> can be obtained as an expression when you
type <literal>.</literal>, the constant <constant>null</constant> or nothing.
By nothing we mean that if you end an expression with
a separator <literal>;</literal>, it is equivalent to ending it with a
separator followed by a <constant>null</constant>.
</para>
<para>Пример: <programlisting>x=5;.
x=5;
</programlisting></para>
<para>Некоторые функции возвращают <constant>null</constant>, если невозможно вернуть значение или произошла ошибка. Также <constant>null</constant> используется как пустой вектор или матрица, или пустая ссылка.</para>
</sect2>
</sect1>
<sect1 id="genius-gel-variables">
<title>Использование переменных</title>
<para>Синтаксис: <programlisting>ИмяПеременной
</programlisting> Пример: <screen><prompt>genius> </prompt><userinput>e</userinput>
= 2.71828182846
</screen></para>
<para>Чтобы вычислить значение переменной, просто введите имя переменной и программа вернёт её значение. Можно использовать переменную в любом месте, где обычно используется число или строка. Кроме того, переменные необходимы при определении функций, принимающих аргументы (см. <xref linkend="genius-gel-functions-defining"/>).</para>
<tip>
<title>Использование автозавершения по клавише Tab</title>
<para>Вы можете использовать автозавершение по нажатию клавиши Tab, чтобы Genius автоматически подставлял полное имя переменной. Попробуйте набрать несколько первых букв имени и нажать <userinput>Tab</userinput>.</para>
</tip>
<important>
<title>Имена переменных чувствительны к регистру</title>
<para>Имена переменных чувствительны к регистру символов. Это означает, что переменные <varname>hello</varname>, <varname>HELLO</varname> и <varname>Hello</varname> — это разные переменные.</para>
</important>
<sect2 id="genius-gel-variables-setting">
<title>Присваивание значения переменным</title>
<para>Синтаксис: <programlisting><![CDATA[<identifier> = <value>
<identifier> := <value>]]>
</programlisting> Пример: <programlisting>x = 3
x := 3
</programlisting></para>
<para>
To assign a value to a variable, use the <literal>=</literal> or <literal>:=</literal> operators. These operators set the value of the variable and return the value you set, so you can do things like
<programlisting>a = b = 5
</programlisting>
This will set <varname>b</varname> to 5 and then also set <varname>a</varname> to 5.
</para>
<para>Для присваивания значения переменным можно использовать любой из операторов <literal>=</literal> и <literal>:=</literal>. Различие между ними в том, что оператор <literal>:=</literal> всегда действует как оператор присваивания, а оператор <literal>=</literal> может интерпретироваться как проверка на равенство там, где ожидается логическое выражение.</para>
<para>
For more information about the scope of variables, that is when are what variables visible, see <xref linkend="genius-gel-variables-global"/>.
</para>
</sect2>
<sect2 id="genius-gel-variables-built-in">
<title>Встроенные переменные</title>
<para>
GEL has a number of built-in ‘variables’, such as
<varname>e</varname>, <varname>pi</varname> or <varname>GoldenRatio</varname>. These are widely used constants with a preset value, and
they cannot be assigned new values.
There are a number of other built-in variables.
See <xref linkend="genius-gel-function-list-constants"/> for a full list. Note that <varname>i</varname> is not by default
the square root of negative one (the imaginary number), and is undefined to allow its use as a counter. If you wish to write the imaginary number you need to
use <userinput>1i</userinput>.
</para>
</sect2>
<sect2 id="genius-gel-previous-result">
<title>Переменные с результатом предыдущего вычисления</title>
<para>Переменные <varname>Ans</varname> и <varname>ans</varname> могут использоваться для получения результата последнего вычисления. Например, чтобы добавить 389 к результату предыдущего вычисления, можно набрать: <programlisting>Ans+389
</programlisting></para>
</sect2>
</sect1>
<sect1 id="genius-gel-functions">
<title>Использование функций</title>
<para>Синтаксис: <programlisting>ИмяФункции(аргумент1, аргумент2, ...)
</programlisting> Пример: <programlisting>Factorial(5)
cos(2*pi)
gcd(921,317)
</programlisting> Чтобы вычислить значение функции, введите имя функции, за которым следуют аргументы функции (если они имеются) в круглых скобках. Программа вернёт результат применения функции к её аргументам. Разумеется, число аргументов может быть разным для разных функций.</para>
<para>
There are many built-in functions, such as <link linkend="gel-function-sin"><function>sin</function></link>, <link linkend="gel-function-cos"><function>cos</function></link> and <link linkend="gel-function-tan"><function>tan</function></link>. You can use the <link linkend="gel-command-help"><function>help</function></link> built-in command to get a list of available functions, or see <xref linkend="genius-gel-function-list"/> for a full listing.
</para>
<tip>
<title>Использование автозавершения по клавише Tab</title>
<para>Можно использовать автозавершение по клавише Tab, чтобы Genius автоматически подставлял имена функций. Попробуйте набрать первые несколько букв имени и нажать <userinput>Tab</userinput>.</para>
</tip>
<important>
<title>Имена функций чувствительны к регистру</title>
<para>Имена функций чувствительны к регистру символов. Это означает, что функции <function>dosomething</function>, <function>DOSOMETHING</function> и <function>DoSomething</function> — это разные функции.</para>
</important>
<sect2 id="genius-gel-functions-defining">
<title>Определение функций</title>
<para>
Syntax:
<programlisting><![CDATA[function <identifier>(<comma separated arguments>) = <function body>
<identifier> = (`() = <function body>)
]]></programlisting>
The <literal>`</literal> is the backquote character, and signifies an anonymous function. By setting it to a variable name you effectively define a function.
</para>
<para>
A function takes zero or more comma separated arguments, and returns the result of the function body. Defining your own functions is primarily a matter of convenience; one possible use is to have sets of functions defined in GEL files that Genius can load in order to make them available.
Example:
<programlisting>function addup(a,b,c) = a+b+c
</programlisting>
then <userinput>addup(1,4,9)</userinput> yields 14
</para>
</sect2>
<sect2 id="genius-gel-functions-variable-argument-lists">
<title>Variable Argument Lists</title>
<para>
If you include <literal>...</literal> after the last argument name in the function declaration, then Genius will allow any number of arguments to be passed in place of that argument. If no arguments were passed then that argument will be set to <constant>null</constant>. Otherwise, it will be a horizontal vector containing all the arguments. For example:
<programlisting>function f(a,b...) = b
</programlisting>
Then <userinput>f(1,2,3)</userinput> yields <computeroutput>[2,3]</computeroutput>, while <userinput>f(1)</userinput> yields a <constant>null</constant>.
</para>
</sect2>
<sect2 id="genius-gel-functions-passing-functions">
<title>Passing Functions to Functions</title>
<para>
In Genius, it is possible to pass a function as an argument to another function. This can be done using either ‘function nodes’ or anonymous functions.
</para>
<para>
If you do not enter the parentheses after a function name, instead of being evaluated, the function will instead be returned as a ‘function node’. The function node can then be passed to another function.
Example:
<programlisting>function f(a,b) = a(b)+1;
function b(x) = x*x;
f(b,2)
</programlisting>
</para>
<para>
To pass functions that are not defined,
you can use an anonymous function (see <xref linkend="genius-gel-functions-defining"/>). That is, you want to pass a function without giving it a name.
Syntax:
<programlisting><![CDATA[function(<comma separated arguments>) = <function body>
`(<comma separated arguments>) = <function body>
]]></programlisting>
Example:
<programlisting>function f(a,b) = a(b)+1;
f(`(x) = x*x,2)
</programlisting>
This will return 5.
</para>
</sect2>
<sect2 id="genius-gel-functions-operations">
<title>Операции с функциями</title>
<para>
Some functions allow arithmetic operations, and some single argument functions such as <link linkend="gel-function-exp"><function>exp</function></link> or <link linkend="gel-function-ln"><function>ln</function></link>, to operate on the function. For example,
<programlisting>exp(sin*cos+4)
</programlisting>
will return a function that takes <varname>x</varname> and returns <userinput>exp(sin(x)*cos(x)+4)</userinput>. It is functionally equivalent
to typing
<programlisting>`(x) = exp(sin(x)*cos(x)+4)
</programlisting>
This operation can be useful when quickly defining functions. For example to create a function called <varname>f</varname>
to perform the above operation, you can just type:
<programlisting>f = exp(sin*cos+4)
</programlisting>
It can also be used in plotting. For example, to plot sin squared you can enter:
<programlisting>LinePlot(sin^2)
</programlisting>
</para>
<warning>
<para>
Not all functions can be used in this way. For example, when you use a binary operation the functions must take the same number of arguments.
</para>
</warning>
</sect2>
</sect1>
<sect1 id="genius-gel-separator">
<title>Разделитель</title>
<para>
GEL is somewhat different from other languages in how it deals with multiple commands and functions.
In GEL you must chain commands together with a separator operator.
That is, if you want to type more than one expression you have to use
the <literal>;</literal> operator in between the expressions. This is
a way in which both expressions are evaluated and the result of the second one (or the last one
if there is more than two expressions) is returned.
Suppose you type the following:
<programlisting>3 ; 5
</programlisting>
This expression will yield 5.
</para>
<para>
This will require some parenthesizing to make it unambiguous sometimes,
especially if the <literal>;</literal> is not the top most primitive. This slightly differs from
other programming languages where the <literal>;</literal> is a terminator of statements, whereas
in GEL it’s actually a binary operator. If you are familiar with pascal
this should be second nature. However genius can let you pretend it is a
terminator to some degree. If a <literal>;</literal> is found at the end of a parenthesis or a block,
genius will append a null to it as if you would have written
<userinput>;null</userinput>.
This is useful in case you do not want to return a value from say a loop,
or if you handle the return differently. Note that it will slightly slow down
the code if it is executed too often as there is one more operator involved.
</para>
<para>
If you are typing expressions in a program you do not have to add a semicolon. In this case
genius will simply print the return value whenever it executes the expression. However, do note that if you are defining a
function, the body of the function is a single expression.
</para>
</sect1>
<sect1 id="genius-gel-comments">
<title>Comments</title>
<para>
GEL is similar to other scripting languages in that <literal>#</literal> denotes
a comment, that is text that is not meant to be evaluated. Everything beyond the
pound sign till the end of line will just be ignored. For example,
<programlisting># This is just a comment
# every line in a comment must have its own pound sign
# in the next line we set x to the value 123
x=123;
</programlisting>
</para>
</sect1>
<sect1 id="genius-gel-modular-evaluation">
<title>Modular Evaluation</title>
<para>
Genius implements modular arithmetic.
To use it you just add "mod <integer>" after
the expression. Example:
<userinput>2^(5!) * 3^(6!) mod 5</userinput>
It could be possible to do modular arithmetic by computing with integers and then modding in the end with
the <literal>%</literal> operator, which simply gives the remainder, but
that may be time consuming if not impossible when working with larger numbers.
For example, <userinput>10^(10^10) % 6</userinput> will simply not work (the exponent
will be too large), while
<userinput>10^(10^10) mod 6</userinput> is instantaneous. The first expression first tries to compute the integer
<userinput>10^(10^10)</userinput> and then find remainder after division by 6, while the second expression evaluates
everything modulo 6 to begin with.
</para>
<para>
You can calculate the inverses of numbers mod some integer by just using
rational numbers (of course the inverse has to exist).
Examples:
<programlisting>10^-1 mod 101
1/10 mod 101</programlisting>
You can also do modular evaluation with matrices including taking inverses,
powers and dividing.
Example:
<programlisting>A = [1,2;3,4]
B = A^-1 mod 5
A*B mod 5</programlisting>
This should yield the identity matrix as B will be the inverse of A mod 5.
</para>
<para>
Some functions such as
<link linkend="gel-function-sqrt"><function>sqrt</function></link> or
<link linkend="gel-function-log"><function>log</function></link>
work in a different way when in modulo mode. These will then work like their
discrete versions working within the ring of integers you selected. For
example:
<programlisting>genius> sqrt(4) mod 7
=
[2, 5]
genius> 2*2 mod 7
= 4</programlisting>
<function>sqrt</function> will actually return all the possible square
roots.
</para>
<para>
Do not chain mod operators, simply place it at the end of the computation, all computations in the expression on the left
will be carried out in mod arithmetic. If you place a mod inside
a mod, you will get unexpected results. If you simply want to
mod a single number and control exactly when remainders are
taken, best to use the <literal>%</literal> operator. When you
need to chain several expressions in modular arithmetic with
different divisors, it may be best to just split up the expression into several and use
temporary variables to avoid a mod inside a mod.
</para>
</sect1>
<sect1 id="genius-gel-operator-list">
<title>Список операторов GEL</title>
<para>
Everything in GEL is really just an expression. Expressions are stringed together with
different operators. As we have seen, even the separator is simply a binary operator
in GEL. Here is a list of the operators in GEL.
</para>
<variablelist>
<varlistentry>
<term><userinput>a;b</userinput></term>
<listitem>
<para>Разделитель, просто вычисляющий как <varname>a</varname>, так и <varname>b</varname>, но возвращающий только результат <varname>b</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a=b</userinput></term>
<listitem>
<para>
The assignment operator. This assigns <varname>b</varname> to
<varname>a</varname> (<varname>a</varname> must be a valid <link linkend="genius-gel-lvalues">lvalue</link>) (note however that this operator
may be translated to <literal>==</literal> if used in a place where boolean
expression is expected)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a:=b</userinput></term>
<listitem>
<para>
The assignment operator. Assigns <varname>b</varname> to
<varname>a</varname> (<varname>a</varname> must be a valid <link linkend="genius-gel-lvalues">lvalue</link>). This is
different from <literal>=</literal> because it never gets translated to a
<literal>==</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>|a|</userinput></term>
<listitem>
<para>
Absolute value.
In case the expression is a complex number the result will be the modulus
(distance from the origin). For example:
<userinput>|3 * e^(1i*pi)|</userinput>
returns 3.
</para>
<para>Смотрите <ulink url="http://mathworld.wolfram.com/AbsoluteValue.html">Mathworld</ulink> для дополнительной информации.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a^b</userinput></term>
<listitem>
<para>Возводит переменную <varname>a</varname> в степень <varname>b</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a.^b</userinput></term>
<listitem>
<para>Поэлементное возведение в степень. Возводит каждый элемент матрицы <varname>a</varname> в степень <varname>b</varname>. Или, если <varname>b</varname> — матрица той же размерности, что и <varname>a</varname>, выполняет операцию поэлементно. Если <varname>a</varname> — число, а <varname>b</varname> — матрица, то создаёт матрицу той же размерности, что и <varname>b</varname> со значением <varname>a</varname>, возведённым во все степени, содержащиеся в <varname>b</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a+b</userinput></term>
<listitem>
<para>
Addition. Adds two numbers, matrices, functions or strings. If
you add a string to anything the result will just be a string. If one is
a square matrix and the other a number, then the number is multiplied by
the identity matrix.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a-b</userinput></term>
<listitem>
<para>Вычитание. Вычитает два числа, матрицы или функции.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a*b</userinput></term>
<listitem>
<para>Умножение. Это обычное умножение матриц.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a.*b</userinput></term>
<listitem>
<para>Поэлементное умножение, если <varname>a</varname> и <varname>b</varname> являются матрицами.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a/b</userinput></term>
<listitem>
<para>
Division. When <varname>a</varname> and <varname>b</varname> are just numbers
this is the normal division. When they are matrices, then this is
equivalent to <userinput>a*b^-1</userinput>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a./b</userinput></term>
<listitem>
<para>
Element by element division. Same as <userinput>a/b</userinput> for
numbers, but operates element by element on matrices.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a\b</userinput></term>
<listitem>
<para>Обратное деление. Это то же самое, что <userinput>b/a</userinput>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a.\b</userinput></term>
<listitem>
<para>Поэлементное обратное деление.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a%b</userinput></term>
<listitem>
<para>
The mod operator. This does not turn on the <link linkend="genius-gel-modular-evaluation">modular mode</link>, but
just returns the remainder of integer division
<userinput>a/b</userinput>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a.%b</userinput></term>
<listitem>
<para>
Element by element mod operator. Returns the remainder
after element by element integer division
<userinput>a./b</userinput>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a mod b</userinput></term>
<listitem>
<para>
Modular evaluation operator. The expression <varname>a</varname>
is evaluated modulo <varname>b</varname>. See <xref linkend="genius-gel-modular-evaluation"/>.
Some functions and operators behave differently modulo an integer.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a!</userinput></term>
<listitem>
<para>Факториал: <userinput>1*...*(n-2)*(n-1)*n</userinput>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a!!</userinput></term>
<listitem>
<para>Двойной факториал: <userinput>1*...*(n-4)*(n-2)*n</userinput>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a==b</userinput></term>
<listitem>
<para>
Equality operator.
Returns <constant>true</constant> or <constant>false</constant>
depending on <varname>a</varname> and <varname>b</varname> being equal or not.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a!=b</userinput></term>
<listitem>
<para>Оператор неравенства, возвращает <constant>true</constant>, если <varname>a</varname> не равно <varname>b</varname>, в противном случае возвращает <constant>false</constant>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a<>b</userinput></term>
<listitem>
<para>Альтернативный оператор неравенства, возвращает <constant>true</constant>, если <varname>a</varname> не равно <varname>b</varname>, иначе возвращает <constant>false</constant>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a<=b</userinput></term>
<listitem>
<para>
Less than or equal operator,
returns <constant>true</constant> if <varname>a</varname> is
less than or equal to
<varname>b</varname> else returns <constant>false</constant>.
These can be chained as in <userinput>a <= b <= c</userinput> (can
also be combined with the less than operator).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a>=b</userinput></term>
<listitem>
<para>
Greater than or equal operator,
returns <constant>true</constant> if <varname>a</varname> is
greater than or equal to
<varname>b</varname> else returns <constant>false</constant>.
These can be chained as in <userinput>a >= b >= c</userinput>
(and they can also be combined with the greater than operator).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a<b</userinput></term>
<listitem>
<para>
Less than operator,
returns <constant>true</constant> if <varname>a</varname> is
less than
<varname>b</varname> else returns <constant>false</constant>.
These can be chained as in <userinput>a < b < c</userinput>
(they can also be combined with the less than or equal to operator).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a>b</userinput></term>
<listitem>
<para>
Greater than operator,
returns <constant>true</constant> if <varname>a</varname> is
greater than
<varname>b</varname> else returns <constant>false</constant>.
These can be chained as in <userinput>a > b > c</userinput>
(they can also be combined with the greater than or equal to operator).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a<=>b</userinput></term>
<listitem>
<para>Оператор сравнения. Если <varname>a</varname> равно <varname>b</varname>, возвращает 0; если <varname>a</varname> меньше <varname>b</varname>, возвращает -1; если <varname>a</varname> больше <varname>b</varname>, возвращает 1.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a and b</userinput></term>
<listitem>
<para>
Logical and. Returns true if both
<varname>a</varname> and <varname>b</varname> are true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a or b</userinput></term>
<listitem>
<para>
Logical or.
Returns true if either
<varname>a</varname> or <varname>b</varname> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a xor b</userinput></term>
<listitem>
<para>
Logical xor.
Returns true if exactly one of
<varname>a</varname> or <varname>b</varname> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>not a</userinput></term>
<listitem>
<para>
Logical not. Returns the logical negation of <varname>a</varname>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>-a</userinput></term>
<listitem>
<para>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a matrix).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>&a</userinput></term>
<listitem>
<para>
Variable referencing (to pass a reference to a variable).
See <xref linkend="genius-gel-references"/>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>*a</userinput></term>
<listitem>
<para>
Variable dereferencing (to access a referenced variable).
See <xref linkend="genius-gel-references"/>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a'</userinput></term>
<listitem>
<para>
Matrix conjugate transpose. That is, rows and columns get swapped and we take complex conjugate of all entries. That is
if the i,j element of <varname>a</varname> is x+iy, then the j,i element of <userinput>a'</userinput> is x-iy.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a.'</userinput></term>
<listitem>
<para>
Matrix transpose, does not conjugate the entries. That is,
the i,j element of <varname>a</varname> becomes the j,i element of <userinput>a.'</userinput>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(b,c)</userinput></term>
<listitem>
<para>
Get element of a matrix in row <varname>b</varname> and column
<varname>c</varname>. If <varname>b</varname>,
<varname>c</varname> are vectors, then this gets the corresponding
rows, columns or submatrices.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(b,)</userinput></term>
<listitem>
<para>
Get row of a matrix (or multiple rows if <varname>b</varname> is a vector).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(b,:)</userinput></term>
<listitem>
<para>То же, что и выше.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(,c)</userinput></term>
<listitem>
<para>Возвращает столбец матрицы (или столбцы, если <varname>c</varname> является вектором).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(:,c)</userinput></term>
<listitem>
<para>То же, что и выше.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(b)</userinput></term>
<listitem>
<para>
Get an element from a matrix treating it as a vector. This will
traverse the matrix row-wise.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a:b</userinput></term>
<listitem>
<para>
Build a vector from <varname>a</varname> to <varname>b</varname> (or specify a row, column region for the <literal>@</literal> operator). For example to get rows 2 to 4 of matrix <varname>A</varname> we could do
<programlisting>A@(2:4,)
</programlisting>
as <userinput>2:4</userinput> will return a vector
<userinput>[2,3,4]</userinput>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a:b:c</userinput></term>
<listitem>
<para>
Build a vector from <varname>a</varname> to <varname>c</varname>
with <varname>b</varname> as a step. That is for example
<programlisting>genius> 1:2:9
=
`[1, 3, 5, 7, 9]
</programlisting>
</para>
<para>
When the numbers involved are floating point numbers, for example
<userinput>1.0:0.4:3.0</userinput>, the output is what is expected
even though adding 0.4 to 1.0 five times is actually just slightly
more than 3.0 due to the way that floating point numbers are
stored in base 2 (there is no 0.4, the actual number stored is
just ever so slightly bigger). The way this is handled is the
same as in the for, sum, and prod loops. If the end is within
<userinput>2^-20</userinput> times the step size of the endpoint,
the endpoint is used and we assume there were roundoff errors.
This is not perfect, but it handles the majority of the cases.
This check is done only from version 1.0.18 onwards, so execution
of your code may differ on older versions. If you want to avoid
dealing with this issue, use actual rational numbers, possibly
using the <function>float</function> if you wish to get floating
point numbers in the end. For example
<userinput>1:2/5:3</userinput> does the right thing and
<userinput>float(1:2/5:3)</userinput> even gives you floating
point numbers and is ever so slightly more precise than
<userinput>1.0:0.4:3.0</userinput>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>(a)i</userinput></term>
<listitem>
<para>
Make <varname>a</varname> into an imaginary number (multiply <varname>a</varname> by the
imaginary). Normally the imaginary number <varname>i</varname> is
written as <userinput>1i</userinput>. So the above is equal to
<programlisting>(a)*1i
</programlisting>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>`a</userinput></term>
<listitem>
<para>
Quote an identifier so that it doesn't get evaluated. Or
quote a matrix so that it doesn't get expanded.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a swapwith b</userinput></term>
<listitem>
<para>Меняет местами значение <varname>a</varname> со значением <varname>b</varname>. В настоящее время не работает с диапазонами элементов матрицы. Возвращает <constant>null</constant>. Доступен, начиная с версии 1.0.13.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>increment a</userinput></term>
<listitem>
<para>Инкремент переменной <varname>a</varname> на 1. Если <varname>a</varname> — матрица, то инкрементирует каждый элемент. Это эквивалентно <userinput>a=a+1</userinput>, но немного быстрее. Возвращает <constant>null</constant>. Доступен с версии 1.0.13.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>increment a by b</userinput></term>
<listitem>
<para>Инкремент переменной <varname>a</varname> на величину <varname>b</varname>. Если <varname>a</varname> — матрица, то инкрементирует каждый элемент. Это эквивалентно <userinput>a=a+b</userinput>, но немного быстрее. Возвращает <constant>null</constant>. Доступен с версии 1.0.13.</para>
</listitem>
</varlistentry>
</variablelist>
<note>
<para>Оператор @() делает использование оператора : наиболее полезным. С его помощью можно указывать области матрицы. Таким образом, a@(2:4,6) — это строки 2,3,4 столбца 6. Или a@(,1:2) возвращает два первых столбца матрицы. Можно также присваивать значения оператору @(), если правое значение — это матрица, совпадающая по размеру с данной областью, или если это любой другой тип значений.</para>
</note>
<note>
<para>
The comparison operators (except for the <=> operator, which behaves normally), are not strictly binary operators, they can in fact be grouped in the normal mathematical way, e.g.: (1<x<=y<5) is a legal boolean expression and means just what it should, that is (1<x and x≤y and y<5)
</para>
</note>
<note>
<para>
The unitary minus operates in a different fashion depending on where it
appears. If it appears before a number it binds very closely, if it appears in
front of an expression it binds less than the power and factorial operators.
So for example <userinput>-1^k</userinput> is really <userinput>(-1)^k</userinput>,
but <userinput>-foo(1)^k</userinput> is really <userinput>-(foo(1)^k)</userinput>. So
be careful how you use it and if in doubt, add parentheses.
</para>
</note>
</sect1>
</chapter>
<!-- ============= GEL Programming ========================== -->
<chapter id="genius-gel-programming">
<title>Программирование в GEL</title>
<sect1 id="genius-gel-conditionals">
<title>Условные операторы</title>
<para>
Syntax:
<programlisting><![CDATA[if <expression1> then <expression2> [else <expression3>]
]]></programlisting>
If <literal>else</literal> is omitted, then if the <literal>expression1</literal> yields <constant>false</constant> or 0, <literal>NULL</literal> is returned.
</para>
<para>
Examples:
<programlisting><![CDATA[if(a==5)then(a=a-1)
if b<a then b=a
if c>0 then c=c-1 else c=0
a = ( if b>0 then b else 1 )
]]></programlisting>
Note that <literal>=</literal> will be translated to <literal>==</literal> if used inside the expression for <literal>if</literal>, so
<programlisting>if a=5 then a=a-1
</programlisting>
will be interpreted as:
<programlisting>if a==5 then a:=a-1
</programlisting>
</para>
</sect1>
<sect1 id="genius-gel-loops">
<title>Циклы</title>
<sect2 id="genius-gel-loops-while">
<title>Циклы While</title>
<para>
Syntax:
<programlisting><![CDATA[while <expression1> do <expression2>
until <expression1> do <expression2>
do <expression2> while <expression1>
do <expression2> until <expression1>]]></programlisting>
These are similar to other languages. However, as in GEL it is simply an expression that must have some return value, these
constructs will simply return the result of the last iteration or <literal>NULL</literal> if no iteration was done. In the boolean expression, <literal>=</literal> is translated into <literal>==</literal> just as for the <literal>if</literal> statement.
</para>
</sect2>
<sect2 id="genius-gel-loops-for">
<title>Циклы For</title>
<para>
Syntax:
<programlisting><![CDATA[for <identifier> = <from> to <to> do <body>
for <identifier> = <from> to <to> by <increment> do <body>]]></programlisting>
Loop with identifier being set to all values from <literal><from></literal> to <literal><to></literal>, optionally using an increment other than 1. These are faster, nicer and more compact than the normal loops such as above, but less flexible. The identifier must be an identifier and can't be a dereference. The value of identifier is the last value of identifier, or <literal><from></literal> if body was never evaluated. The variable is guaranteed to be initialized after a loop, so you can safely use it. Also the <literal><from></literal>, <literal><to></literal> and <literal><increment></literal> must be non complex values. The <literal><to></literal> is not guaranteed to be hit, but will never be overshot, for example the following prints out odd numbers from 1 to 19:
<programlisting>for i = 1 to 20 by 2 do print(i)
</programlisting>
</para>
<para>
When one of the values is a floating point number, then the
final check is done to within 2^-20 of the step size. That is,
even if we overshoot by 2^-20 times the "by" above, we still execute the last
iteration. This way
<programlisting>for x = 0 to 1 by 0.1 do print(x)
</programlisting>
does the expected even though adding 0.1 ten times becomes just slightly more than 1.0 due to the way that floating point numbers
are stored in base 2 (there is no 0.1, the actual number stored is just ever so slightly bigger). This is not perfect but it handles
the majority of the cases. If you want to avoid dealing with this issue, use actual rational numbers for example:
<programlisting>for x = 0 to 1 by 1/10 do print(x)
</programlisting>
This check is done only from version 1.0.16 onwards, so execution of your code may differ on older versions.
</para>
</sect2>
<sect2 id="genius-gel-loops-foreach">
<title>Циклы Foreach</title>
<para>
Syntax:
<programlisting><![CDATA[for <identifier> in <matrix> do <body>]]></programlisting>
For each element in the matrix, going row by row from left to right we execute the body
with the identifier set to the current element. To
print numbers 1,2,3 and 4 in this order you could do:
<programlisting>for n in [1,2:3,4] do print(n)
</programlisting>
If you wish to run through the rows and columns of a matrix, you can use
the RowsOf and ColumnsOf functions, which return a vector of the rows or
columns of the matrix. So,
<programlisting>for n in RowsOf ([1,2:3,4]) do print(n)
</programlisting>
will print out [1,2] and then [3,4].
</para>
</sect2>
<sect2 id="genius-gel-loops-break-continue">
<title>Break и Continue</title>
<para>В циклах также можно использовать команды <literal>break</literal> и <literal>continue</literal>. Команда <literal>continue</literal> перезапускает текущий цикл с его следующей итерации, а команда <literal>break</literal> позволяет выйти из текущего цикла. <programlisting><![CDATA[while(<expression1>) do (
if(<expression2>) break
else if(<expression3>) continue;
<expression4>
)
]]></programlisting></para>
</sect2>
</sect1>
<sect1 id="genius-gel-sums-products">
<title>Суммы и произведения</title>
<para>Синтаксис: <programlisting><![CDATA[sum <identifier> = <from> to <to> do <body>
sum <identifier> = <from> to <to> by <increment> do <body>
sum <identifier> in <matrix> do <body>
prod <identifier> = <from> to <to> do <body>
prod <identifier> = <from> to <to> by <increment> do <body>
prod <identifier> in <matrix> do <body>]]></programlisting> Если заменить <literal>for</literal> на <literal>sum</literal> или <literal>prod</literal>, то вместо цикла <literal>for</literal> получатся циклы вычисления суммы или произведения. Вместо того, чтобы возвращать последнее значение, эти команды возвращают сумму или произведение значений, соответственно.</para>
<para>Если тело цикла не вычислялось (например, <userinput>sum i=1 to 0 do ...</userinput>), то <literal>sum</literal> возвращает 0, а <literal>prod</literal> возвращает 1.</para>
<para>
For floating point numbers the same roundoff error protection is done as in the for loop.
See <xref linkend="genius-gel-loops-for"/>.
</para>
</sect1>
<sect1 id="genius-gel-comparison-operators">
<title>Операторы сравнения</title>
<para>В GEL поддерживаются следующие стандартные операторы сравнения, имеющие очевидное значение: <literal>==</literal>, <literal>>=</literal>, <literal><=</literal>, <literal>!=</literal>, <literal><></literal>, <literal><</literal>, <literal>></literal>. Они возвращают <constant>true</constant> или <constant>false</constant>. Операторы <literal>!=</literal> и <literal><></literal> эквивалентны и означают «не равно». GEL также поддерживает оператор <literal><=></literal>, который возвращает -1, если левая сторона меньше, 0 при равенстве обеих сторон и 1, если левая сторона больше.</para>
<para>
Normally <literal>=</literal> is translated to <literal>==</literal> if
it happens to be somewhere where GEL is expecting a condition such as
in the if condition. For example
<programlisting>if a=b then c
if a==b then c
</programlisting>
are the same thing in GEL. However you should really use
<literal>==</literal> or <literal>:=</literal> when you want to compare
or assign respectively if you want your code to be easy to read and
to avoid mistakes.
</para>
<para>
All the comparison operators (except for the
<literal><=></literal> operator, which
behaves normally), are not strictly binary operators, they can in fact
be grouped in the normal mathematical way, e.g.:
(<literal>1<x<=y<5</literal>) is
a legal boolean expression and means just what it should, that is
(1<x and x≤y and y<5)
</para>
<para>
To build up logical expressions use the words <literal>not</literal>,
<literal>and</literal>, <literal>or</literal>, <literal>xor</literal>.
The operators <literal>or</literal> and <literal>and</literal> are
special beasts as they evaluate their arguments one by one, so the usual trick
for conditional evaluation works here as well. For example, <literal>1 or a=1</literal> will not set
<literal>a=1</literal> since the first argument was true.
</para>
</sect1>
<sect1 id="genius-gel-variables-global">
<title>Глобальные переменные и область видимости переменных</title>
<para>
GEL is a
<ulink url="https://en.wikipedia.org/wiki/Scope_%28programming%29">
dynamically scoped language</ulink>. We will explain what this
means below. That is, normal variables and functions are dynamically
scoped. The exception are
<link linkend="genius-gel-parameters">parameter variables</link>,
which are always global.
</para>
<para>
Like most programming languages, GEL has different types
of variables. Normally when a variable is defined in a function,
it is visible from that function and from all functions that are
called (all higher contexts). For example, suppose a function
<function>f</function> defines a variable <varname>a</varname>
and then calls function <function>g</function>. Then
function <function>g</function> can reference
<varname>a</varname>. But once <function>f</function> returns,
the variable <varname>a</varname> goes out of scope.
For example, the following code will print out 5.
The function <function>g</function> cannot be called on the
top level (outside <function>f</function> as <varname>a</varname>
will not be defined).
<programlisting>function f() = (a:=5; g());
function g() = print(a);
f();
</programlisting>
</para>
<para>
If you define a variable inside a function it will override
any variables defined in calling functions. For example,
we modify the above code and write:
<programlisting>function f() = (a:=5; g());
function g() = print(a);
a:=10;
f();
</programlisting>
This code will still print out 5. But if you call
<function>g</function> outside of <function>f</function> then
you will get a printout of 10. Note that
setting <varname>a</varname>
to 5 inside <function>f</function> does not change
the value of <varname>a</varname> at the top (global) level,
so if you now check the value of <varname>a</varname> it will
still be 10.
</para>
<para>
Function arguments are exactly like variables defined inside
the function, except that they are initialized with the value
that was passed to the function. Other than this point, they are
treated just like all other variables defined inside the
function.
</para>
<para>
Functions are treated exactly like variables. Hence you can
locally redefine functions. Normally (on the top level) you
cannot redefine protected variables and functions. But locally
you can do this. Consider the following session:
<screen><prompt>genius> </prompt><userinput>function f(x) = sin(x)^2</userinput>
= (`(x)=(sin(x)^2))
<prompt>genius> </prompt><userinput>function f(x) = sin(x)^2</userinput>
= (`(x)=(sin(x)^2))
<prompt>genius> </prompt><userinput>function g(x) = ((function sin(x)=x^10);f(x))</userinput>
= (`(x)=((sin:=(`(x)=(x^10)));f(x)))
<prompt>genius> </prompt><userinput>g(10)</userinput>
= 1e20
</screen>
</para>
<para>
Functions and variables defined at the top level are
considered global. They are visible from anywhere. As we
said the following function <function>f</function>
will not change the value of <varname>a</varname> to 5.
<programlisting>a=6;
function f() = (a:=5);
f();
</programlisting>
Sometimes, however, it is necessary to set
a global variable from inside a function. When this behavior is needed,
use the
<link linkend="gel-function-set"><function>set</function></link> function. Passing a string or a quoted identifier to
this function sets the variable globally (on the top level).
For example, to set
<varname>a</varname> to the value 3 you could call:
<programlisting>set(`a,3)
</programlisting>
or:
<programlisting>set("a",3)
</programlisting>
</para>
<para>
The <function>set</function> function always sets the toplevel
global. There is no way to set a local variable in some function
from a subroutine. If this is required, must use passing by
reference.
</para>
<para>
See also the
<link linkend="gel-function-SetElement"><function>SetElement</function></link> and
<link linkend="gel-function-SetVElement"><function>SetVElement</function></link> functions.
</para>
<para>
So to recap in a more technical language: Genius operates with
different numbered contexts. The top level is the context 0
(zero). Whenever a function is entered, the context is raised,
and when the function returns the context is lowered. A function
or a variable is always visible from all higher numbered contexts.
When a variable was defined in a lower numbered context, then
setting this variable has the effect of creating a new local
variable in the current context number and this variable
will now be visible from all higher numbered contexts.
</para>
<para>
There are also true local variables that are not seen from
anywhere but the current context. Also when returning functions
by value it may reference variables not visible from higher context
and this may be a problem. See the sections
<link linkend="genius-gel-true-local-variables">True
Local Variables</link> and
<link linkend="genius-gel-returning-functions">Returning
Functions</link>.
</para>
</sect1>
<sect1 id="genius-gel-parameters">
<title>Parameter variables</title>
<para>
As we said before, there exist special variables called parameters
that exist in all scopes. To declare a parameter called
<varname>foo</varname> with the initial value 1, we write
<programlisting><![CDATA[parameter foo = 1
]]></programlisting>
From then on, <varname>foo</varname> is a strictly global variable.
Setting <varname>foo</varname> inside any function will modify the
variable in all contexts, that is, functions do not have a private
copy of parameters.
</para>
<para>
When you undefine a parameter using the
<link linkend="gel-function-undefine">
<function>undefine</function></link> function, it stops being
a parameter.
</para>
<para>
Some parameters are built-in and modify the behavior of genius.
</para>
</sect1>
<sect1 id="genius-gel-returning">
<title>Returning</title>
<para>
Normally a function is one or several expressions separated by a
semicolon, and the value of the last expression is returned. This is fine for
simple functions, but
sometimes you do not want a function to return the last thing calculated. You may, for example, want to return from a middle of a function. In this case, you can use the <literal>return</literal> keyword. <literal>return</literal> takes one argument, which is the value to be returned.
</para>
<para>Пример: <programlisting><![CDATA[function f(x) = (
y=1;
while true do (
if x>50 then return y;
y=y+1;
x=x+1
)
)
]]></programlisting></para>
</sect1>
<sect1 id="genius-gel-references">
<title>References</title>
<para>
It may be necessary for some functions to return more than one value.
This may be accomplished by returning a vector of values, but many
times it is convenient to use passing a reference to a variable.
You pass a reference to a variable to a function, and the function
will set the variable for you using a dereference. You do not have
to use references only for this purpose, but this is their main use.
</para>
<para>
When using functions that return values through references
in the argument list, just pass the variable name with an ampersand.
For example the following code will compute an eigenvalue of a matrix
<varname>A</varname> with initial eigenvector guess
<varname>x</varname>, and store the computed eigenvector
into the variable named <varname>v</varname>:
<programlisting><![CDATA[RayleighQuotientIteration (A,x,0.001,100,&v)
]]></programlisting>
</para>
<para>
The details of how references work and the syntax is similar to the C language.
The operator
<literal>&</literal> references a variable
and <literal>*</literal> dereferences a variable. Both can only be applied to an identifier,
so <literal>**a</literal> is not a legal expression in GEL.
</para>
<para>
References are best explained by an example:
<programlisting><![CDATA[a=1;
b=&a;
*b=2;
]]></programlisting>
now <varname>a</varname> contains 2. You can also reference functions:
<programlisting><![CDATA[function f(x) = x+1;
t=&f;
*t(3)
]]></programlisting>
gives us 4.
</para>
</sect1>
<sect1 id="genius-gel-lvalues">
<title>Lvalues</title>
<para>
An lvalue is the left hand side of an assignment. In other words, an
lvalue is what you assign something to. Valid lvalues are:
<variablelist>
<varlistentry>
<term><userinput>a</userinput></term>
<listitem>
<para>
Identifier. Here we would be setting the variable of name
<varname>a</varname>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>*a</userinput></term>
<listitem>
<para>
Dereference of an identifier. This will set whatever variable
<varname>a</varname> points to.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><userinput>a@(<region>)</userinput></term>
<listitem>
<para>
A region of a matrix. Here the region is specified normally as with
the regular @() operator, and can be a single entry, or an entire
region of the matrix.
</para>
</listitem>
</varlistentry>
</variablelist>
</para>
<para>
Examples:
<programlisting>a:=4
*tmp := 89
a@(1,1) := 5
a@(4:8,3) := [1,2,3,4,5]'
</programlisting>
Note that both <literal>:=</literal> and <literal>=</literal> can be used
interchangeably. Except if the assignment appears in a condition.
It is thus always safer to just use
<literal>:=</literal> when you mean assignment, and <literal>==</literal>
when you mean comparison.
</para>
</sect1>
</chapter>
<chapter id="genius-gel-programming-advanced">
<title>Advanced Programming with GEL</title>
<sect1 id="genius-gel-error-handling">
<title>Обработка ошибок</title>
<para>
If you detect an error in your function, you can bail out of it. For normal
errors, such as wrong types of arguments, you can fail to compute the function
by adding the statement <literal>bailout</literal>. If something went
really wrong and you want to completely kill the current computation, you can
use <literal>exception</literal>.
</para>
<para>
For example if you want to check for arguments in your function. You
could use the following code.
<programlisting>function f(M) = (
if not IsMatrix (M) then (
error ("M not a matrix!");
bailout
);
...
)
</programlisting>
</para>
</sect1>
<sect1 id="genius-gel-toplevel-syntax">
<title>Toplevel Syntax</title>
<para>
The syntax is slightly different if you enter statements on
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
parentheses). When a statement does not end with a separator on the
top level, the result is printed after being executed.
</para>
<para>
For example,
<programlisting>function f(x)=x^2
f(3)
</programlisting>
will print first the result of setting a function (a representation of
the function, in this case <computeroutput>(`(x)=(x^2))</computeroutput>)
and then the expected 9. To avoid this, enter a separator
after the function definition.
<programlisting>function f(x)=x^2;
f(3)
</programlisting>
If you need to put a separator into your function then you have to surround with
parenthesis. For example:
<programlisting>function f(x)=(
y=1;
for j=1 to x do
y = y+j;
y^2
);
</programlisting>
</para>
<para>
The following code will produce an error when entered on the top
level of a program, while it will work just fine in a function.
<programlisting>if Something() then
DoSomething()
else
DoSomethingElse()
</programlisting>
</para>
<para>
The problem is that after <application>Genius Mathematics Tool</application> sees the end of line after the
second line, it will decide that we have whole statement and
it will execute it. After the execution is done, <application>Genius Mathematics Tool</application> will
go on to the next
line, it will see <literal>else</literal>, and it will produce
a parsing error. To fix this, use parentheses. <application>Genius Mathematics Tool</application> will not
be satisfied until it has found that all parentheses are closed.
<programlisting>if Something() then (
DoSomething()
) else (
DoSomethingElse()
)
</programlisting>
</para>
</sect1>
<sect1 id="genius-gel-returning-functions">
<title>Returning Functions</title>
<para>
It is possible to return functions as value. This way you can
build functions that construct special purpose functions according
to some parameters. The tricky bit is what variables does the
function see. The way this works in GEL is that when a function
returns another function, all identifiers referenced in the
function body that went out of scope
are prepended a private dictionary of the returned
function. So the function will see all variables that were in
scope
when it was defined. For example, we define a function that
returns a function that adds 5 to its argument.
<programlisting>function f() = (
k = 5;
`(x) = (x+k)
)
</programlisting>
Notice that the function adds <varname>k</varname> to
<varname>x</varname>. You could use this as follows.
<programlisting>g = f();
g(5)
</programlisting>
And <userinput>g(5)</userinput> should return 10.
</para>
<para>
One thing to note is that the value of <varname>k</varname>
that is used is the one that's in effect when the
<function>f</function> returns. For example:
<programlisting>function f() = (
k := 5;
function r(x) = (x+k);
k := 10;
r
)
</programlisting>
will return a function that adds 10 to its argument rather than
5. This is because the extra dictionary is created only when
the context
in which the function was defined ends, which is when the function
<function>f</function> returns. This is consistent with how you
would expect the function <function>r</function> to work inside
the function <function>f</function> according to the rules of
scope of variables in GEL. Only those variables are added to the
extra dictionary that are in the context that just ended and
no longer exists. Variables
used in the function that are in still valid contexts will work
as usual, using the current value of the variable.
The only difference is with global variables and functions.
All identifiers that referenced global variables at time of
the function definition are not added to the private dictionary.
This is to avoid much unnecessary work when returning functions
and would rarely be a problem. For example, suppose that you
delete the "k=5" from the function <function>f</function>,
and at the top level you define <varname>k</varname> to be
say 5. Then when you run <function>f</function>, the function
<function>r</function> will not put <varname>k</varname> into
the private dictionary because it was global (toplevel)
at the time of definition of <function>r</function>.
</para>
<para>
Sometimes it is better to have more control over how variables
are copied into the private dictionary. Since version 1.0.7,
you can specify which
variables are copied into the private dictionary by putting
extra square brackets after the arguments with the list of
variables to be copied separated by commas.
If you do this, then variables are
copied into the private dictionary at time of the function
definition, and the private dictionary is not touched afterwards.
For example
<programlisting>function f() = (
k := 5;
function r(x) [k] = (x+k);
k := 10;
r
)
</programlisting>
will return a function that when called will add 5 to its
argument. The local copy of <varname>k</varname> was created
when the function was defined.
</para>
<para>
When you want the function to not have any private dictionary
then put empty square brackets after the argument list. Then
no private dictionary will be created at all. Doing this is
good to increase efficiency when a private dictionary is not
needed or when you want the function to lookup all variables
as it sees them when called. For example suppose you want
the function returned from <function>f</function> to see
the value of <varname>k</varname> from the toplevel despite
there being a local variable of the same name during definition.
So the code
<programlisting>function f() = (
k := 5;
function r(x) [] = (x+k);
r
);
k := 10;
g = f();
g(10)
</programlisting>
will return 20 and not 15, which would happen if
<varname>k</varname> with a value of 5 was added to the private
dictionary.
</para>
</sect1>
<sect1 id="genius-gel-true-local-variables">
<title>True Local Variables</title>
<para>
When passing functions into other functions, the normal scoping of
variables might be undesired. For example:
<programlisting>k := 10;
function r(x) = (x+k);
function f(g,x) = (
k := 5;
g(x)
);
f(r,1)
</programlisting>
you probably want the function <function>r</function>
when passed as <function>g</function> into <function>f</function>
to see <varname>k</varname> as 10 rather than 5, so that
the code returns 11 and not 6. However, as written, the function
when executed will see the <varname>k</varname> that is
equal to 5. There are two ways to solve this. One would be
to have <function>r</function> get <varname>k</varname> in a
private dictionary using the square bracket notation section
<link linkend="genius-gel-returning-functions">Returning
Functions</link>.
</para>
<para>
But there is another solution. Since version 1.0.7 there are
true local variables. These are variables that are visible only
from the current context and not from any called functions.
We could define <varname>k</varname> as a local variable in the
function <function>f</function>. To do this add a
<command>local</command> statement as the first statement in the
function (it must always be the first statement in the function).
You can also make any arguments be local variables as well.
That is,
<programlisting>function f(g,x) = (
local g,x,k;
k := 5;
g(x)
);
</programlisting>
Then the code will work as expected and prints out 11.
Note that the <command>local</command> statement initializes
all the referenced variables (except for function arguments) to
a <constant>null</constant>.
</para>
<para>
If all variables are to be created as locals you can just pass an
asterisk instead of a list of variables. In this case the variables
will not be initialized until they are actually set of course.
So the following definition of <function>f</function>
will also work:
<programlisting>function f(g,x) = (
local *;
k := 5;
g(x)
);
</programlisting>
</para>
<para>
It is good practice that all functions that take other functions
as arguments use local variables. This way the passed function
does not see implementation details and get confused.
</para>
</sect1>
<sect1 id="genius-gel-startup-procedure">
<title>GEL Startup Procedure</title>
<para>
First the program looks for the installed library file (the compiled version <filename>lib.cgel</filename>) in the installed directory, then it looks into the current directory, and then it tries to load an uncompiled file called
<filename>~/.geniusinit</filename>.
</para>
<para>
If you ever change the library in its installed place, you’ll have to
first compile it with <command>genius --compile loader.gel > lib.cgel</command>
</para>
</sect1>
<sect1 id="genius-gel-loading-programs">
<title>Загрузка программ</title>
<para>
Sometimes you have a larger program you wrote into a file and want to read that file into <application>Genius Mathematics Tool</application>. In these situations, you have two options. You can keep the functions you use most inside the <filename>~/.geniusinit</filename> file. Or if you want to load up a file in a middle of a session (or from within another file), you can type <command>load <list of filenames></command> at the prompt. This has to be done on the top level and not inside any function or whatnot, and it cannot be part of any expression. It also has a slightly different syntax than the rest of genius, more similar to a shell. You can enter the file in quotes. If you use the '' quotes, you will get exactly the string that you typed, if you use the "" quotes, special characters will be unescaped as they are for strings. Example:
<programlisting>load program1.gel program2.gel
load "Причудливое имя файла с ПРОБЕЛАМИ.gel"
</programlisting>
There are also <command>cd</command>, <command>pwd</command> and <command>ls</command> commands built in. <command>cd</command> will take one argument, <command>ls</command> will take an argument that is like the glob in the UNIX shell (i.e., you can use wildcards). <command>pwd</command> takes no arguments. For example:
<programlisting>cd каталог_с_программами_gel
ls *.gel
</programlisting>
</para>
</sect1>
</chapter>
<!-- ============= Matrices ================================= -->
<chapter id="genius-gel-matrices">
<title>Матрицы в GEL</title>
<para>
Genius has support for vectors and matrices and possesses a sizable library of
matrix manipulation and linear algebra functions.
</para>
<sect1 id="genius-gel-matrix-support">
<title>Ввод матриц</title>
<para>
To enter matrices, you can use one of the following two syntaxes. You can either enter
the matrix on one line, separating values by commas and rows by semicolons. Or you
can enter each row on one line, separating
values by commas.
You can also just combine the two methods.
So to enter a 3x3 matrix
of numbers 1-9 you could do
<programlisting>[1,2,3;4,5,6;7,8,9]
</programlisting>
or
<programlisting>[1, 2, 3
4, 5, 6
7, 8, 9]
</programlisting>
Do not use both ';' and return at once on the same line though.
</para>
<para>
You can also use the matrix expansion functionality to enter matrices.
For example you can do:
<programlisting>a = [ 1, 2, 3
4, 5, 6
7, 8, 9]
b = [ a, 10
11, 12]
</programlisting>
and you should get
<programlisting>[1, 2, 3, 10
4, 5, 6, 10
7, 8, 9, 10
11, 11, 11, 12]
</programlisting>
similarly you can build matrices out of vectors and other stuff like that.
</para>
<para>
Another thing is that non-specified spots are initialized to 0, so
<programlisting>[1, 2, 3
4, 5
6]
</programlisting>
will end up being
<programlisting>
[1, 2, 3
4, 5, 0
6, 0, 0]
</programlisting>
</para>
<para>
When matrices are evaluated, they are evaluated and traversed row-wise. This is just
like the <literal>M@(j)</literal> operator, which traverses the matrix row-wise.
</para>
<note>
<para>
Be careful about using returns for expressions inside the
<literal>[ ]</literal> brackets, as they have a slightly different meaning
there. You will start a new row.
</para>
</note>
</sect1>
<sect1 id="genius-gel-matrix-transpose">
<title>Conjugate Transpose and Transpose Operator</title>
<para>
You can conjugate transpose a matrix by using the <literal>'</literal> operator. That is
the entry in the
<varname>i</varname>th column and the <varname>j</varname>th row will be
the complex conjugate of the entry in the
<varname>j</varname>th column and the <varname>i</varname>th row of the original matrix.
For example:
<programlisting>[1,2,3]*[4,5,6]'
</programlisting>
We transpose the second vector to make matrix multiplication possible.
If you just want to transpose a matrix without conjugating it, you would
use the <literal>.'</literal> operator. For example:
<programlisting>[1,2,3]*[4,5,6i].'
</programlisting>
</para>
<para>
Note that normal transpose, that is the <literal>.'</literal> operator, is much faster
and will not create a new copy of the matrix in memory. The conjugate transpose does
create a new copy unfortunately.
It is recommended to always use the <literal>.'</literal> operator when working with real
matrices and vectors.
</para>
</sect1>
<sect1 id="genius-gel-matrix-linalg">
<title>Линейная алгебра</title>
<para>
Genius implements many useful linear algebra and matrix manipulation
routines. See the <link linkend="genius-gel-function-list-linear-algebra">Linear Algebra</link> and
<link linkend="genius-gel-function-list-matrix">Matrix Manipulation</link>
sections of the GEL function listing.
</para>
<para>
The linear algebra routines implemented in GEL do not currently come
from a well tested numerical package, and thus should not be used for critical
numerical computation. On the other hand, Genius implements very well many
linear algebra operations with rational and integer coefficients. These are
inherently exact and in fact will give you much better results than common
double precision routines for linear algebra.
</para>
<para>
For example, it is pointless to compute the rank and nullspace of a
floating point matrix since for all practical purposes, we need to consider the
matrix as having some slight errors. You are likely to get a different result
than you expect. The problem is that under a small perturbation every matrix
is of full rank and invertible. If the matrix however is of rational numbers,
then the rank and nullspace are always exact.
</para>
<para>
In general when Genius computes the basis of a certain vectorspace
(for example with the <link linkend="gel-function-NullSpace"><function>NullSpace</function></link>) it will give the basis as
a matrix, in which the columns are the vectors of the basis. That is, when
Genius talks of a linear subspace it means a matrix whose column space is
the given linear subspace.
</para>
<para>
It should be noted that Genius can remember certain properties of a
matrix. For example, it will remember that a matrix is in row reduced form.
If many calls are made to functions that internally use row reduced form of
the matrix, we can just row reduce the matrix beforehand once. Successive
calls to <link linkend="gel-function-rref"><function>rref</function></link> will be very fast.
</para>
</sect1>
</chapter>
<!-- ============= Polynomials ============================== -->
<chapter id="genius-gel-polynomials">
<title>Многочлены в GEL</title>
<para>В настоящее время Genius может работать с многочленами одной переменной, записанными в виде векторов, и выполнять некоторые основные операции с ними. В будущем планируется расширить их поддержку.</para>
<sect1 id="genius-gel-polynomials-using">
<title>Использование многочленов</title>
<para>
Currently
polynomials in one variable are just horizontal vectors with value only nodes.
The power of the term is the position in the vector, with the first position
being 0. So,
<programlisting>[1,2,3]
</programlisting>
translates to a polynomial of
<programlisting>1 + 2*x + 3*x^2
</programlisting>
</para>
<para>
You can add, subtract and multiply polynomials using the
<link linkend="gel-function-AddPoly"><function>AddPoly</function></link>,
<link linkend="gel-function-SubtractPoly"><function>SubtractPoly</function></link>, and
<link linkend="gel-function-MultiplyPoly"><function>MultiplyPoly</function></link> functions respectively.
You can print a polynomial using the
<link linkend="gel-function-PolyToString"><function>PolyToString</function></link>
function.
For example,
<programlisting>PolyToString([1,2,3],"y")
</programlisting>
gives
<programlisting>3*y^2 + 2*y + 1
</programlisting>
You can also get a function representation of the polynomial so that you can
evaluate it. This is done by using
<link linkend="gel-function-PolyToFunction"><function>PolyToFunction</function></link>,
which
returns an anonymous function.
<programlisting>f = PolyToFunction([0,1,1])
f(2)
</programlisting>
</para>
<para>
It is also possible to find roots of polynomials of degrees 1 through 4 by using the
function
<link linkend="gel-function-PolynomialRoots"><function>PolynomialRoots</function></link>,
which calls the appropriate formula function. Higher degree polynomials must be converted to
functions and solved
numerically using a function such as
<link linkend="gel-function-FindRootBisection"><function>FindRootBisection</function></link>,
<link linkend="gel-function-FindRootFalsePosition"><function>FindRootFalsePosition</function></link>,
<link linkend="gel-function-FindRootMullersMethod"><function>FindRootMullersMethod</function></link>, or
<link linkend="gel-function-FindRootSecant"><function>FindRootSecant</function></link>.
</para>
<para>
See <xref linkend="genius-gel-function-list-polynomials"/> in the function list
for the rest of functions acting on polynomials.
</para>
</sect1>
</chapter>
<!-- ============= Set Theory ============================== -->
<chapter id="genius-gel-settheory">
<title>Теория множеств в GEL</title>
<para>
Genius has some basic set theoretic functionality built in. Currently a set is
just a vector (or a matrix). Every distinct object is treated as a different element.
</para>
<sect1 id="genius-gel-sets-using">
<title>Using Sets</title>
<para>
Just like vectors, objects
in sets can include numbers, strings, <constant>null</constant>, matrices and vectors. It is
planned in the future to have a dedicated type for sets, rather than using vectors.
Note that floating point numbers are distinct from integers, even if they appear the same.
That is, Genius will treat <constant>0</constant> and <constant>0.0</constant>
as two distinct elements. The <constant>null</constant> is treated as an empty set.
</para>
<para>
To build a set out of a vector, use the
<link linkend="gel-function-MakeSet"><function>MakeSet</function></link> function.
Currently, it will just return a new vector where every element is unique.
<screen><prompt>genius> </prompt><userinput>MakeSet([1,2,2,3])</userinput>
= [1, 2, 3]
</screen>
</para>
<para>
Similarly there are functions
<link linkend="gel-function-Union"><function>Union</function></link>,
<link linkend="gel-function-Intersection"><function>Intersection</function></link>,
<link linkend="gel-function-SetMinus"><function>SetMinus</function></link>, which
are rather self explanatory. For example:
<screen><prompt>genius> </prompt><userinput>Union([1,2,3], [1,2,4])</userinput>
= [1, 2, 4, 3]
</screen>
Note that no order is guaranteed for the return values. If you wish to sort the vector you
should use the
<link linkend="gel-function-SortVector"><function>SortVector</function></link> function.
</para>
<para>
For testing membership, there are functions
<link linkend="gel-function-IsIn"><function>IsIn</function></link> and
<link linkend="gel-function-IsSubset"><function>IsSubset</function></link>,
which return a boolean value. For example:
<screen><prompt>genius> </prompt><userinput>IsIn (1, [0,1,2])</userinput>
= true
</screen>
The input <userinput>IsIn(x,X)</userinput> is of course equivalent to
<userinput>IsSubset([x],X)</userinput>. Note that since the empty set is a subset
of every set, <userinput>IsSubset(null,X)</userinput> is always true.
</para>
</sect1>
</chapter>
<!-- ============= GEL function list ======================== -->
<chapter id="genius-gel-function-list">
<title>Список функций GEL</title>
<!--&gel-function-list;-->
<para>Для получения справки по определённой функции, наберите в консоли: <programlisting>help ИмяФункции
</programlisting></para>
<sect1 id="genius-gel-function-list-commands">
<title>Команды</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-command-help"/>help</term>
<listitem>
<synopsis>help</synopsis>
<synopsis>help ИмяФункции</synopsis>
<para>Показывает справку (или справку по функции/команде).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-command-load"/>load</term>
<listitem>
<synopsis>load "file.gel"</synopsis>
<para>Load a file into the interpreter. The file will execute
as if it were typed onto the command line.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-command-cd"/>cd</term>
<listitem>
<synopsis>cd /каталог/имя</synopsis>
<para>Меняет рабочий каталог на <filename>/каталог/имя</filename>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-command-pwd"/>pwd</term>
<listitem>
<synopsis>pwd</synopsis>
<para>Выводит текущий рабочий каталог.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-command-ls"/>ls</term>
<listitem>
<synopsis>ls</synopsis>
<para>Показывает список файлов в текущем каталоге.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-command-plugin"/>plugin</term>
<listitem>
<synopsis>plugin plugin_name</synopsis>
<para>Load a plugin. Plugin of that name must be installed on the system
in the proper directory.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-basic">
<title>Основные</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-AskButtons"/>AskButtons</term>
<listitem>
<synopsis>AskButtons (вопрос)</synopsis>
<synopsis>AskButtons (вопрос, кнопка1, ...)</synopsis>
<para>Задаёт вопрос и предлагает пользователю список кнопок (или меню вариантов в текстовом режиме). Возвращает отсчитываемый с 1 индекс нажатой кнопки. То есть 1, если нажата первая кнопка, 2 — если нажата вторая и т.д. Если пользователь закрыл окно (или просто нажал Enter в текстовом режиме), то возвращает <constant>null</constant>. Выполнение программы останавливается, пока пользователь не ответит.</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-AskString"/>AskString</term>
<listitem>
<synopsis>AskString (query)</synopsis>
<synopsis>AskString (query, default)</synopsis>
<para>Asks a question and lets the user enter a string, which
it then returns. If the user cancels or closes the window, then
<constant>null</constant> is returned. The execution of the program
is blocked until the user responds. If <varname>default</varname> is given, then it is pre-typed in for the user to just press enter on (version 1.0.6 onwards).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Compose"/>Compose</term>
<listitem>
<synopsis>Compose (f,g)</synopsis>
<para>Compose two functions and return a function that is the composition of <function>f</function> and <function>g</function>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ComposePower"/>ComposePower</term>
<listitem>
<synopsis>ComposePower (f,n,x)</synopsis>
<para>Compose and execute a function with itself <varname>n</varname> times, passing <varname>x</varname> as argument. Returning <varname>x</varname> if
<varname>n</varname> equals 0.
Example:
<screen><prompt>genius></prompt> <userinput>function f(x) = x^2 ;</userinput>
<prompt>genius></prompt> <userinput>ComposePower (f,3,7)</userinput>
= 5764801
<prompt>genius></prompt> <userinput>f(f(f(7)))</userinput>
= 5764801
</screen>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Evaluate"/>Evaluate</term>
<listitem>
<synopsis>Evaluate (str)</synopsis>
<para>Parses and evaluates a string.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GetCurrentModulo"/>GetCurrentModulo</term>
<listitem>
<synopsis>GetCurrentModulo</synopsis>
<para>Get current modulo from the context outside the function. That is, if outside of
the function was executed in modulo (using <literal>mod</literal>) then this returns what
this modulo was. Normally the body of the function called is not executed in modular arithmetic,
and this builtin function makes it possible to make GEL functions aware of modular arithmetic.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Identity"/>Identity</term>
<listitem>
<synopsis>Identity (x)</synopsis>
<para>Identity function, returns its argument. It is equivalent to <userinput>function Identity(x)=x</userinput>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IntegerFromBoolean"/>IntegerFromBoolean</term>
<listitem>
<synopsis>IntegerFromBoolean (bval)</synopsis>
<para>Преобразует логическое значение в целое число (0 для <constant>false</constant> или 1 для <constant>true</constant>). <varname>bval</varname> может также быть числом, в этом случае ненулевое значение интерпретируется как <constant>true</constant>, а нулевое — как <constant>false</constant>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsBoolean"/>IsBoolean</term>
<listitem>
<synopsis>IsBoolean (arg)</synopsis>
<para>Проверяет, является аргумент логическим значением (а не числом).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsDefined"/>IsDefined</term>
<listitem>
<synopsis>IsDefined (id)</synopsis>
<para>Check if an id is defined. You should pass a string or
and identifier. If you pass a matrix, each entry will be
evaluated separately and the matrix should contain strings
or identifiers.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsFunction"/>IsFunction</term>
<listitem>
<synopsis>IsFunction (arg)</synopsis>
<para>Проверяет, является ли аргумент функцией.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsFunctionOrIdentifier"/>IsFunctionOrIdentifier</term>
<listitem>
<synopsis>IsFunctionOrIdentifier (arg)</synopsis>
<para>Check if argument is a function or an identifier.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsFunctionRef"/>IsFunctionRef</term>
<listitem>
<synopsis>IsFunctionRef (arg)</synopsis>
<para>Check if argument is a function reference. This includes variable
references.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrix"/>IsMatrix</term>
<listitem>
<synopsis>IsMatrix (arg)</synopsis>
<para>Проверяет, является ли аргумент матрицей. Хотя <constant>null</constant> иногда используют вместо пустой матрицы, функция <function>IsMatrix</function> не считает <constant>null</constant> матрицей.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsNull"/>IsNull</term>
<listitem>
<synopsis>IsNull (arg)</synopsis>
<para>Проверяет, имеет ли аргумент значение <constant>null</constant>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsString"/>IsString</term>
<listitem>
<synopsis>IsString (arg)</synopsis>
<para>Проверяет, является ли аргумент текстовой строкой.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsValue"/>IsValue</term>
<listitem>
<synopsis>IsValue (arg)</synopsis>
<para>Проверяет, является ли аргумент числом.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Parse"/>Parse</term>
<listitem>
<synopsis>Parse (str)</synopsis>
<para>Parses but does not evaluate a string. Note that certain
pre-computation is done during the parsing stage.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetFunctionFlags"/>SetFunctionFlags</term>
<listitem>
<synopsis>SetFunctionFlags (id,flags...)</synopsis>
<para>Set flags for a function, currently <literal>"PropagateMod"</literal> and <literal>"NoModuloArguments"</literal>.
If <literal>"PropagateMod"</literal> is set, then the body of the function is evaluated in modular arithmetic when the function
is called inside a block that was evaluated using modular arithmetic (using <literal>mod</literal>). If
<literal>"NoModuloArguments"</literal>, then the arguments of the function are never evaluated using modular arithmetic.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetHelp"/>SetHelp</term>
<listitem>
<synopsis>SetHelp (id,category,desc)</synopsis>
<para>Set the category and help description line for a function.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetHelpAlias"/>SetHelpAlias</term>
<listitem>
<synopsis>SetHelpAlias (id,alias)</synopsis>
<para>Sets up a help alias.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-chdir"/>chdir</term>
<listitem>
<synopsis>chdir (dir)</synopsis>
<para>Изменяет текущий каталог. То же, что и <command>cd</command>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CurrentTime"/>CurrentTime</term>
<listitem>
<synopsis>CurrentTime</synopsis>
<para>Возвращает текущее время UNIX с точностью до микросекунд в виде числа с плавающей точкой. То есть возвращает число секунд с 1 января 1970 г.</para>
<para>Version 1.0.15 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-display"/>display</term>
<listitem>
<synopsis>display (str,expr)</synopsis>
<para>Display a string and an expression with a colon to separate them.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DisplayVariables"/>DisplayVariables</term>
<listitem>
<synopsis>DisplayVariables (var1,var2,...)</synopsis>
<para>Display set of variables. The variables can be given as
strings or identifiers. For example:
<programlisting>DisplayVariables(`x,`y,`z)
</programlisting>
</para>
<para>
If called without arguments (must supply empty argument list) as
<programlisting>DisplayVariables()
</programlisting>
then all variables are printed including a stacktrace similar to
<guilabel>Show user variables</guilabel> in the graphical version.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-error"/>error</term>
<listitem>
<synopsis>error (str)</synopsis>
<para>Выводит строку в поток ошибок (на консоль).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-exit"/>exit</term>
<listitem>
<synopsis>exit</synopsis>
<para>Псевдоним: <function>quit</function></para>
<para>Завершает работу программы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-false"/>false</term>
<listitem>
<synopsis>false</synopsis>
<para>Псевдонимы: <function>False</function><function>FALSE</function></para>
<para>Логическое значение <constant>false</constant> (ложь).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-manual"/>manual</term>
<listitem>
<synopsis>manual</synopsis>
<para>Показывает руководство пользователя.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-print"/>print</term>
<listitem>
<synopsis>print (str)</synopsis>
<para>Выводит выражение и выполняет переход на новую строку. Аргумент <varname>str</varname> может быть любым выражением. Он преобразуется в строку перед выводом.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-printn"/>printn</term>
<listitem>
<synopsis>printn (str)</synopsis>
<para>Выводит выражение без перехода на новую строку. Аргумент <varname>str</varname> может быть любым выражением. Он преобразуется в строку перед выводом.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PrintTable"/>PrintTable</term>
<listitem>
<synopsis>PrintTable (f,v)</synopsis>
<para>Print a table of values for a function. The values are in the
vector <varname>v</varname>. You can use the vector
building notation as follows:
<programlisting>PrintTable (f,[0:10])
</programlisting>
If <varname>v</varname> is a positive integer, then the table of
integers from 1 up to and including v will be used.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-protect"/>protect</term>
<listitem>
<synopsis>protect (id)</synopsis>
<para>Protect a variable from being modified. This is used on the internal GEL functions to
avoid them being accidentally overridden.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ProtectAll"/>ProtectAll</term>
<listitem>
<synopsis>ProtectAll ()</synopsis>
<para>Protect all currently defined variables, parameters and
functions from being modified. This is used on the internal GEL functions to
avoid them being accidentally overridden. Normally <application>Genius Mathematics Tool</application> considers
unprotected variables as user defined.</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-set"/>set</term>
<listitem>
<synopsis>set (id,val)</synopsis>
<para>Set a global variable. The <varname>id</varname>
can be either a string or a quoted identifier.
For example:
<programlisting>set(`x,1)
</programlisting>
will set the global variable <varname>x</varname> to the value 1.
</para>
<para>The function returns the <varname>val</varname>, to be
usable in chaining.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetElement"/>SetElement</term>
<listitem>
<synopsis>SetElement (id,row,col,val)</synopsis>
<para>Set an element of a global variable which is a matrix.
The <varname>id</varname>
can be either a string or a quoted identifier.
For example:
<programlisting>SetElement(`x,2,3,1)
</programlisting>
will set the second row third column element of the global variable <varname>x</varname> to the value 1. If no global variable of the name exists, or if it is set to something that's not a matrix, a new zero matrix of appropriate size will be created.
</para>
<para>The <varname>row</varname> and <varname>col</varname> can also be ranges, and the semantics are the same as for regular setting of the elements with an equals sign.
</para>
<para>The function returns the <varname>val</varname>, to be
usable in chaining.</para>
<para>Available from 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetVElement"/>SetVElement</term>
<listitem>
<synopsis>SetElement (id,elt,val)</synopsis>
<para>Set an element of a global variable which is a vector.
The <varname>id</varname>
can be either a string or a quoted identifier.
For example:
<programlisting>SetElement(`x,2,1)
</programlisting>
will set the second element of the global vector variable <varname>x</varname> to the value 1. If no global variable of the name exists, or if it is set to something that's not a vector (matrix), a new zero row vector of appropriate size will be created.
</para>
<para>The <varname>elt</varname> can also be a range, and the semantics are the same as for regular setting of the elements with an equals sign.
</para>
<para>The function returns the <varname>val</varname>, to be
usable in chaining.</para>
<para>Available from 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-string"/>string</term>
<listitem>
<synopsis>string (s)</synopsis>
<para>Преобразует аргумент любого типа в строку.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-true"/>true</term>
<listitem>
<synopsis>true</synopsis>
<para>Псевдонимы: <function>True</function><function>TRUE</function></para>
<para>Логическое значение <constant>true</constant> (истина).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-undefine"/>undefine</term>
<listitem>
<synopsis>undefine (id)</synopsis>
<para>Alias: <function>Undefine</function></para>
<para>Undefine a variable. This includes locals and globals,
every value on all context levels is wiped. This function
should really not be used on local variables. A vector of
identifiers can also be passed to undefine several variables.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-UndefineAll"/>UndefineAll</term>
<listitem>
<synopsis>UndefineAll ()</synopsis>
<para>Undefine all unprotected global variables
(including functions and parameters). Normally <application>Genius Mathematics Tool</application>
considers protected variables as system defined functions
and variables. Note that <function>UndefineAll</function>
only removes the global definition of symbols not local ones,
so that it may be run from inside other functions safely.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-unprotect"/>unprotect</term>
<listitem>
<synopsis>unprotect (id)</synopsis>
<para>Unprotect a variable from being modified.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-UserVariables"/>UserVariables</term>
<listitem>
<synopsis>UserVariables ()</synopsis>
<para>Возвращает вектор идентификаторов определённых пользователем (незащищённых) глобальных переменных.</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-wait"/>wait</term>
<listitem>
<synopsis>wait (secs)</synopsis>
<para>Waits a specified number of seconds. <varname>secs</varname>
must be non-negative. Zero is accepted and nothing happens in this case,
except possibly user interface events are processed.</para>
<para>Since version 1.0.18, <varname>secs</varname> can be a noninteger number, so
<userinput>wait(0.1)</userinput> will wait for one tenth of a second.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-version"/>version</term>
<listitem>
<synopsis>version</synopsis>
<para>Returns the version of Genius as a horizontal 3-vector with
major version first, then minor version and finally the patch level.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-warranty"/>warranty</term>
<listitem>
<synopsis>warranty</synopsis>
<para>Gives the warranty information.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-parameters">
<title>Параметры</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-ChopTolerance"/>ChopTolerance</term>
<listitem>
<synopsis>ChopTolerance = number</synopsis>
<para>Tolerance of the <function>Chop</function> function.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ContinuousNumberOfTries"/>ContinuousNumberOfTries</term>
<listitem>
<synopsis>ContinuousNumberOfTries = number</synopsis>
<para>How many iterations to try to find the limit for continuity and limits.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ContinuousSFS"/>ContinuousSFS</term>
<listitem>
<synopsis>ContinuousSFS = number</synopsis>
<para>How many successive steps to be within tolerance for calculation of continuity.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ContinuousTolerance"/>ContinuousTolerance</term>
<listitem>
<synopsis>ContinuousTolerance = number</synopsis>
<para>Tolerance for continuity of functions and for calculating the limit.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DerivativeNumberOfTries"/>DerivativeNumberOfTries</term>
<listitem>
<synopsis>DerivativeNumberOfTries = number</synopsis>
<para>How many iterations to try to find the limit for derivative.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DerivativeSFS"/>DerivativeSFS</term>
<listitem>
<synopsis>DerivativeSFS = number</synopsis>
<para>How many successive steps to be within tolerance for calculation of derivative.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DerivativeTolerance"/>DerivativeTolerance</term>
<listitem>
<synopsis>DerivativeTolerance = number</synopsis>
<para>Tolerance for calculating the derivatives of functions.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ErrorFunctionTolerance"/>ErrorFunctionTolerance</term>
<listitem>
<synopsis>ErrorFunctionTolerance = number</synopsis>
<para>Tolerance of the <link linkend="gel-function-ErrorFunction"><function>ErrorFunction</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FloatPrecision"/>FloatPrecision</term>
<listitem>
<synopsis>FloatPrecision = число</synopsis>
<para>Точность чисел с плавающей точкой.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FullExpressions"/>FullExpressions</term>
<listitem>
<synopsis>FullExpressions = логическое значение</synopsis>
<para>Выводить полные выражения, даже если они превышают длину строки.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GaussDistributionTolerance"/>GaussDistributionTolerance</term>
<listitem>
<synopsis>GaussDistributionTolerance = number</synopsis>
<para>Tolerance of the <link linkend="gel-function-GaussDistribution"><function>GaussDistribution</function></link> function.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IntegerOutputBase"/>IntegerOutputBase</term>
<listitem>
<synopsis>IntegerOutputBase = number</synopsis>
<para>Integer output base.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPrimeMillerRabinReps"/>IsPrimeMillerRabinReps</term>
<listitem>
<synopsis>IsPrimeMillerRabinReps = number</synopsis>
<para>Number of extra Miller-Rabin tests to run on a number before declaring it a prime in <link linkend="gel-function-IsPrime"><function>IsPrime</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotDrawLegends"/>LinePlotDrawLegends</term>
<listitem>
<synopsis>LinePlotDrawLegends = true</synopsis>
<para>Tells genius to draw the legends for <link linkend="genius-gel-function-list-plotting">line plotting
functions</link> such as <link linkend="gel-function-LinePlot"><function>LinePlot</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotDrawAxisLabels"/>LinePlotDrawAxisLabels</term>
<listitem>
<synopsis>LinePlotDrawAxisLabels = true</synopsis>
<para>Tells genius to draw the axis labels for <link linkend="genius-gel-function-list-plotting">line plotting
functions</link> such as <link linkend="gel-function-LinePlot"><function>LinePlot</function></link>.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotVariableNames"/>LinePlotVariableNames</term>
<listitem>
<synopsis>LinePlotVariableNames = ["x","y","z","t"]</synopsis>
<para>Tells genius which variable names are used as default names for <link linkend="genius-gel-function-list-plotting">line plotting
functions</link> such as <link linkend="gel-function-LinePlot"><function>LinePlot</function></link> and friends.
</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotWindow"/>LinePlotWindow</term>
<listitem>
<synopsis>LinePlotWindow = [x1,x2,y1,y2]</synopsis>
<para>Sets the limits for <link linkend="genius-gel-function-list-plotting">line plotting
functions</link> such as <link linkend="gel-function-LinePlot"><function>LinePlot</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MaxDigits"/>MaxDigits</term>
<listitem>
<synopsis>MaxDigits = number</synopsis>
<para>Maximum digits to display.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MaxErrors"/>MaxErrors</term>
<listitem>
<synopsis>MaxErrors = number</synopsis>
<para>Maximum errors to display.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MixedFractions"/>MixedFractions</term>
<listitem>
<synopsis>MixedFractions = логическое значение</synopsis>
<para>Если true, выводятся смешанные дроби.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalIntegralFunction"/>NumericalIntegralFunction</term>
<listitem>
<synopsis>NumericalIntegralFunction = function</synopsis>
<para>The function used for numerical integration in <link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalIntegralSteps"/>NumericalIntegralSteps</term>
<listitem>
<synopsis>NumericalIntegralSteps = number</synopsis>
<para>Steps to perform in <link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OutputChopExponent"/>OutputChopExponent</term>
<listitem>
<synopsis>OutputChopExponent = number</synopsis>
<para>When another number in the object being printed (a matrix or a
value) is greater than
10<superscript>-OutputChopWhenExponent</superscript>, and
the number being printed is less than
10<superscript>-OutputChopExponent</superscript>, then
display <computeroutput>0.0</computeroutput> instead of the number.
</para>
<para>
Output is never chopped if <function>OutputChopExponent</function> is zero.
It must be a non-negative integer.
</para>
<para>
If you want output always chopped according to
<function>OutputChopExponent</function>, then set
<function>OutputChopWhenExponent</function>, to something
greater than or equal to
<function>OutputChopExponent</function>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OutputChopWhenExponent"/>OutputChopWhenExponent</term>
<listitem>
<synopsis>OutputChopWhenExponent = number</synopsis>
<para>When to chop output. See
<link linkend="gel-function-OutputChopExponent"><function>OutputChopExponent</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OutputStyle"/>OutputStyle</term>
<listitem>
<synopsis>OutputStyle = строка</synopsis>
<para>Стиль вывода, может быть <literal>normal</literal>, <literal>latex</literal>, <literal>mathml</literal> или <literal>troff</literal>.</para>
<para>В основном влияет на то, как выводятся матрицы и дроби, и может быть полезно для копирования и вставки в документы. Например, задать вывод в стиле latex можно с помощью: <programlisting>OutputStyle = "latex"
</programlisting></para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ResultsAsFloats"/>ResultsAsFloats</term>
<listitem>
<synopsis>ResultsAsFloats = boolean</synopsis>
<para>Convert all results to floats before printing.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ScientificNotation"/>ScientificNotation</term>
<listitem>
<synopsis>ScientificNotation = boolean</synopsis>
<para>Use scientific notation.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SlopefieldTicks"/>SlopefieldTicks</term>
<listitem>
<synopsis>SlopefieldTicks = [vertical,horizontal]</synopsis>
<para>Sets the number of vertical and horizontal ticks in a
slopefield plot. (See <link linkend="gel-function-SlopefieldPlot"><function>SlopefieldPlot</function></link>).
</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SumProductNumberOfTries"/>SumProductNumberOfTries</term>
<listitem>
<synopsis>SumProductNumberOfTries = number</synopsis>
<para>How many iterations to try for <link linkend="gel-function-InfiniteSum"><function>InfiniteSum</function></link> and <link linkend="gel-function-InfiniteProduct"><function>InfiniteProduct</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SumProductSFS"/>SumProductSFS</term>
<listitem>
<synopsis>SumProductSFS = number</synopsis>
<para>How many successive steps to be within tolerance for <link linkend="gel-function-InfiniteSum"><function>InfiniteSum</function></link> and <link linkend="gel-function-InfiniteProduct"><function>InfiniteProduct</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SumProductTolerance"/>SumProductTolerance</term>
<listitem>
<synopsis>SumProductTolerance = number</synopsis>
<para>Tolerance for <link linkend="gel-function-InfiniteSum"><function>InfiniteSum</function></link> and <link linkend="gel-function-InfiniteProduct"><function>InfiniteProduct</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotDrawLegends"/>SurfacePlotDrawLegends</term>
<listitem>
<synopsis>SurfacePlotDrawLegends = true</synopsis>
<para>Tells genius to draw the legends for <link linkend="genius-gel-function-list-plotting">surface plotting
functions</link> such as <link linkend="gel-function-SurfacePlot"><function>SurfacePlot</function></link>.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotVariableNames"/>SurfacePlotVariableNames</term>
<listitem>
<synopsis>SurfacePlotVariableNames = ["x","y","z"]</synopsis>
<para>Tells genius which variable names are used as default names for <link linkend="genius-gel-function-list-plotting">surface plotting
functions</link> using <link linkend="gel-function-SurfacePlot"><function>SurfacePlot</function></link>.
Note that the <varname>z</varname> does not refer to the dependent (vertical) axis, but to the independent complex variable
<userinput>z=x+iy</userinput>.
</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotWindow"/>SurfacePlotWindow</term>
<listitem>
<synopsis>SurfacePlotWindow = [x1,x2,y1,y2,z1,z2]</synopsis>
<para>Sets the limits for surface plotting (See <link linkend="gel-function-SurfacePlot"><function>SurfacePlot</function></link>).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorfieldNormalized"/>VectorfieldNormalized</term>
<listitem>
<synopsis>VectorfieldNormalized = true</synopsis>
<para>Should the vectorfield plotting have normalized arrow length. If true, vector fields will only show direction
and not magnitude. (See <link linkend="gel-function-VectorfieldPlot"><function>VectorfieldPlot</function></link>).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorfieldTicks"/>VectorfieldTicks</term>
<listitem>
<synopsis>VectorfieldTicks = [vertical,horizontal]</synopsis>
<para>Sets the number of vertical and horizontal ticks in a
vectorfield plot. (See <link linkend="gel-function-VectorfieldPlot"><function>VectorfieldPlot</function></link>).
</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-constants">
<title>Константы</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-CatalanConstant"/>CatalanConstant</term>
<listitem>
<synopsis>CatalanConstant</synopsis>
<para>
Catalan's Constant, approximately 0.915... It is defined to be the series where terms are <userinput>(-1^k)/((2*k+1)^2)</userinput>, where <varname>k</varname> ranges from 0 to infinity.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Catalan%27s_constant">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/CatalansConstant.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-EulerConstant"/>EulerConstant</term>
<listitem>
<synopsis>EulerConstant</synopsis>
<para>Aliases: <function>gamma</function></para>
<para>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MascheroniConstant">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GoldenRatio"/>GoldenRatio</term>
<listitem>
<synopsis>GoldenRatio</synopsis>
<para>The Golden Ratio.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Golden_ratio">Wikipedia</ulink> or
<ulink url="http://planetmath.org/GoldenRatio">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/GoldenRatio.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Gravity"/>Gravity</term>
<listitem>
<synopsis>Gravity</synopsis>
<para>Free fall acceleration at sea level in meters per second squared. This is the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the fact that the earth is not perfectly
round and uniform.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Standard_gravity">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-e"/>e</term>
<listitem>
<synopsis>e</synopsis>
<para>
The base of the natural logarithm. <userinput>e^x</userinput>
is the exponential function
<link linkend="gel-function-exp"><function>exp</function></link>. It is approximately
2.71828182846... This number is sometimes called Euler's number, although there are
several numbers that are also called Euler's. An example is the gamma constant: <link linkend="gel-function-EulerConstant"><function>EulerConstant</function></link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/E_(mathematical_constant)">Wikipedia</ulink> or
<ulink url="http://planetmath.org/E">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/e.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-pi"/>pi</term>
<listitem>
<synopsis>pi</synopsis>
<para>Число «пи» — отношение длины окружности к её диаметру. Значение приблизительно равно 3.14159265359...</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Pi">Wikipedia</ulink> or
<ulink url="http://planetmath.org/Pi">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/Pi.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-numeric">
<title>Числовые</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-AbsoluteValue"/>AbsoluteValue</term>
<listitem>
<synopsis>AbsoluteValue (x)</synopsis>
<para>Синонимы: <function>abs</function></para>
<para>
Absolute value of a number and if <varname>x</varname> is
a complex value the modulus of <varname>x</varname>. I.e. this
the distance of <varname>x</varname> to the origin. This is equivalent
to <userinput>|x|</userinput>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Absolute_value">Wikipedia</ulink>,
<ulink url="http://planetmath.org/AbsoluteValue">Planetmath (absolute value)</ulink>,
<ulink url="http://planetmath.org/ModulusOfComplexNumber">Planetmath (modulus)</ulink>,
<ulink url="http://mathworld.wolfram.com/AbsoluteValue.html">Mathworld (absolute value)</ulink> or
<ulink url="http://mathworld.wolfram.com/ComplexModulus.html">Mathworld (complex modulus)</ulink>
for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Chop"/>Chop</term>
<listitem>
<synopsis>Chop (x)</synopsis>
<para>Заменяет очень малое число нулём.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ComplexConjugate"/>ComplexConjugate</term>
<listitem>
<synopsis>ComplexConjugate (z)</synopsis>
<para>Aliases: <function>conj</function> <function>Conj</function></para>
<para>Calculates the complex conjugate of the complex number <varname>z</varname>. If <varname>z</varname> is a vector or matrix,
all its elements are conjugated.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Complex_conjugate">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Denominator"/>Denominator</term>
<listitem>
<synopsis>Denominator (x)</synopsis>
<para>Возвращает знаменатель рационального числа.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Denominator">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FractionalPart"/>FractionalPart</term>
<listitem>
<synopsis>FractionalPart (x)</synopsis>
<para>Возвращает дробную часть числа.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fractional_part">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Im"/>Im</term>
<listitem>
<synopsis>Im (z)</synopsis>
<para>Синонимы: <function>ImaginaryPart</function></para>
<para>Get the imaginary part of a complex number. For example <userinput>Re(3+4i)</userinput> yields 4.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Imaginary_part">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IntegerQuotient"/>IntegerQuotient</term>
<listitem>
<synopsis>IntegerQuotient (m,n)</synopsis>
<para>Деление без остатка.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsComplex"/>IsComplex</term>
<listitem>
<synopsis>IsComplex (num)</synopsis>
<para>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number. That is,
<userinput>IsComplex(3)</userinput> yields false, while
<userinput>IsComplex(3-1i)</userinput> yields true.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsComplexRational"/>IsComplexRational</term>
<listitem>
<synopsis>IsComplexRational (num)</synopsis>
<para>Check if argument is a possibly complex rational number. That is, if both real and imaginary parts are
given as rational numbers. Of course rational simply means "not stored as a floating point number."</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsFloat"/>IsFloat</term>
<listitem>
<synopsis>IsFloat (num)</synopsis>
<para>Check if argument is a real floating point number (non-complex).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsGaussInteger"/>IsGaussInteger</term>
<listitem>
<synopsis>IsGaussInteger (num)</synopsis>
<para>Aliases: <function>IsComplexInteger</function></para>
<para>Check if argument is a possibly complex integer. That is a complex integer is a number of
the form <userinput>n+1i*m</userinput> where <varname>n</varname> and <varname>m</varname>
are integers.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsInteger"/>IsInteger</term>
<listitem>
<synopsis>IsInteger (num)</synopsis>
<para>Проверяет, является ли аргумент целым числом (не комплексным).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsNonNegativeInteger"/>IsNonNegativeInteger</term>
<listitem>
<synopsis>IsNonNegativeInteger (num)</synopsis>
<para>Check if argument is a non-negative real integer. That is, either a positive integer or zero.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPositiveInteger"/>IsPositiveInteger</term>
<listitem>
<synopsis>IsPositiveInteger (num)</synopsis>
<para>Синонимы: <function>IsNaturalNumber</function></para>
<para>Проверяет, является ли аргумент положительным действительным целым числом. Обратите внимание, что мы придерживаемся соглашения о том, что 0 не является натуральным числом.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsRational"/>IsRational</term>
<listitem>
<synopsis>IsRational (num)</synopsis>
<para>Проверяет, является ли аргумент рациональным числом (не комплексным). Разумеется, «рациональное» означает просто «не хранящееся в виде числа с плавающей точкой».</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsReal"/>IsReal</term>
<listitem>
<synopsis>IsReal (num)</synopsis>
<para>Проверяет, является ли аргумент действительным числом.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Numerator"/>Numerator</term>
<listitem>
<synopsis>Numerator (x)</synopsis>
<para>Возвращает числитель рационального числа.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Numerator">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Re"/>Re</term>
<listitem>
<synopsis>Re (z)</synopsis>
<para>Синонимы: <function>RealPart</function></para>
<para>Get the real part of a complex number. For example <userinput>Re(3+4i)</userinput> yields 3.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Real_part">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Sign"/>Sign</term>
<listitem>
<synopsis>Sign (x)</synopsis>
<para>Синонимы: <function>sign</function></para>
<para>Возвращает знак числа. То есть, возвращает <literal>-1</literal>, если значение отрицательно, <literal>0</literal>, если рано нулю и <literal>1</literal>, если значение положительно. Если <varname>x</varname> — комплексное число, то <function>Sign</function> возвращает направление на числовой оси (положительное или отрицательное) или 0.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ceil"/>ceil</term>
<listitem>
<synopsis>ceil (x)</synopsis>
<para>Синонимы: <function>Ceiling</function></para>
<para>Возвращает наименьшее целое число, которое больше или равно <varname>n</varname>. Примеры: <screen><prompt>genius></prompt> <userinput>ceil(1.1)</userinput>
= 2
<prompt>genius></prompt> <userinput>ceil(-1.1)</userinput>
= -1
</screen></para>
<para>Note that you should be careful and notice that floating point
numbers are stored in binary and so may not be what you
expect. For example <userinput>ceil(420/4.2)</userinput>
returns 101 instead of the expected 100. This is because
4.2 is actually very slightly less than 4.2. Use rational
representation <userinput>42/10</userinput> if you want
exact arithmetic.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-exp"/>exp</term>
<listitem>
<synopsis>exp (x)</synopsis>
<para>Экспоненциальная функция. Это функция <userinput>e^x</userinput>, где <varname>e</varname> — <link linkend="gel-function-e">основание натурального логарифма</link>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Exponential_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/LogarithmFunction">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/ExponentialFunction.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-float"/>float</term>
<listitem>
<synopsis>float (x)</synopsis>
<para>Возвращает представление числа <varname>x</varname> в виде числа с плавающей точкой.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-floor"/>floor</term>
<listitem>
<synopsis>floor (x)</synopsis>
<para>Синонимы: <function>Floor</function></para>
<para>Возвращает наибольшее целое число, которое меньше или равно <varname>n</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ln"/>ln</term>
<listitem>
<synopsis>ln (x)</synopsis>
<para>Натуральный логарифм (логарифм по основанию <varname>e</varname>).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Natural_logarithm">Wikipedia</ulink> or
<ulink url="http://planetmath.org/LogarithmFunction">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/NaturalLogarithm.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-log"/>log</term>
<listitem>
<synopsis>log (x)</synopsis>
<synopsis>log (x,b)</synopsis>
<para>Logarithm of <varname>x</varname> base <varname>b</varname> (calls <link linkend="gel-function-DiscreteLog"><function>DiscreteLog</function></link> if in modulo mode), if base is not given, <link linkend="gel-function-e"><varname>e</varname></link> is used.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-log10"/>log10</term>
<listitem>
<synopsis>log10 (x)</synopsis>
<para>Логарифм <varname>x</varname> по основанию 10.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-log2"/>log2</term>
<listitem>
<synopsis>log2 (x)</synopsis>
<para>Синоним: <function>lg</function></para>
<para>Логарифм <varname>x</varname> по основанию 2.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-max"/>max</term>
<listitem>
<synopsis>max (a,args...)</synopsis>
<para>Псевдонимы: <function>Max</function><function>Maximum</function></para>
<para>Возвращает максимальный из аргументов или элементов матрицы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-min"/>min</term>
<listitem>
<synopsis>min (a,args...)</synopsis>
<para>Псевдонимы: <function>Min</function><function>Minimum</function></para>
<para>Возвращает минимальный из аргументов или элементов матрицы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-rand"/>rand</term>
<listitem>
<synopsis>rand (size...)</synopsis>
<para>Генерирует случайное число с плавающей точкой в диапазоне <literal>[0,1)</literal>. Если задан аргумент size, то может возвращать матрицу (если указано два числа) или вектор (если указано одно число) заданной размерности.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-randint"/>randint</term>
<listitem>
<synopsis>randint (max,size...)</synopsis>
<para>Генерирует случайное целое число в диапазоне <literal>[0,max)</literal>. Если задан аргумент size, возвращает матрицу (если указано два числа) или вектор (если указано одно число) заданной размерности. Например, <screen><prompt>genius></prompt> <userinput>randint(4)</userinput>
= 3
<prompt>genius></prompt> <userinput>randint(4,2)</userinput>
=
[0 1]
<prompt>genius></prompt> <userinput>randint(4,2,3)</userinput>
=
[2 2 1
0 0 3]
</screen></para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-round"/>round</term>
<listitem>
<synopsis>round (x)</synopsis>
<para>Синонимы: <function>Round</function></para>
<para>Округляет число.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-sqrt"/>sqrt</term>
<listitem>
<synopsis>sqrt (x)</synopsis>
<para>Синонимы: <function>SquareRoot</function></para>
<para>The square root. When operating modulo some integer will return either a <constant>null</constant> or a vector of the square roots. Examples:
<screen><prompt>genius></prompt> <userinput>sqrt(2)</userinput>
= 1.41421356237
<prompt>genius></prompt> <userinput>sqrt(-1)</userinput>
= 1i
<prompt>genius></prompt> <userinput>sqrt(4) mod 7</userinput>
=
[2 5]
<prompt>genius></prompt> <userinput>2*2 mod 7</userinput>
= 4
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Square_root">Wikipedia</ulink> or
<ulink url="http://planetmath.org/SquareRoot">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-trunc"/>trunc</term>
<listitem>
<synopsis>trunc (x)</synopsis>
<para>Синонимы: <function>Truncate</function><function>IntegerPart</function></para>
<para>Усекает число до целого (возвращает целую часть).</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-trigonometry">
<title>Тригонометрические</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-acos"/>acos</term>
<listitem>
<synopsis>acos (x)</synopsis>
<para>Синонимы: <function>arccos</function></para>
<para>Функция arccos (арккосинус, обратный косинус).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-acosh"/>acosh</term>
<listitem>
<synopsis>acosh (x)</synopsis>
<para>Синонимы: <function>arccosh</function></para>
<para>Функция arccosh (обратный гиперболический косинус).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-acot"/>acot</term>
<listitem>
<synopsis>acot (x)</synopsis>
<para>Синонимы: <function>arccot</function></para>
<para>Фунция arccot (арккотангенс, обратный котангенс).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-acoth"/>acoth</term>
<listitem>
<synopsis>acoth (x)</synopsis>
<para>Синонимы: <function>arccoth</function></para>
<para>Функция arccoth (обратный гиперболический котангенс).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-acsc"/>acsc</term>
<listitem>
<synopsis>acsc (x)</synopsis>
<para>Синонимы: <function>arccsc</function></para>
<para>Обратный косеканс.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-acsch"/>acsch</term>
<listitem>
<synopsis>acsch (x)</synopsis>
<para>Синонимы: <function>arccsch</function></para>
<para>Обратный гиперболический косеканс.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-asec"/>asec</term>
<listitem>
<synopsis>asec (x)</synopsis>
<para>Синонимы: <function>arcsec</function></para>
<para>Обратный секанс.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-asech"/>asech</term>
<listitem>
<synopsis>asech (x)</synopsis>
<para>Синонимы: <function>arcsech</function></para>
<para>Обратный гиперболический секанс.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-asin"/>asin</term>
<listitem>
<synopsis>asin (x)</synopsis>
<para>Синонимы: <function>arcsin</function></para>
<para>Функция arcsin (арксинус, обратный синус).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-asinh"/>asinh</term>
<listitem>
<synopsis>asinh (x)</synopsis>
<para>Синонимы: <function>arcsinh</function></para>
<para>Фунция arcsinh (обратный гиперболический синус).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-atan"/>atan</term>
<listitem>
<synopsis>atan (x)</synopsis>
<para>Синонимы: <function>arctan</function></para>
<para>Вычисляет функцию arctan (арктангенс, обратный тангенс).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Arctangent">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/InverseTangent.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-atanh"/>atanh</term>
<listitem>
<synopsis>atanh (x)</synopsis>
<para>Синонимы: <function>arctanh</function></para>
<para>Функция arctanh (обратный гиперболический тангенс).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-atan2"/>atan2</term>
<listitem>
<synopsis>atan2 (y, x)</synopsis>
<para>Синонимы: <function>arctan2</function></para>
<para>Calculates the arctan2 function. If
<userinput>x>0</userinput> then it returns
<userinput>atan(y/x)</userinput>. If <userinput>x<0</userinput>
then it returns <userinput>sign(y) * (pi - atan(|y/x|)</userinput>.
When <userinput>x=0</userinput> it returns <userinput>sign(y) *
pi/2</userinput>. <userinput>atan2(0,0)</userinput> returns 0
rather than failing.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Atan2">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/InverseTangent.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-cos"/>cos</term>
<listitem>
<synopsis>cos (x)</synopsis>
<para>Вычисляет косинус.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trigonometric_functions">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DefinitionsInTrigonometry">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-cosh"/>cosh</term>
<listitem>
<synopsis>cosh (x)</synopsis>
<para>Вычисляет гиперболический косинус.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hyperbolic_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HyperbolicFunctions">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-cot"/>cot</term>
<listitem>
<synopsis>cot (x)</synopsis>
<para>Вычисляет котангенс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trigonometric_functions">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DefinitionsInTrigonometry">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-coth"/>coth</term>
<listitem>
<synopsis>coth (x)</synopsis>
<para>Вычисляет гиперболический котангенс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hyperbolic_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HyperbolicFunctions">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-csc"/>csc</term>
<listitem>
<synopsis>csc (x)</synopsis>
<para>Вычисляет косеканс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trigonometric_functions">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DefinitionsInTrigonometry">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-csch"/>csch</term>
<listitem>
<synopsis>csch (x)</synopsis>
<para>Вычисляет гиперболический косеканс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hyperbolic_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HyperbolicFunctions">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-sec"/>sec</term>
<listitem>
<synopsis>sec (x)</synopsis>
<para>Вычисляет секанс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trigonometric_functions">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DefinitionsInTrigonometry">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-sech"/>sech</term>
<listitem>
<synopsis>sech (x)</synopsis>
<para>Вычисляет гиперболический секанс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hyperbolic_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HyperbolicFunctions">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-sin"/>sin</term>
<listitem>
<synopsis>sin (x)</synopsis>
<para>Вычисляет синус.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trigonometric_functions">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DefinitionsInTrigonometry">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-sinh"/>sinh</term>
<listitem>
<synopsis>sinh (x)</synopsis>
<para>Вычисляет гиперболический синус.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hyperbolic_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HyperbolicFunctions">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-tan"/>tan</term>
<listitem>
<synopsis>tan (x)</synopsis>
<para>Вычисляет тангенс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trigonometric_functions">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DefinitionsInTrigonometry">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-tanh"/>tanh</term>
<listitem>
<synopsis>tanh (x)</synopsis>
<para>Вычисляет гиперболический тангенс.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hyperbolic_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HyperbolicFunctions">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-number-theory">
<title>Теория чисел</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-AreRelativelyPrime"/>AreRelativelyPrime</term>
<listitem>
<synopsis>AreRelativelyPrime (a,b)</synopsis>
<para>
Are the real integers <varname>a</varname> and <varname>b</varname> relatively prime?
Returns <constant>true</constant> or <constant>false</constant>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Coprime_integers">Wikipedia</ulink> or
<ulink url="http://planetmath.org/RelativelyPrime">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/RelativelyPrime.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BernoulliNumber"/>BernoulliNumber</term>
<listitem>
<synopsis>BernoulliNumber (n)</synopsis>
<para>Return the <varname>n</varname>th Bernoulli number.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bernoulli_number">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/BernoulliNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ChineseRemainder"/>ChineseRemainder</term>
<listitem>
<synopsis>ChineseRemainder (a,m)</synopsis>
<para>Aliases: <function>CRT</function></para>
<para>Find the <varname>x</varname> that solves the system given by
the vector <varname>a</varname> and modulo the elements of
<varname>m</varname>, using the Chinese Remainder Theorem.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Chinese_remainder_theorem">Wikipedia</ulink> or
<ulink url="http://planetmath.org/ChineseRemainderTheorem">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/ChineseRemainderTheorem.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CombineFactorizations"/>CombineFactorizations</term>
<listitem>
<synopsis>CombineFactorizations (a,b)</synopsis>
<para>Given two factorizations, give the factorization of the
product.</para>
<para>See <link linkend="gel-function-Factorize">Factorize</link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ConvertFromBase"/>ConvertFromBase</term>
<listitem>
<synopsis>ConvertFromBase (v,b)</synopsis>
<para>Convert a vector of values indicating powers of b to a number.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ConvertToBase"/>ConvertToBase</term>
<listitem>
<synopsis>ConvertToBase (n,b)</synopsis>
<para>Convert a number to a vector of powers for elements in base <varname>b</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DiscreteLog"/>DiscreteLog</term>
<listitem>
<synopsis>DiscreteLog (n,b,q)</synopsis>
<para>Find discrete log of <varname>n</varname> base <varname>b</varname> in
F<subscript>q</subscript>, the finite field of order <varname>q</varname>, where <varname>q</varname>
is a prime, using the Silver-Pohlig-Hellman algorithm.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Discrete_logarithm">Wikipedia</ulink>,
<ulink url="http://planetmath.org/DiscreteLogarithm">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/DiscreteLogarithm.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Divides"/>Divides</term>
<listitem>
<synopsis>Divides (m,n)</synopsis>
<para>Проверяет делимость (делится ли <varname>n</varname> на <varname>m</varname>).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-EulerPhi"/>EulerPhi</term>
<listitem>
<synopsis>EulerPhi (n)</synopsis>
<para>
Compute the Euler phi function for <varname>n</varname>, that is
the number of integers between 1 and <varname>n</varname>
relatively prime to <varname>n</varname>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Euler_phi">Wikipedia</ulink>,
<ulink url="http://planetmath.org/EulerPhifunction">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/TotientFunction.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ExactDivision"/>ExactDivision</term>
<listitem>
<synopsis>ExactDivision (n,d)</synopsis>
<para>Возвращает <userinput>n/d</userinput>, но только если <varname>n</varname> делится на <varname>d</varname>. Если не делится, то функция возвращает мусор. Для очень больших чисел это гораздо быстрее, чем операция <userinput>n/d</userinput>, но, разумеется, полезно только в том случае, если вы точно знаете, что числа делятся без остатка.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Factorize"/>Factorize</term>
<listitem>
<synopsis>Factorize (n)</synopsis>
<para>
Return factorization of a number as a matrix. The first
row is the primes in the factorization (including 1) and the
second row are the powers. So for example:
<screen><prompt>genius></prompt> <userinput>Factorize(11*11*13)</userinput>
=
[1 11 13
1 2 1]</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Factorization">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Factors"/>Factors</term>
<listitem>
<synopsis>Factors (n)</synopsis>
<para>
Return all factors of <varname>n</varname> in a vector. This
includes all the non-prime factors as well. It includes 1 and the
number itself. So for example to print all the perfect numbers
(those that are sums of their factors) up to the number 1000 you
could do (this is of course very inefficient)
<programlisting>for n=1 to 1000 do (
if MatrixSum (Factors(n)) == 2*n then
print(n)
)
</programlisting>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FermatFactorization"/>FermatFactorization</term>
<listitem>
<synopsis>FermatFactorization (n,tries)</synopsis>
<para>
Attempt Fermat factorization of <varname>n</varname> into
<userinput>(t-s)*(t+s)</userinput>, returns <varname>t</varname>
and <varname>s</varname> as a vector if possible, <constant>null</constant> otherwise.
<varname>tries</varname> specifies the number of tries before
giving up.
</para>
<para>
This is a fairly good factorization if your number is the product
of two factors that are very close to each other.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fermat_factorization">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FindPrimitiveElementMod"/>FindPrimitiveElementMod</term>
<listitem>
<synopsis>FindPrimitiveElementMod (q)</synopsis>
<para>Find the first primitive element in F<subscript>q</subscript>, the finite
group of order <varname>q</varname>. Of course <varname>q</varname> must be a prime.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FindRandomPrimitiveElementMod"/>FindRandomPrimitiveElementMod</term>
<listitem>
<synopsis>FindRandomPrimitiveElementMod (q)</synopsis>
<para>Find a random primitive element in F<subscript>q</subscript>, the finite
group of order <varname>q</varname> (q must be a prime).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IndexCalculus"/>IndexCalculus</term>
<listitem>
<synopsis>IndexCalculus (n,b,q,S)</synopsis>
<para>Compute discrete log base <varname>b</varname> of n in F<subscript>q</subscript>, the finite
group of order <varname>q</varname> (<varname>q</varname> a prime), using the
factor base <varname>S</varname>. <varname>S</varname> should be a column of
primes possibly with second column precalculated by
<link linkend="gel-function-IndexCalculusPrecalculation"><function>IndexCalculusPrecalculation</function></link>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IndexCalculusPrecalculation"/>IndexCalculusPrecalculation</term>
<listitem>
<synopsis>IndexCalculusPrecalculation (b,q,S)</synopsis>
<para>Run the precalculation step of
<link linkend="gel-function-IndexCalculus"><function>IndexCalculus</function></link> for logarithms base <varname>b</varname> in
F<subscript>q</subscript>, the finite group of order <varname>q</varname>
(<varname>q</varname> a prime), for the factor base <varname>S</varname> (where
<varname>S</varname> is a column vector of primes). The logs will be
precalculated and returned in the second column.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsEven"/>IsEven</term>
<listitem>
<synopsis>IsEven (n)</synopsis>
<para>Проверяет, является ли целое число чётным.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMersennePrimeExponent"/>IsMersennePrimeExponent</term>
<listitem>
<synopsis>IsMersennePrimeExponent (p)</synopsis>
<para>
Tests if a positive integer <varname>p</varname> is a
Mersenne prime exponent. That is if
2<superscript>p</superscript>-1 is a prime. It does this
by looking it up in a table of known values, which is relatively
short.
See also
<link linkend="gel-function-MersennePrimeExponents">MersennePrimeExponents</link>
and
<link linkend="gel-function-LucasLehmer">LucasLehmer</link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Mersenne_prime">Wikipedia</ulink>,
<ulink url="http://planetmath.org/MersenneNumbers">Planetmath</ulink>,
<ulink url="http://mathworld.wolfram.com/MersennePrime.html">Mathworld</ulink> or
<ulink url="http://www.mersenne.org/">GIMPS</ulink>
for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsNthPower"/>IsNthPower</term>
<listitem>
<synopsis>IsNthPower (m,n)</synopsis>
<para>
Tests if a rational number <varname>m</varname> is a perfect
<varname>n</varname>th power. See also
<link linkend="gel-function-IsPerfectPower">IsPerfectPower</link>
and
<link linkend="gel-function-IsPerfectSquare">IsPerfectSquare</link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsOdd"/>IsOdd</term>
<listitem>
<synopsis>IsOdd (n)</synopsis>
<para>Проверяет, является ли целое число нечётным.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPerfectPower"/>IsPerfectPower</term>
<listitem>
<synopsis>IsPerfectPower (n)</synopsis>
<para>Check an integer for being any perfect power, a<superscript>b</superscript>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPerfectSquare"/>IsPerfectSquare</term>
<listitem>
<synopsis>IsPerfectSquare (n)</synopsis>
<para>
Check an integer for being a perfect square of an integer. The number must
be an integer. Negative integers are of course never perfect
squares of integers.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPrime"/>IsPrime</term>
<listitem>
<synopsis>IsPrime (n)</synopsis>
<para>
Tests primality of integers, for numbers less than 2.5e10 the
answer is deterministic (if Riemann hypothesis is true). For
numbers larger, the probability of a false positive
depends on
<link linkend="gel-function-IsPrimeMillerRabinReps">
<function>IsPrimeMillerRabinReps</function></link>. That
is the probability of false positive is 1/4 to the power
<function>IsPrimeMillerRabinReps</function>. The default
value of 22 yields a probability of about 5.7e-14.
</para>
<para>
If <constant>false</constant> is returned, you can be sure that
the number is a composite. If you want to be absolutely sure
that you have a prime you can use
<link linkend="gel-function-MillerRabinTestSure">
<function>MillerRabinTestSure</function></link> but it may take
a lot longer.
</para>
<para>
See
<ulink url="http://planetmath.org/PrimeNumber">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/PrimeNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPrimitiveMod"/>IsPrimitiveMod</term>
<listitem>
<synopsis>IsPrimitiveMod (g,q)</synopsis>
<para>Check if <varname>g</varname> is primitive in F<subscript>q</subscript>, the finite
group of order <varname>q</varname>, where <varname>q</varname> is a prime. If <varname>q</varname> is not prime results are bogus.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPrimitiveModWithPrimeFactors"/>IsPrimitiveModWithPrimeFactors</term>
<listitem>
<synopsis>IsPrimitiveModWithPrimeFactors (g,q,f)</synopsis>
<para>Check if <varname>g</varname> is primitive in F<subscript>q</subscript>, the finite
group of order <varname>q</varname>, where <varname>q</varname> is a prime and
<varname>f</varname> is a vector of prime factors of <varname>q</varname>-1.
If <varname>q</varname> is not prime results are bogus.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPseudoprime"/>IsPseudoprime</term>
<listitem>
<synopsis>IsPseudoprime (n,b)</synopsis>
<para>If <varname>n</varname> is a pseudoprime base <varname>b</varname> but not a prime,
that is if <userinput>b^(n-1) == 1 mod n</userinput>. This calls the <link linkend="gel-function-PseudoprimeTest"><function>PseudoprimeTest</function></link></para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsStrongPseudoprime"/>IsStrongPseudoprime</term>
<listitem>
<synopsis>IsStrongPseudoprime (n,b)</synopsis>
<para>Test if <varname>n</varname> is a strong pseudoprime to base <varname>b</varname> but not a prime.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Jacobi"/>Jacobi</term>
<listitem>
<synopsis>Jacobi (a,b)</synopsis>
<para>Синонимы: <function>JacobiSymbol</function></para>
<para>Вычисляет символ Якоби (a/b) (b должно быть нечётным).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-JacobiKronecker"/>JacobiKronecker</term>
<listitem>
<synopsis>JacobiKronecker (a,b)</synopsis>
<para>Синонимы: <function>JacobiKroneckerSymbol</function></para>
<para>Вычисляет символ Якоби (a/b) с дополнением Кронекера (a/2)=(2/a), если нечётное или (a/2)=0, если чётное.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LeastAbsoluteResidue"/>LeastAbsoluteResidue</term>
<listitem>
<synopsis>LeastAbsoluteResidue (a,n)</synopsis>
<para>Return the residue of <varname>a</varname> mod <varname>n</varname> with the least absolute value (in the interval -n/2 to n/2).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Legendre"/>Legendre</term>
<listitem>
<synopsis>Legendre (a,p)</synopsis>
<para>Aliases: <function>LegendreSymbol</function></para>
<para>Calculate the Legendre symbol (a/p).</para>
<para>
See
<ulink url="http://planetmath.org/LegendreSymbol">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/LegendreSymbol.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LucasLehmer"/>LucasLehmer</term>
<listitem>
<synopsis>LucasLehmer (p)</synopsis>
<para>Test if 2<superscript>p</superscript>-1 is a Mersenne prime using the Lucas-Lehmer test.
See also
<link linkend="gel-function-MersennePrimeExponents">MersennePrimeExponents</link>
and
<link linkend="gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test">Wikipedia</ulink>,
<ulink url="http://planetmath.org/LucasLhemer">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/Lucas-LehmerTest.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LucasNumber"/>LucasNumber</term>
<listitem>
<synopsis>LucasNumber (n)</synopsis>
<para>Returns the <varname>n</varname>th Lucas number.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Lucas_number">Wikipedia</ulink>,
<ulink url="http://planetmath.org/LucasNumbers">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/LucasNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MaximalPrimePowerFactors"/>MaximalPrimePowerFactors</term>
<listitem>
<synopsis>MaximalPrimePowerFactors (n)</synopsis>
<para>Return all maximal prime power factors of a number.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MersennePrimeExponents"/>MersennePrimeExponents</term>
<listitem>
<synopsis>MersennePrimeExponents</synopsis>
<para>
A vector of known Mersenne prime exponents, that is
a list of positive integers
<varname>p</varname> such that
2<superscript>p</superscript>-1 is a prime.
See also
<link linkend="gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</link>
and
<link linkend="gel-function-LucasLehmer">LucasLehmer</link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Mersenne_prime">Wikipedia</ulink>,
<ulink url="http://planetmath.org/MersenneNumbers">Planetmath</ulink>,
<ulink url="http://mathworld.wolfram.com/MersennePrime.html">Mathworld</ulink> or
<ulink url="http://www.mersenne.org/">GIMPS</ulink>
for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MillerRabinTest"/>MillerRabinTest</term>
<listitem>
<synopsis>MillerRabinTest (n,reps)</synopsis>
<para>
Use the Miller-Rabin primality test on <varname>n</varname>,
<varname>reps</varname> number of times. The probability of false
positive is <userinput>(1/4)^reps</userinput>. It is probably
usually better to use
<link linkend="gel-function-IsPrime">
<function>IsPrime</function></link> since that is faster and
better on smaller integers.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MillerRabinPrimeTest">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MillerRabinTestSure"/>MillerRabinTestSure</term>
<listitem>
<synopsis>MillerRabinTestSure (n)</synopsis>
<para>
Use the Miller-Rabin primality test on <varname>n</varname> with
enough bases that assuming the Generalized Riemann Hypothesis the
result is deterministic.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test">Wikipedia</ulink>,
<ulink url="http://planetmath.org/MillerRabinPrimeTest">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ModInvert"/>ModInvert</term>
<listitem>
<synopsis>ModInvert (n,m)</synopsis>
<para>Returns inverse of n mod m.</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/ModularInverse.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MoebiusMu"/>MoebiusMu</term>
<listitem>
<synopsis>MoebiusMu (n)</synopsis>
<para>
Return the Moebius mu function evaluated in <varname>n</varname>.
That is, it returns 0 if <varname>n</varname> is not a product
of distinct primes and <userinput>(-1)^k</userinput> if it is
a product of <varname>k</varname> distinct primes.
</para>
<para>
See
<ulink url="http://planetmath.org/MoebiusFunction">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/MoebiusFunction.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NextPrime"/>NextPrime</term>
<listitem>
<synopsis>NextPrime (n)</synopsis>
<para>
Returns the least prime greater than <varname>n</varname>.
Negatives of primes are considered prime and so to get the
previous prime you can use <userinput>-NextPrime(-n)</userinput>.
</para>
<para>
This function uses the GMPs <function>mpz_nextprime</function>,
which in turn uses the probabilistic Miller-Rabin test
(See also <link linkend="gel-function-MillerRabinTest"><function>MillerRabinTest</function></link>).
The probability
of false positive is not tunable, but is low enough
for all practical purposes.
</para>
<para>
See
<ulink url="http://planetmath.org/PrimeNumber">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/PrimeNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PadicValuation"/>PadicValuation</term>
<listitem>
<synopsis>PadicValuation (n,p)</synopsis>
<para>Returns the p-adic valuation (number of trailing zeros in base <varname>p</varname>).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/P-adic_order">Wikipedia</ulink> or
<ulink url="http://planetmath.org/PAdicValuation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PowerMod"/>PowerMod</term>
<listitem>
<synopsis>PowerMod (a,b,m)</synopsis>
<para>
Compute <userinput>a^b mod m</userinput>. The
<varname>b</varname>'s power of <varname>a</varname> modulo
<varname>m</varname>. It is not necessary to use this function
as it is automatically used in modulo mode. Hence
<userinput>a^b mod m</userinput> is just as fast.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Prime"/>Prime</term>
<listitem>
<synopsis>Prime (n)</synopsis>
<para>Aliases: <function>prime</function></para>
<para>Return the <varname>n</varname>th prime (up to a limit).</para>
<para>
See
<ulink url="http://planetmath.org/PrimeNumber">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/PrimeNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PrimeFactors"/>PrimeFactors</term>
<listitem>
<synopsis>PrimeFactors (n)</synopsis>
<para>Return all prime factors of a number as a vector.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Prime_factor">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/PrimeFactor.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PseudoprimeTest"/>PseudoprimeTest</term>
<listitem>
<synopsis>PseudoprimeTest (n,b)</synopsis>
<para>Pseudoprime test, returns <constant>true</constant> if and only if
<userinput>b^(n-1) == 1 mod n</userinput></para>
<para>
See
<ulink url="http://planetmath.org/Pseudoprime">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/Pseudoprime.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RemoveFactor"/>RemoveFactor</term>
<listitem>
<synopsis>RemoveFactor (n,m)</synopsis>
<para>Removes all instances of the factor <varname>m</varname> from the number <varname>n</varname>. That is divides by the largest power of <varname>m</varname>, that divides <varname>n</varname>.</para>
<para>
See
<ulink url="http://planetmath.org/Divisibility">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/Factor.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SilverPohligHellmanWithFactorization"/>SilverPohligHellmanWithFactorization</term>
<listitem>
<synopsis>SilverPohligHellmanWithFactorization (n,b,q,f)</synopsis>
<para>Find discrete log of <varname>n</varname> base <varname>b</varname> in F<subscript>q</subscript>, the finite group of order <varname>q</varname>, where <varname>q</varname> is a prime using the Silver-Pohlig-Hellman algorithm, given <varname>f</varname> being the factorization of <varname>q</varname>-1.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SqrtModPrime"/>SqrtModPrime</term>
<listitem>
<synopsis>SqrtModPrime (n,p)</synopsis>
<para>Find square root of <varname>n</varname> modulo <varname>p</varname> (where <varname>p</varname> is a prime). Null is returned if not a quadratic residue.</para>
<para>
See
<ulink url="http://planetmath.org/QuadraticResidue">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/QuadraticResidue.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StrongPseudoprimeTest"/>StrongPseudoprimeTest</term>
<listitem>
<synopsis>StrongPseudoprimeTest (n,b)</synopsis>
<para>Run the strong pseudoprime test base <varname>b</varname> on <varname>n</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Strong_pseudoprime">Wikipedia</ulink>,
<ulink url="http://planetmath.org/StrongPseudoprime">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/StrongPseudoprime.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-gcd"/>gcd</term>
<listitem>
<synopsis>gcd (a,args...)</synopsis>
<para>Aliases: <function>GCD</function></para>
<para>
Greatest common divisor of integers. You can enter as many
integers as you want in the argument list, or you can give
a vector or a matrix of integers. If you give more than
one matrix of the same size then GCD is done element by
element.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Greatest_common_divisor">Wikipedia</ulink>,
<ulink url="http://planetmath.org/GreatestCommonDivisor">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/GreatestCommonDivisor.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-lcm"/>lcm</term>
<listitem>
<synopsis>lcm (a,args...)</synopsis>
<para>Aliases: <function>LCM</function></para>
<para>
Least common multiplier of integers. You can enter as many
integers as you want in the argument list, or you can give a
vector or a matrix of integers. If you give more than one
matrix of the same size then LCM is done element by element.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Least_common_multiple">Wikipedia</ulink>,
<ulink url="http://planetmath.org/LeastCommonMultiple">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/LeastCommonMultiple.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-matrix">
<title>Операции с матрицами</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-ApplyOverMatrix"/>ApplyOverMatrix</term>
<listitem>
<synopsis>ApplyOverMatrix (a,func)</synopsis>
<para>Применяет функцию к каждому элементу матрицы и возвращает матрицу результатов.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ApplyOverMatrix2"/>ApplyOverMatrix2</term>
<listitem>
<synopsis>ApplyOverMatrix2 (a,b,func)</synopsis>
<para>Применяет функцию к каждому элементу двух матриц (или 1 значению и 1 матрице) и возвращает матрицу результатов.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ColumnsOf"/>ColumnsOf</term>
<listitem>
<synopsis>ColumnsOf (M)</synopsis>
<para>Возвращает столбцы матрицы в виде горизонтального вектора.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ComplementSubmatrix"/>ComplementSubmatrix</term>
<listitem>
<synopsis>ComplementSubmatrix (m,r,c)</synopsis>
<para>Remove column(s) and row(s) from a matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CompoundMatrix"/>CompoundMatrix</term>
<listitem>
<synopsis>CompoundMatrix (k,A)</synopsis>
<para>Calculate the kth compound matrix of A.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CountZeroColumns"/>CountZeroColumns</term>
<listitem>
<synopsis>CountZeroColumns (M)</synopsis>
<para>
Count the number of zero columns in a matrix. For example,
once you column-reduce a matrix, you can use this to find
the nullity. See <link linkend="gel-function-cref"><function>cref</function></link>
and <link linkend="gel-function-Nullity"><function>Nullity</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DeleteColumn"/>DeleteColumn</term>
<listitem>
<synopsis>DeleteColumn (M,столбец)</synopsis>
<para>Удаляет столбец матрицы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DeleteRow"/>DeleteRow</term>
<listitem>
<synopsis>DeleteRow (M,строка)</synopsis>
<para>Удаляет строку матрицы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DiagonalOf"/>DiagonalOf</term>
<listitem>
<synopsis>DiagonalOf (M)</synopsis>
<para>Gets the diagonal entries of a matrix as a column vector.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DotProduct"/>DotProduct</term>
<listitem>
<synopsis>DotProduct (u,v)</synopsis>
<para>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <link linkend="gel-function-HermitianProduct">HermitianProduct</link> for the standard sesquilinear inner product.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Dot_product">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DotProduct">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ExpandMatrix"/>ExpandMatrix</term>
<listitem>
<synopsis>ExpandMatrix (M)</synopsis>
<para>
Expands a matrix just like we do on unquoted matrix input.
That is we expand any internal matrices as blocks. This is
a way to construct matrices out of smaller ones and this is
normally done automatically on input unless the matrix is quoted.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-HermitianProduct"/>HermitianProduct</term>
<listitem>
<synopsis>HermitianProduct (u,v)</synopsis>
<para>Aliases: <function>InnerProduct</function></para>
<para>Get the Hermitian product of two vectors. The vectors must be of the same size. This is a sesquilinear form using the identity matrix.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Sesquilinear_form">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/HermitianInnerProduct.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-I"/>I</term>
<listitem>
<synopsis>I (n)</synopsis>
<para>Aliases: <function>eye</function></para>
<para>Return an identity matrix of a given size, that is <varname>n</varname> by <varname>n</varname>. If <varname>n</varname> is zero, returns <constant>null</constant>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Identity_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/IdentityMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IndexComplement"/>IndexComplement</term>
<listitem>
<synopsis>IndexComplement (vec,msize)</synopsis>
<para>Return the index complement of a vector of indexes. Everything is one based. For example for vector <userinput>[2,3]</userinput> and size
<userinput>5</userinput>, we return <userinput>[1,4,5]</userinput>. If
<varname>msize</varname> is 0, we always return <constant>null</constant>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsDiagonal"/>IsDiagonal</term>
<listitem>
<synopsis>IsDiagonal (M)</synopsis>
<para>Является ли матрица диагональной.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Diagonal_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DiagonalMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsIdentity"/>IsIdentity</term>
<listitem>
<synopsis>IsIdentity (x)</synopsis>
<para>Check if a matrix is the identity matrix. Automatically returns <constant>false</constant>
if the matrix is not square. Also works on numbers, in which
case it is equivalent to <userinput>x==1</userinput>. When <varname>x</varname> is
<constant>null</constant> (we could think of that as a 0 by 0 matrix),
no error is generated and <constant>false</constant> is returned.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsLowerTriangular"/>IsLowerTriangular</term>
<listitem>
<synopsis>IsLowerTriangular (M)</synopsis>
<para>Является ли матрица нижнетреугольной, то есть все её элементы над диагональю равны нулю.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrixInteger"/>IsMatrixInteger</term>
<listitem>
<synopsis>IsMatrixInteger (M)</synopsis>
<para>Check if a matrix is a matrix of integers (non-complex).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrixNonnegative"/>IsMatrixNonnegative</term>
<listitem>
<synopsis>IsMatrixNonnegative (M)</synopsis>
<para>Check if a matrix is non-negative, that is if each element is non-negative.
Do not confuse positive matrices with positive semi-definite matrices.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Positive_matrix">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrixPositive"/>IsMatrixPositive</term>
<listitem>
<synopsis>IsMatrixPositive (M)</synopsis>
<para>Check if a matrix is positive, that is if each element is
positive (and hence real). In particular, no element is 0. Do not confuse
positive matrices with positive definite matrices.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Positive_matrix">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrixRational"/>IsMatrixRational</term>
<listitem>
<synopsis>IsMatrixRational (M)</synopsis>
<para>Проверяет, является ли матрица матрицей из рациональных (не комплексных) чисел.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrixReal"/>IsMatrixReal</term>
<listitem>
<synopsis>IsMatrixReal (M)</synopsis>
<para>Проверяет, является ли матрица матрицей из действительных (не комплексных) чисел.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsMatrixSquare"/>IsMatrixSquare</term>
<listitem>
<synopsis>IsMatrixSquare (M)</synopsis>
<para>Проверяет, является ли матрица квадратной, то есть её ширина равна высоте.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsUpperTriangular"/>IsUpperTriangular</term>
<listitem>
<synopsis>IsUpperTriangular (M)</synopsis>
<para>Is a matrix upper triangular? That is, a matrix is upper triangular if all the entries below the diagonal are zero.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsValueOnly"/>IsValueOnly</term>
<listitem>
<synopsis>IsValueOnly (M)</synopsis>
<para>Проверяет, состоит ли матрица только из чисел. Многие встроенные функции делают эту проверку. Значения могут быть любыми числами, включая комплексные.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsVector"/>IsVector</term>
<listitem>
<synopsis>IsVector (v)</synopsis>
<para>Является ли аргумент горизонтальным или вертикальным вектором. Genius не делает различий между матрицей и вектором: вектор — это просто матрица 1 на <varname>n</varname> или <varname>n</varname> на 1.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsZero"/>IsZero</term>
<listitem>
<synopsis>IsZero (x)</synopsis>
<para>Проверяет, состоит ли матрица из одних нулей. Также работает с числами, в этом случае эквивалентна выражению <userinput>x==0</userinput>. Если переменная <varname>x</varname> равна <constant>null</constant> (можно представить это, как матрицу 0 на 0 элементов), ошибка не генерируется и возвращается <constant>true</constant>, так как условие является бессмысленным.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LowerTriangular"/>LowerTriangular</term>
<listitem>
<synopsis>LowerTriangular (M)</synopsis>
<para>Возвращает копию матрицы <varname>M</varname>, в которой все элементы под диагональю заменены нулями.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MakeDiagonal"/>MakeDiagonal</term>
<listitem>
<synopsis>MakeDiagonal (v,arg...)</synopsis>
<para>Псевдоним: <function>diag</function></para>
<para>Создаёт диагональную матрицу из вектора. Значения для диагонали также могут быть переданы в виде аргументов функции. Поэтому <userinput>MakeDiagonal([1,2,3])</userinput> — то же самое, что и <userinput>MakeDiagonal(1,2,3)</userinput>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Diagonal_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/DiagonalMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MakeVector"/>MakeVector</term>
<listitem>
<synopsis>MakeVector (A)</synopsis>
<para>Make column vector out of matrix by putting columns above
each other. Returns <constant>null</constant> when given <constant>null</constant>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MatrixProduct"/>MatrixProduct</term>
<listitem>
<synopsis>MatrixProduct (A)</synopsis>
<para>Вычисляет произведение всех элементов матрицы или вектора. То есть, умножает друг на друга все элементы и возвращает число, являющееся их произведением.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MatrixSum"/>MatrixSum</term>
<listitem>
<synopsis>MatrixSum (A)</synopsis>
<para>Вычисляет сумму всех элементов матрицы или вектора. То есть, складывает все элементы и возвращает число, являющееся их суммой.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MatrixSumSquares"/>MatrixSumSquares</term>
<listitem>
<synopsis>MatrixSumSquares (A)</synopsis>
<para>Вычисляет сумму квадратов всех элементов матрицы или вектора.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NonzeroColumns"/>NonzeroColumns</term>
<listitem>
<synopsis>NonzeroColumns (M)</synopsis>
<para>Returns a row vector of the indices of nonzero columns in the matrix <varname>M</varname>.</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NonzeroElements"/>NonzeroElements</term>
<listitem>
<synopsis>NonzeroElements (v)</synopsis>
<para>Returns a row vector of the indices of nonzero elements in the vector <varname>v</varname>.</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OuterProduct"/>OuterProduct</term>
<listitem>
<synopsis>OuterProduct (u,v)</synopsis>
<para>Get the outer product of two vectors. That is, suppose that
<varname>u</varname> and <varname>v</varname> are vertical vectors, then
the outer product is <userinput>v * u.'</userinput>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ReverseVector"/>ReverseVector</term>
<listitem>
<synopsis>ReverseVector (v)</synopsis>
<para>Reverse elements in a vector. Return <constant>null</constant> if given <constant>null</constant></para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowSum"/>RowSum</term>
<listitem>
<synopsis>RowSum (m)</synopsis>
<para>Вычисляет суммы элементов в каждой строке матрицы и возвращает вертикальный вектор с результатом.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowSumSquares"/>RowSumSquares</term>
<listitem>
<synopsis>RowSumSquares (m)</synopsis>
<para>Вычисляет суммы квадратов элементов в каждой строке матрицы и возвращает вертикальный вектор с результатами.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowsOf"/>RowsOf</term>
<listitem>
<synopsis>RowsOf (M)</synopsis>
<para>Gets the rows of a matrix as a vertical vector. Each element
of the vector is a horizontal vector that is the corresponding row of
<varname>M</varname>. This function is useful if you wish to loop over the
rows of a matrix. For example, as <userinput>for r in RowsOf(M) do
something(r)</userinput>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetMatrixSize"/>SetMatrixSize</term>
<listitem>
<synopsis>SetMatrixSize (M,строки,столбцы)</synopsis>
<para>Make new matrix of given size from old one. That is, a new
matrix will be returned to which the old one is copied. Entries that
don't fit are clipped and extra space is filled with zeros.
If <varname>rows</varname> or <varname>columns</varname> are zero
then <constant>null</constant> is returned.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ShuffleVector"/>ShuffleVector</term>
<listitem>
<synopsis>ShuffleVector (v)</synopsis>
<para>Shuffle elements in a vector. Return <constant>null</constant> if given <constant>null</constant>.</para>
<para>Version 1.0.13 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SortVector"/>SortVector</term>
<listitem>
<synopsis>SortVector (v)</synopsis>
<para>Sort vector elements in an increasing order.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StripZeroColumns"/>StripZeroColumns</term>
<listitem>
<synopsis>StripZeroColumns (M)</synopsis>
<para>Удаляет все состоящие только из нулей столбцы матрицы <varname>M</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StripZeroRows"/>StripZeroRows</term>
<listitem>
<synopsis>StripZeroRows (M)</synopsis>
<para>Удаляет все состоящие только из нулей строки матрицы <varname>M</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Submatrix"/>Submatrix</term>
<listitem>
<synopsis>Submatrix (m,r,c)</synopsis>
<para>Return column(s) and row(s) from a matrix. This is
just equivalent to <userinput>m@(r,c)</userinput>. <varname>r</varname>
and <varname>c</varname> should be vectors of rows and columns (or single numbers if only one row or column is needed).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SwapRows"/>SwapRows</term>
<listitem>
<synopsis>SwapRows (m,строка1,строка2)</synopsis>
<para>Меняет местами две строки матрицы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-UpperTriangular"/>UpperTriangular</term>
<listitem>
<synopsis>UpperTriangular (M)</synopsis>
<para>Возвращает копию матрицы <varname>M</varname>, в которой все элементы под диагональю заменены нулями.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-columns"/>columns</term>
<listitem>
<synopsis>columns (M)</synopsis>
<para>Возвращает число столбцов в матрице.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-elements"/>elements</term>
<listitem>
<synopsis>elements (M)</synopsis>
<para>Возвращает общее число элементов в матрице. Это число столбцов, умноженное на число строк.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ones"/>ones</term>
<listitem>
<synopsis>ones (строки,столбцы...)</synopsis>
<para>Создаёт матрицу, состоящую только из единиц (или строчный вектор, если задан только один аргумент). Возвращает <constant>null</constant>, если задано число строк или столбцов, равное нулю.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-rows"/>rows</term>
<listitem>
<synopsis>rows (M)</synopsis>
<para>Возвращает число строк в матрице.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-zeros"/>zeros</term>
<listitem>
<synopsis>zeros (строки,столбцы...)</synopsis>
<para>Создаёт матрицу, состоящую только из единиц (или строчный вектор, если задан только один аргумент). Возвращает <constant>null</constant>, если задано число строк или столбцов, равное нулю.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-linear-algebra">
<title>Линейная алгебра</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-AuxiliaryUnitMatrix"/>AuxiliaryUnitMatrix</term>
<listitem>
<synopsis>AuxiliaryUnitMatrix (n)</synopsis>
<para>Get the auxiliary unit matrix of size <varname>n</varname>. This is a square matrix with that is all zero except the
superdiagonal being all ones. It is the Jordan block matrix of one zero eigenvalue.</para>
<para>
See
<ulink url="http://planetmath.org/JordanCanonicalFormTheorem">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/JordanBlock.html">Mathworld</ulink> for more information on Jordan Canonical Form.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BilinearForm"/>BilinearForm</term>
<listitem>
<synopsis>BilinearForm (v,A,w)</synopsis>
<para>Evaluate (v,w) with respect to the bilinear form given by the matrix A.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BilinearFormFunction"/>BilinearFormFunction</term>
<listitem>
<synopsis>BilinearFormFunction (A)</synopsis>
<para>Return a function that evaluates two vectors with respect to the bilinear form given by A.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CharacteristicPolynomial"/>CharacteristicPolynomial</term>
<listitem>
<synopsis>CharacteristicPolynomial (M)</synopsis>
<para>Aliases: <function>CharPoly</function></para>
<para>Get the characteristic polynomial as a vector. That is, return
the coefficients of the polynomial starting with the constant term. This is
the polynomial defined by <userinput>det(M-xI)</userinput>. The roots of this
polynomial are the eigenvalues of <varname>M</varname>.
See also <link linkend="gel-function-CharacteristicPolynomialFunction">CharacteristicPolynomialFunction</link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Characteristic_polynomial">Wikipedia</ulink> or
<ulink url="http://planetmath.org/CharacteristicEquation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CharacteristicPolynomialFunction"/>CharacteristicPolynomialFunction</term>
<listitem>
<synopsis>CharacteristicPolynomialFunction (M)</synopsis>
<para>Get the characteristic polynomial as a function. This is
the polynomial defined by <userinput>det(M-xI)</userinput>. The roots of this
polynomial are the eigenvalues of <varname>M</varname>.
See also <link linkend="gel-function-CharacteristicPolynomial">CharacteristicPolynomial</link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Characteristic_polynomial">Wikipedia</ulink> or
<ulink url="http://planetmath.org/CharacteristicEquation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ColumnSpace"/>ColumnSpace</term>
<listitem>
<synopsis>ColumnSpace (M)</synopsis>
<para>Get a basis matrix for the columnspace of a matrix. That is,
return a matrix whose columns are the basis for the column space of
<varname>M</varname>. That is the space spanned by the columns of
<varname>M</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Row_and_column_spaces">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CommutationMatrix"/>CommutationMatrix</term>
<listitem>
<synopsis>CommutationMatrix (m, n)</synopsis>
<para>Return the commutation matrix <userinput>K(m,n)</userinput>, which is the unique <userinput>m*n</userinput> by
<userinput>m*n</userinput> matrix such that <userinput>K(m,n) * MakeVector(A) = MakeVector(A.')</userinput> for all <varname>m</varname> by <varname>n</varname>
matrices <varname>A</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CompanionMatrix"/>CompanionMatrix</term>
<listitem>
<synopsis>CompanionMatrix (p)</synopsis>
<para>Companion matrix of a polynomial (as vector).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ConjugateTranspose"/>ConjugateTranspose</term>
<listitem>
<synopsis>ConjugateTranspose (M)</synopsis>
<para>Conjugate transpose of a matrix (adjoint). This is the
same as the <userinput>'</userinput> operator.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Conjugate_transpose">Wikipedia</ulink> or
<ulink url="http://planetmath.org/ConjugateTranspose">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Convolution"/>Convolution</term>
<listitem>
<synopsis>Convolution (a,b)</synopsis>
<para>Aliases: <function>convol</function></para>
<para>Calculate convolution of two horizontal vectors.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ConvolutionVector"/>ConvolutionVector</term>
<listitem>
<synopsis>ConvolutionVector (a,b)</synopsis>
<para>Calculate convolution of two horizontal vectors. Return
result as a vector and not added together.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CrossProduct"/>CrossProduct</term>
<listitem>
<synopsis>CrossProduct (v,w)</synopsis>
<para>CrossProduct of two vectors in R<superscript>3</superscript> as
a column vector.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Cross_product">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DeterminantalDivisorsInteger"/>DeterminantalDivisorsInteger</term>
<listitem>
<synopsis>DeterminantalDivisorsInteger (M)</synopsis>
<para>Get the determinantal divisors of an integer matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DirectSum"/>DirectSum</term>
<listitem>
<synopsis>DirectSum (M,N...)</synopsis>
<para>Direct sum of matrices.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Matrix_addition#directsum">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DirectSumMatrixVector"/>DirectSumMatrixVector</term>
<listitem>
<synopsis>DirectSumMatrixVector (v)</synopsis>
<para>Direct sum of a vector of matrices.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Matrix_addition#directsum">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Eigenvalues"/>Eigenvalues</term>
<listitem>
<synopsis>Eigenvalues (M)</synopsis>
<para>Aliases: <function>eig</function></para>
<para>Get the eigenvalues of a square matrix.
Currently only works for matrices of size up to 4 by 4, or for
triangular matrices (for which the eigenvalues are on the
diagonal).
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Eigenvalue">Wikipedia</ulink>,
<ulink url="http://planetmath.org/Eigenvalue">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/Eigenvalue.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Eigenvectors"/>Eigenvectors</term>
<listitem>
<synopsis>Eigenvectors (M)</synopsis>
<synopsis>Eigenvectors (M, &eigenvalues)</synopsis>
<synopsis>Eigenvectors (M, &eigenvalues, &multiplicities)</synopsis>
<para>Get the eigenvectors of a square matrix. Optionally get also
the eigenvalues and their algebraic multiplicities.
Currently only works for matrices of size up to 2 by 2.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Eigenvector">Wikipedia</ulink>,
<ulink url="http://planetmath.org/Eigenvector">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/Eigenvector.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GramSchmidt"/>GramSchmidt</term>
<listitem>
<synopsis>GramSchmidt (v,B...)</synopsis>
<para>Apply the Gram-Schmidt process (to the columns) with respect to
inner product given by <varname>B</varname>. If <varname>B</varname> is not
given then the standard Hermitian product is used. <varname>B</varname> can
either be a sesquilinear function of two arguments or it can be a matrix giving
a sesquilinear form. The vectors will be made orthonormal with respect to
<varname>B</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Wikipedia</ulink> or
<ulink url="http://planetmath.org/GramSchmidtOrthogonalization">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-HankelMatrix"/>HankelMatrix</term>
<listitem>
<synopsis>HankelMatrix (c,r)</synopsis>
<para>Hankel matrix, a matrix whose skew-diagonals are constant. <varname>c</varname> is the first row and <varname>r</varname> is the
last column. It is assumed that both arguments are vectors and the last element of <varname>c</varname> is the same
as the first element of <varname>r</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hankel_matrix">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-HilbertMatrix"/>HilbertMatrix</term>
<listitem>
<synopsis>HilbertMatrix (n)</synopsis>
<para>Hilbert matrix of order <varname>n</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hilbert_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HilbertMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Image"/>Image</term>
<listitem>
<synopsis>Image (T)</synopsis>
<para>Get the image (columnspace) of a linear transform.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Row_and_column_spaces">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InfNorm"/>InfNorm</term>
<listitem>
<synopsis>InfNorm (v)</synopsis>
<para>Get the Inf Norm of a vector, sometimes called the sup norm or the max norm.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InvariantFactorsInteger"/>InvariantFactorsInteger</term>
<listitem>
<synopsis>InvariantFactorsInteger (M)</synopsis>
<para>Get the invariant factors of a square integer matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InverseHilbertMatrix"/>InverseHilbertMatrix</term>
<listitem>
<synopsis>InverseHilbertMatrix (n)</synopsis>
<para>Inverse Hilbert matrix of order <varname>n</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hilbert_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HilbertMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsHermitian"/>IsHermitian</term>
<listitem>
<synopsis>IsHermitian (M)</synopsis>
<para>Is a matrix Hermitian. That is, is it equal to its conjugate transpose.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hermitian_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/HermitianMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsInSubspace"/>IsInSubspace</term>
<listitem>
<synopsis>IsInSubspace (v,W)</synopsis>
<para>Test if a vector is in a subspace.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsInvertible"/>IsInvertible</term>
<listitem>
<synopsis>IsInvertible (n)</synopsis>
<para>Is a matrix (or number) invertible (Integer matrix is invertible if and only if it is invertible over the integers).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsInvertibleField"/>IsInvertibleField</term>
<listitem>
<synopsis>IsInvertibleField (n)</synopsis>
<para>Is a matrix (or number) invertible over a field.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsNormal"/>IsNormal</term>
<listitem>
<synopsis>IsNormal (M)</synopsis>
<para>Is <varname>M</varname> a normal matrix. That is,
does <userinput>M*M' == M'*M</userinput>.</para>
<para>
See
<ulink url="http://planetmath.org/NormalMatrix">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/NormalMatrix.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPositiveDefinite"/>IsPositiveDefinite</term>
<listitem>
<synopsis>IsPositiveDefinite (M)</synopsis>
<para>Is <varname>M</varname> a Hermitian positive definite matrix. That is if
<userinput>HermitianProduct(M*v,v)</userinput> is always strictly positive for
any vector <varname>v</varname>.
<varname>M</varname> must be square and Hermitian to be positive definite.
The check that is performed is that every principal submatrix has a non-negative
determinant.
(See <link linkend="gel-function-HermitianProduct">HermitianProduct</link>)</para>
<para>
Note that some authors (for example Mathworld) do not require that
<varname>M</varname> be Hermitian, and then the condition is
on the real part of the inner product, but we do not take this
view. If you wish to perform this check, just check the
Hermitian part of the matrix <varname>M</varname> as follows:
<userinput>IsPositiveDefinite(M+M')</userinput>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Positive-definite_matrix">Wikipedia</ulink>,
<ulink url="http://planetmath.org/PositiveDefinite">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPositiveSemidefinite"/>IsPositiveSemidefinite</term>
<listitem>
<synopsis>IsPositiveSemidefinite (M)</synopsis>
<para>Is <varname>M</varname> a Hermitian positive semidefinite matrix. That is if
<userinput>HermitianProduct(M*v,v)</userinput> is always non-negative for
any vector <varname>v</varname>.
<varname>M</varname> must be square and Hermitian to be positive semidefinite.
The check that is performed is that every principal submatrix has a non-negative
determinant.
(See <link linkend="gel-function-HermitianProduct">HermitianProduct</link>)</para>
<para>
Note that some authors do not require that
<varname>M</varname> be Hermitian, and then the condition is
on the real part of the inner product, but we do not take this
view. If you wish to perform this check, just check the
Hermitian part of the matrix <varname>M</varname> as follows:
<userinput>IsPositiveSemidefinite(M+M')</userinput>.
</para>
<para>
See
<ulink url="http://planetmath.org/PositiveSemidefinite">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsSkewHermitian"/>IsSkewHermitian</term>
<listitem>
<synopsis>IsSkewHermitian (M)</synopsis>
<para>Is a matrix skew-Hermitian. That is, is the conjugate transpose equal to negative of the matrix.</para>
<para>
See
<ulink url="http://planetmath.org/SkewHermitianMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsUnitary"/>IsUnitary</term>
<listitem>
<synopsis>IsUnitary (M)</synopsis>
<para>Is a matrix unitary? That is, does
<userinput>M'*M</userinput> and <userinput>M*M'</userinput>
equal the identity.</para>
<para>
See
<ulink url="http://planetmath.org/UnitaryTransformation">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/UnitaryMatrix.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-JordanBlock"/>JordanBlock</term>
<listitem>
<synopsis>JordanBlock (n,lambda)</synopsis>
<para>Aliases: <function>J</function></para>
<para>Get the Jordan block corresponding to the eigenvalue
<varname>lambda</varname> with multiplicity <varname>n</varname>.
</para>
<para>
See
<ulink url="http://planetmath.org/JordanCanonicalFormTheorem">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/JordanBlock.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Kernel"/>Kernel</term>
<listitem>
<synopsis>Kernel (T)</synopsis>
<para>Get the kernel (nullspace) of a linear transform.</para>
<para>
(See <link linkend="gel-function-NullSpace">NullSpace</link>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-KroneckerProduct"/>KroneckerProduct</term>
<listitem>
<synopsis>KroneckerProduct (M, N)</synopsis>
<para>Aliases: <function>TensorProduct</function></para>
<para>
Compute the Kronecker product (tensor product in standard basis)
of two matrices.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Kronecker_product">Wikipedia</ulink>,
<ulink url="http://planetmath.org/KroneckerProduct">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/KroneckerProduct.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LUDecomposition"/>LUDecomposition</term>
<listitem>
<synopsis>LUDecomposition (A, L, U)</synopsis>
<para>
Get the LU decomposition of <varname>A</varname>, that is
find a lower triangular matrix and upper triangular
matrix whose product is <varname>A</varname>.
Store the result in the <varname>L</varname> and
<varname>U</varname>, which should be references. It returns <constant>true</constant>
if successful.
For example suppose that A is a square matrix, then after running:
<screen><prompt>genius></prompt> <userinput>LUDecomposition(A,&L,&U)</userinput>
</screen>
You will have the lower matrix stored in a variable called
<varname>L</varname> and the upper matrix in a variable called
<varname>U</varname>.
</para>
<para>
This is the LU decomposition of a matrix aka Crout and/or Cholesky
reduction.
(ISBN 0-201-11577-8 pp.99-103)
The upper triangular matrix features a diagonal
of values 1 (one). This is not Doolittle's Method, which features
the 1's diagonal on the lower matrix.
</para>
<para>
Not all matrices have LU decompositions, for example
<userinput>[0,1;1,0]</userinput> does not and this function returns
<constant>false</constant> in this case and sets <varname>L</varname>
and <varname>U</varname> to <constant>null</constant>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/LU_decomposition">Wikipedia</ulink>,
<ulink url="http://planetmath.org/LUDecomposition">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/LUDecomposition.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Minor"/>Minor</term>
<listitem>
<synopsis>Minor (M,i,j)</synopsis>
<para>Get the <varname>i</varname>-<varname>j</varname> minor of a matrix.</para>
<para>
See
<ulink url="http://planetmath.org/Minor">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NonPivotColumns"/>NonPivotColumns</term>
<listitem>
<synopsis>NonPivotColumns (M)</synopsis>
<para>Return the columns that are not the pivot columns of a matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Norm"/>Norm</term>
<listitem>
<synopsis>Norm (v,p...)</synopsis>
<para>Aliases: <function>norm</function></para>
<para>Get the p Norm (or 2 Norm if no p is supplied) of a vector.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NullSpace"/>NullSpace</term>
<listitem>
<synopsis>NullSpace (T)</synopsis>
<para>Get the nullspace of a matrix. That is the kernel of the
linear mapping that the matrix represents. This is returned
as a matrix whose column space is the nullspace of
<varname>T</varname>.</para>
<para>
See
<ulink url="http://planetmath.org/Nullspace">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Nullity"/>Nullity</term>
<listitem>
<synopsis>Nullity (M)</synopsis>
<para>Aliases: <function>nullity</function></para>
<para>Get the nullity of a matrix. That is, return the dimension of
the nullspace; the dimension of the kernel of <varname>M</varname>.</para>
<para>
See
<ulink url="http://planetmath.org/Nullity">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OrthogonalComplement"/>OrthogonalComplement</term>
<listitem>
<synopsis>OrthogonalComplement (M)</synopsis>
<para>Get the orthogonal complement of the columnspace.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PivotColumns"/>PivotColumns</term>
<listitem>
<synopsis>PivotColumns (M)</synopsis>
<para>Return pivot columns of a matrix, that is columns that have a leading 1 in row reduced form. Also returns the row where they occur.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Projection"/>Projection</term>
<listitem>
<synopsis>Projection (v,W,B...)</synopsis>
<para>Projection of vector <varname>v</varname> onto subspace
<varname>W</varname> with respect to inner product given by
<varname>B</varname>. If <varname>B</varname> is not given then the standard
Hermitian product is used. <varname>B</varname> can either be a sesquilinear
function of two arguments or it can be a matrix giving a sesquilinear form.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-QRDecomposition"/>QRDecomposition</term>
<listitem>
<synopsis>QRDecomposition (A, Q)</synopsis>
<para>
Get the QR decomposition of a square matrix <varname>A</varname>,
returns the upper triangular matrix <varname>R</varname>
and sets <varname>Q</varname> to the orthogonal (unitary) matrix.
<varname>Q</varname> should be a reference or <constant>null</constant> if you don't
want any return.
For example:
<screen><prompt>genius></prompt> <userinput>R = QRDecomposition(A,&Q)</userinput>
</screen>
You will have the upper triangular matrix stored in
a variable called
<varname>R</varname> and the orthogonal (unitary) matrix stored in
<varname>Q</varname>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/QR_decomposition">Wikipedia</ulink> or
<ulink url="http://planetmath.org/QRDecomposition">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/QRDecomposition.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RayleighQuotient"/>RayleighQuotient</term>
<listitem>
<synopsis>RayleighQuotient (A,x)</synopsis>
<para>Return the Rayleigh quotient (also called the Rayleigh-Ritz quotient or ratio) of a matrix and a vector.</para>
<para>
See
<ulink url="http://planetmath.org/RayleighQuotient">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RayleighQuotientIteration"/>RayleighQuotientIteration</term>
<listitem>
<synopsis>RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)</synopsis>
<para>Find eigenvalues of <varname>A</varname> using the Rayleigh
quotient iteration method. <varname>x</varname> is a guess
at a eigenvector and could be random. It should have
nonzero imaginary part if it will have any chance at finding
complex eigenvalues. The code will run at most
<varname>maxiter</varname> iterations and return <constant>null</constant>
if we cannot get within an error of <varname>epsilon</varname>.
<varname>vecref</varname> should either be <constant>null</constant> or a reference
to a variable where the eigenvector should be stored.
</para>
<para>
See
<ulink url="http://planetmath.org/RayleighQuotient">Planetmath</ulink> for more information on Rayleigh quotient.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Rank"/>Rank</term>
<listitem>
<synopsis>Rank (M)</synopsis>
<para>Синонимы: <function>rank</function></para>
<para>Возвращает ранг матрицы.</para>
<para>
See
<ulink url="http://planetmath.org/SylvestersLaw">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RosserMatrix"/>RosserMatrix</term>
<listitem>
<synopsis>RosserMatrix ()</synopsis>
<para>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Rotation2D"/>Rotation2D</term>
<listitem>
<synopsis>Rotation2D (angle)</synopsis>
<para>Aliases: <function>RotationMatrix</function></para>
<para>Return the matrix corresponding to rotation around origin in R<superscript>2</superscript>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Rotation3DX"/>Rotation3DX</term>
<listitem>
<synopsis>Rotation3DX (angle)</synopsis>
<para>Return the matrix corresponding to rotation around origin in R<superscript>3</superscript> about the x-axis.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Rotation3DY"/>Rotation3DY</term>
<listitem>
<synopsis>Rotation3DY (angle)</synopsis>
<para>Return the matrix corresponding to rotation around origin in R<superscript>3</superscript> about the y-axis.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Rotation3DZ"/>Rotation3DZ</term>
<listitem>
<synopsis>Rotation3DZ (angle)</synopsis>
<para>Return the matrix corresponding to rotation around origin in R<superscript>3</superscript> about the z-axis.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowSpace"/>RowSpace</term>
<listitem>
<synopsis>RowSpace (M)</synopsis>
<para>Get a basis matrix for the rowspace of a matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SesquilinearForm"/>SesquilinearForm</term>
<listitem>
<synopsis>SesquilinearForm (v,A,w)</synopsis>
<para>Evaluate (v,w) with respect to the sesquilinear form given by the matrix A.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SesquilinearFormFunction"/>SesquilinearFormFunction</term>
<listitem>
<synopsis>SesquilinearFormFunction (A)</synopsis>
<para>Return a function that evaluates two vectors with respect to the sesquilinear form given by A.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SmithNormalFormField"/>SmithNormalFormField</term>
<listitem>
<synopsis>SmithNormalFormField (A)</synopsis>
<para>Returns the Smith normal form of a matrix over fields (will end up with 1's on the diagonal).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Smith_normal_form">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SmithNormalFormInteger"/>SmithNormalFormInteger</term>
<listitem>
<synopsis>SmithNormalFormInteger (M)</synopsis>
<para>Return the Smith normal form for square integer matrices over integers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Smith_normal_form">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SolveLinearSystem"/>SolveLinearSystem</term>
<listitem>
<synopsis>SolveLinearSystem (M,V,args...)</synopsis>
<para>Solve linear system Mx=V, return solution V if there is a unique solution, <constant>null</constant> otherwise. Extra two reference parameters can optionally be used to get the reduced M and V.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ToeplitzMatrix"/>ToeplitzMatrix</term>
<listitem>
<synopsis>ToeplitzMatrix (c, r...)</synopsis>
<para>Return the Toeplitz matrix constructed given the first column c
and (optionally) the first row r. If only the column c is given then it is
conjugated and the nonconjugated version is used for the first row to give a
Hermitian matrix (if the first element is real of course).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Toeplitz_matrix">Wikipedia</ulink> or
<ulink url="http://planetmath.org/ToeplitzMatrix">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Trace"/>Trace</term>
<listitem>
<synopsis>Trace (M)</synopsis>
<para>Aliases: <function>trace</function></para>
<para>Calculate the trace of a matrix. That is the sum of the diagonal elements.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Trace_(linear_algebra)">Wikipedia</ulink> or
<ulink url="http://planetmath.org/Trace">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Transpose"/>Transpose</term>
<listitem>
<synopsis>Transpose (M)</synopsis>
<para>Транспозиция матрицы. То же самое, что оператор <userinput>.'</userinput>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Transpose">Wikipedia</ulink> or
<ulink url="http://planetmath.org/Transpose">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VandermondeMatrix"/>VandermondeMatrix</term>
<listitem>
<synopsis>VandermondeMatrix (v)</synopsis>
<para>Aliases: <function>vander</function></para>
<para>Return the Vandermonde matrix.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Vandermonde_matrix">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorAngle"/>VectorAngle</term>
<listitem>
<synopsis>VectorAngle (v,w,B...)</synopsis>
<para>The angle of two vectors with respect to inner product given by
<varname>B</varname>. If <varname>B</varname> is not given then the standard
Hermitian product is used. <varname>B</varname> can either be a sesquilinear
function of two arguments or it can be a matrix giving a sesquilinear form.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorSpaceDirectSum"/>VectorSpaceDirectSum</term>
<listitem>
<synopsis>VectorSpaceDirectSum (M,N)</synopsis>
<para>The direct sum of the vector spaces M and N.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorSubspaceIntersection"/>VectorSubspaceIntersection</term>
<listitem>
<synopsis>VectorSubspaceIntersection (M,N)</synopsis>
<para>Intersection of the subspaces given by M and N.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorSubspaceSum"/>VectorSubspaceSum</term>
<listitem>
<synopsis>VectorSubspaceSum (M,N)</synopsis>
<para>The sum of the vector spaces M and N, that is {w | w=m+n, m in M, n in N}.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-adj"/>adj</term>
<listitem>
<synopsis>adj (m)</synopsis>
<para>Aliases: <function>Adjugate</function></para>
<para>Get the classical adjoint (adjugate) of a matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-cref"/>cref</term>
<listitem>
<synopsis>cref (M)</synopsis>
<para>Aliases: <function>CREF</function> <function>ColumnReducedEchelonForm</function></para>
<para>Compute the Column Reduced Echelon Form.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-det"/>det</term>
<listitem>
<synopsis>det (M)</synopsis>
<para>Aliases: <function>Determinant</function></para>
<para>Get the determinant of a matrix.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Determinant">Wikipedia</ulink> or
<ulink url="http://planetmath.org/Determinant2">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ref"/>ref</term>
<listitem>
<synopsis>ref (M)</synopsis>
<para>Aliases: <function>REF</function> <function>RowEchelonForm</function></para>
<para>Get the row echelon form of a matrix. That is, apply gaussian
elimination but not backaddition to <varname>M</varname>. The pivot rows are
divided to make all pivots 1.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Row_echelon_form">Wikipedia</ulink> or
<ulink url="http://planetmath.org/RowEchelonForm">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-rref"/>rref</term>
<listitem>
<synopsis>rref (M)</synopsis>
<para>Aliases: <function>RREF</function> <function>ReducedRowEchelonForm</function></para>
<para>Get the reduced row echelon form of a matrix. That is, apply gaussian elimination together with backaddition to <varname>M</varname>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Reduced_row_echelon_form">Wikipedia</ulink> or
<ulink url="http://planetmath.org/ReducedRowEchelonForm">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-combinatorics">
<title>Комбинаторика</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-Catalan"/>Catalan</term>
<listitem>
<synopsis>Catalan (n)</synopsis>
<para>Возвращает <varname>n</varname>-ое число Каталана.</para>
<para>
See
<ulink url="http://planetmath.org/CatalanNumbers">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Combinations"/>Combinations</term>
<listitem>
<synopsis>Combinations (k,n)</synopsis>
<para>Get all combinations of k numbers from 1 to n as a vector of vectors.
(See also <link linkend="gel-function-NextCombination">NextCombination</link>)
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Combination">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DoubleFactorial"/>DoubleFactorial</term>
<listitem>
<synopsis>DoubleFactorial (n)</synopsis>
<para>Двойной факториал: <userinput>n(n-2)(n-4)...</userinput></para>
<para>
See
<ulink url="http://planetmath.org/DoubleFactorial">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Factorial"/>Factorial</term>
<listitem>
<synopsis>Factorial (n)</synopsis>
<para>Факториал: <userinput>n(n-1)(n-2)...</userinput></para>
<para>
See
<ulink url="http://planetmath.org/Factorial">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FallingFactorial"/>FallingFactorial</term>
<listitem>
<synopsis>FallingFactorial (n,k)</synopsis>
<para>Убывающий факториал: <userinput>(n)_k = n(n-1)...(n-(k-1))</userinput></para>
<para>
See
<ulink url="http://planetmath.org/FallingFactorial">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Fibonacci"/>Fibonacci</term>
<listitem>
<synopsis>Fibonacci (x)</synopsis>
<para>Синонимы: <function>fib</function></para>
<para>Вычисляет <varname>n</varname>-ое число Фибоначчи. Это число, вычисляемое рекурсивно по формулам <userinput>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</userinput> и <userinput>Fibonacci(1) = Fibonacci(2) = 1</userinput>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fibonacci_number">Wikipedia</ulink> or
<ulink url="http://planetmath.org/FibonacciSequence">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/FibonacciNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FrobeniusNumber"/>FrobeniusNumber</term>
<listitem>
<synopsis>FrobeniusNumber (v,arg...)</synopsis>
<para>
Calculate the Frobenius number. That is calculate largest
number that cannot be given as a non-negative integer linear
combination of a given vector of non-negative integers.
The vector can be given as separate numbers or a single vector.
All the numbers given should have GCD of 1.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Coin_problem">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FrobeniusNumber.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GaloisMatrix"/>GaloisMatrix</term>
<listitem>
<synopsis>GaloisMatrix (combining_rule)</synopsis>
<para>Galois matrix given a linear combining rule (a_1*x_1+...+a_n*x_n=x_(n+1)).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GreedyAlgorithm"/>GreedyAlgorithm</term>
<listitem>
<synopsis>GreedyAlgorithm (n,v)</synopsis>
<para>
Find the vector <varname>c</varname> of non-negative integers
such that taking the dot product with <varname>v</varname> is
equal to n. If not possible returns <constant>null</constant>. <varname>v</varname>
should be given sorted in increasing order and should consist
of non-negative integers.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Greedy_algorithm">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/GreedyAlgorithm.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-HarmonicNumber"/>HarmonicNumber</term>
<listitem>
<synopsis>HarmonicNumber (n,r)</synopsis>
<para>Aliases: <function>HarmonicH</function></para>
<para>Harmonic Number, the <varname>n</varname>th harmonic number of order <varname>r</varname>.
That is, it is the sum of <userinput>1/k^r</userinput> for <varname>k</varname>
from 1 to n. Equivalent to <userinput>sum k = 1 to n do 1/k^r</userinput>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Harmonic_number">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Hofstadter"/>Hofstadter</term>
<listitem>
<synopsis>Hofstadter (n)</synopsis>
<para>Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Hofstadter_sequence">Wikipedia</ulink> for more information.
The sequence is <ulink url="https://oeis.org/A005185">A005185 in OEIS</ulink>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinearRecursiveSequence"/>LinearRecursiveSequence</term>
<listitem>
<synopsis>LinearRecursiveSequence (seed_values,combining_rule,n)</synopsis>
<para>Compute linear recursive sequence using Galois stepping.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Multinomial"/>Multinomial</term>
<listitem>
<synopsis>Multinomial (v,arg...)</synopsis>
<para>Calculate multinomial coefficients. Takes a vector of
<varname>k</varname>
non-negative integers and computes the multinomial coefficient.
This corresponds to the coefficient in the homogeneous polynomial
in <varname>k</varname> variables with the corresponding powers.
</para>
<para>
The formula for <userinput>Multinomial(a,b,c)</userinput>
can be written as:
<programlisting>(a+b+c)! / (a!b!c!)
</programlisting>
In other words, if we would have only two elements, then
<userinput>Multinomial(a,b)</userinput> is the same thing as
<userinput>Binomial(a+b,a)</userinput> or
<userinput>Binomial(a+b,b)</userinput>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Multinomial_theorem">Wikipedia</ulink>,
<ulink url="http://planetmath.org/MultinomialTheorem">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/MultinomialCoefficient.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NextCombination"/>NextCombination</term>
<listitem>
<synopsis>NextCombination (v,n)</synopsis>
<para>Get combination that would come after v in call to
combinations, first combination should be <userinput>[1:k]</userinput>. This
function is useful if you have many combinations to go through and you don't
want to waste memory to store them all.
</para>
<para>
For example with Combinations you would normally write a loop like:
<screen><userinput>for n in Combinations (4,6) do (
SomeFunction (n)
);</userinput>
</screen>
But with NextCombination you would write something like:
<screen><userinput>n:=[1:4];
do (
SomeFunction (n)
) while not IsNull(n:=NextCombination(n,6));</userinput>
</screen>
See also <link linkend="gel-function-Combinations">Combinations</link>.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Combination">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Pascal"/>Pascal</term>
<listitem>
<synopsis>Pascal (i)</synopsis>
<para>Get the Pascal's triangle as a matrix. This will return
an <varname>i</varname>+1 by <varname>i</varname>+1 lower diagonal
matrix that is the Pascal's triangle after <varname>i</varname>
iterations.</para>
<para>
See
<ulink url="http://planetmath.org/PascalsTriangle">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Permutations"/>Permutations</term>
<listitem>
<synopsis>Permutations (k,n)</synopsis>
<para>Get all permutations of <varname>k</varname> numbers from 1 to <varname>n</varname> as a vector of vectors.</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/Permutation.html">Mathworld</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Permutation">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RisingFactorial"/>RisingFactorial</term>
<listitem>
<synopsis>RisingFactorial (n,k)</synopsis>
<para>Aliases: <function>Pochhammer</function></para>
<para>(Pochhammer) Rising factorial: (n)_k = n(n+1)...(n+(k-1)).</para>
<para>
See
<ulink url="http://planetmath.org/RisingFactorial">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StirlingNumberFirst"/>StirlingNumberFirst</term>
<listitem>
<synopsis>StirlingNumberFirst (n,m)</synopsis>
<para>Aliases: <function>StirlingS1</function></para>
<para>Stirling number of the first kind.</para>
<para>
See
<ulink url="http://planetmath.org/StirlingNumbersOfTheFirstKind">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StirlingNumberSecond"/>StirlingNumberSecond</term>
<listitem>
<synopsis>StirlingNumberSecond (n,m)</synopsis>
<para>Aliases: <function>StirlingS2</function></para>
<para>Stirling number of the second kind.</para>
<para>
See
<ulink url="http://planetmath.org/StirlingNumbersSecondKind">Planetmath</ulink> or
<ulink url="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Subfactorial"/>Subfactorial</term>
<listitem>
<synopsis>Subfactorial (n)</synopsis>
<para>Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Triangular"/>Triangular</term>
<listitem>
<synopsis>Triangular (nth)</synopsis>
<para>Calculate the <varname>n</varname>th triangular number.</para>
<para>
See
<ulink url="http://planetmath.org/TriangularNumbers">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-nCr"/>nCr</term>
<listitem>
<synopsis>nCr (n,r)</synopsis>
<para>Aliases: <function>Binomial</function></para>
<para>Calculate combinations, that is, the binomial coefficient.
<varname>n</varname> can be any real number.</para>
<para>
See
<ulink url="http://planetmath.org/Choose">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-nPr"/>nPr</term>
<listitem>
<synopsis>nPr (n,r)</synopsis>
<para>Calculate the number of permutations of size
<varname>r</varname> of numbers from 1 to <varname>n</varname>.</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/Permutation.html">Mathworld</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Permutation">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-calculus">
<title>Calculus</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-CompositeSimpsonsRule"/>CompositeSimpsonsRule</term>
<listitem>
<synopsis>CompositeSimpsonsRule (f,a,b,n)</synopsis>
<para>Integration of f by Composite Simpson's Rule on the interval [a,b] with n subintervals with error of max(f'''')*h^4*(b-a)/180, note that n should be even.</para>
<para>
See
<ulink url="http://planetmath.org/SimpsonsRule">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-CompositeSimpsonsRuleTolerance"/>CompositeSimpsonsRuleTolerance</term>
<listitem>
<synopsis>CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)</synopsis>
<para>Integration of f by Composite Simpson's Rule on the interval [a,b] with the number of steps calculated by the fourth derivative bound and the desired tolerance.</para>
<para>
See
<ulink url="http://planetmath.org/SimpsonsRule">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Derivative"/>Derivative</term>
<listitem>
<synopsis>Derivative (f,x0)</synopsis>
<para>Attempt to calculate derivative by trying first symbolically and then numerically.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Derivative">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-EvenPeriodicExtension"/>EvenPeriodicExtension</term>
<listitem>
<synopsis>EvenPeriodicExtension (f,L)</synopsis>
<para>Return a function that is the even periodic extension of
<function>f</function> with half period <varname>L</varname>. That
is a function defined on the interval <userinput>[0,L]</userinput>
extended to be even on <userinput>[-L,L]</userinput> and then
extended to be periodic with period <userinput>2*L</userinput>.</para>
<para>
See also
<link linkend="gel-function-OddPeriodicExtension">OddPeriodicExtension</link>
and
<link linkend="gel-function-PeriodicExtension">PeriodicExtension</link>.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FourierSeriesFunction"/>FourierSeriesFunction</term>
<listitem>
<synopsis>FourierSeriesFunction (a,b,L)</synopsis>
<para>Return a function that is a Fourier series with the
coefficients given by the vectors <varname>a</varname> (sines) and
<varname>b</varname> (cosines). Note that <userinput>a@(1)</userinput> is
the constant coefficient! That is, <userinput>a@(n)</userinput> refers to
the term <userinput>cos(x*(n-1)*pi/L)</userinput>, while
<userinput>b@(n)</userinput> refers to the term
<userinput>sin(x*n*pi/L)</userinput>. Either <varname>a</varname>
or <varname>b</varname> can be <constant>null</constant>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierSeries.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InfiniteProduct"/>InfiniteProduct</term>
<listitem>
<synopsis>InfiniteProduct (func,start,inc)</synopsis>
<para>Try to calculate an infinite product for a single parameter function.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InfiniteProduct2"/>InfiniteProduct2</term>
<listitem>
<synopsis>InfiniteProduct2 (func,arg,start,inc)</synopsis>
<para>Try to calculate an infinite product for a double parameter function with func(arg,n).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InfiniteSum"/>InfiniteSum</term>
<listitem>
<synopsis>InfiniteSum (func,start,inc)</synopsis>
<para>Try to calculate an infinite sum for a single parameter function.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-InfiniteSum2"/>InfiniteSum2</term>
<listitem>
<synopsis>InfiniteSum2 (func,arg,start,inc)</synopsis>
<para>Try to calculate an infinite sum for a double parameter function with func(arg,n).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsContinuous"/>IsContinuous</term>
<listitem>
<synopsis>IsContinuous (f,x0)</synopsis>
<para>Try and see if a real-valued function is continuous at x0 by calculating the limit there.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsDifferentiable"/>IsDifferentiable</term>
<listitem>
<synopsis>IsDifferentiable (f,x0)</synopsis>
<para>Test for differentiability by approximating the left and right limits and comparing.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LeftLimit"/>LeftLimit</term>
<listitem>
<synopsis>LeftLimit (f,x0)</synopsis>
<para>Calculate the left limit of a real-valued function at x0.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Limit"/>Limit</term>
<listitem>
<synopsis>Limit (f,x0)</synopsis>
<para>Calculate the limit of a real-valued function at x0. Tries to calculate both left and right limits.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MidpointRule"/>MidpointRule</term>
<listitem>
<synopsis>MidpointRule (f,a,b,n)</synopsis>
<para>Integration by midpoint rule.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalDerivative"/>NumericalDerivative</term>
<listitem>
<synopsis>NumericalDerivative (f,x0)</synopsis>
<para>Aliases: <function>NDerivative</function></para>
<para>Attempt to calculate numerical derivative.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Derivative">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalFourierSeriesCoefficients"/>NumericalFourierSeriesCoefficients</term>
<listitem>
<synopsis>NumericalFourierSeriesCoefficients (f,L,N)</synopsis>
<para>Return a vector of vectors <userinput>[a,b]</userinput>
where <varname>a</varname> are the cosine coefficients and
<varname>b</varname> are the sine coefficients of
the Fourier series of
<function>f</function> with half-period <varname>L</varname> (that is defined
on <userinput>[-L,L]</userinput> and extended periodically) with coefficients
up to <varname>N</varname>th harmonic computed numerically. The coefficients are
computed by numerical integration using
<link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierSeries.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalFourierSeriesFunction"/>NumericalFourierSeriesFunction</term>
<listitem>
<synopsis>NumericalFourierSeriesFunction (f,L,N)</synopsis>
<para>Return a function that is the Fourier series of
<function>f</function> with half-period <varname>L</varname> (that is defined
on <userinput>[-L,L]</userinput> and extended periodically) with coefficients
up to <varname>N</varname>th harmonic computed numerically. This is the
trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
<link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierSeries.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalFourierCosineSeriesCoefficients"/>NumericalFourierCosineSeriesCoefficients</term>
<listitem>
<synopsis>NumericalFourierCosineSeriesCoefficients (f,L,N)</synopsis>
<para>Return a vector of coefficients of
the cosine Fourier series of
<function>f</function> with half-period <varname>L</varname>. That is,
we take <function>f</function> defined on <userinput>[0,L]</userinput>
take the even periodic extension and compute the Fourier series, which
only has cosine terms. The series is computed up to the
<varname>N</varname>th harmonic. The coefficients are
computed by numerical integration using
<link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.
Note that <userinput>a@(1)</userinput> is
the constant coefficient! That is, <userinput>a@(n)</userinput> refers to
the term <userinput>cos(x*(n-1)*pi/L)</userinput>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierCosineSeries.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalFourierCosineSeriesFunction"/>NumericalFourierCosineSeriesFunction</term>
<listitem>
<synopsis>NumericalFourierCosineSeriesFunction (f,L,N)</synopsis>
<para>Return a function that is the cosine Fourier series of
<function>f</function> with half-period <varname>L</varname>. That is,
we take <function>f</function> defined on <userinput>[0,L]</userinput>
take the even periodic extension and compute the Fourier series, which
only has cosine terms. The series is computed up to the
<varname>N</varname>th harmonic. The coefficients are
computed by numerical integration using
<link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierCosineSeries.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalFourierSineSeriesCoefficients"/>NumericalFourierSineSeriesCoefficients</term>
<listitem>
<synopsis>NumericalFourierSineSeriesCoefficients (f,L,N)</synopsis>
<para>Return a vector of coefficients of
the sine Fourier series of
<function>f</function> with half-period <varname>L</varname>. That is,
we take <function>f</function> defined on <userinput>[0,L]</userinput>
take the odd periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the
<varname>N</varname>th harmonic. The coefficients are
computed by numerical integration using
<link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierSineSeries.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalFourierSineSeriesFunction"/>NumericalFourierSineSeriesFunction</term>
<listitem>
<synopsis>NumericalFourierSineSeriesFunction (f,L,N)</synopsis>
<para>Return a function that is the sine Fourier series of
<function>f</function> with half-period <varname>L</varname>. That is,
we take <function>f</function> defined on <userinput>[0,L]</userinput>
take the odd periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the
<varname>N</varname>th harmonic. The coefficients are
computed by numerical integration using
<link linkend="gel-function-NumericalIntegral"><function>NumericalIntegral</function></link>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Fourier_series">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/FourierSineSeries.html">Mathworld</ulink> for more information.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalIntegral"/>NumericalIntegral</term>
<listitem>
<synopsis>NumericalIntegral (f,a,b)</synopsis>
<para>Integration by rule set in NumericalIntegralFunction of f from a to b using NumericalIntegralSteps steps.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalLeftDerivative"/>NumericalLeftDerivative</term>
<listitem>
<synopsis>NumericalLeftDerivative (f,x0)</synopsis>
<para>Attempt to calculate numerical left derivative.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalLimitAtInfinity"/>NumericalLimitAtInfinity</term>
<listitem>
<synopsis>NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</synopsis>
<para>Attempt to calculate the limit of f(step_fun(i)) as i goes from 1 to N.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NumericalRightDerivative"/>NumericalRightDerivative</term>
<listitem>
<synopsis>NumericalRightDerivative (f,x0)</synopsis>
<para>Attempt to calculate numerical right derivative.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OddPeriodicExtension"/>OddPeriodicExtension</term>
<listitem>
<synopsis>OddPeriodicExtension (f,L)</synopsis>
<para>Return a function that is the odd periodic extension of
<function>f</function> with half period <varname>L</varname>. That
is a function defined on the interval <userinput>[0,L]</userinput>
extended to be odd on <userinput>[-L,L]</userinput> and then
extended to be periodic with period <userinput>2*L</userinput>.</para>
<para>
See also
<link linkend="gel-function-EvenPeriodicExtension">EvenPeriodicExtension</link>
and
<link linkend="gel-function-PeriodicExtension">PeriodicExtension</link>.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OneSidedFivePointFormula"/>OneSidedFivePointFormula</term>
<listitem>
<synopsis>OneSidedFivePointFormula (f,x0,h)</synopsis>
<para>Compute one-sided derivative using five point formula.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-OneSidedThreePointFormula"/>OneSidedThreePointFormula</term>
<listitem>
<synopsis>OneSidedThreePointFormula (f,x0,h)</synopsis>
<para>Compute one-sided derivative using three-point formula.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PeriodicExtension"/>PeriodicExtension</term>
<listitem>
<synopsis>PeriodicExtension (f,a,b)</synopsis>
<para>Return a function that is the periodic extension of
<function>f</function> defined on the interval <userinput>[a,b]</userinput>
and has period <userinput>b-a</userinput>.</para>
<para>
See also
<link linkend="gel-function-OddPeriodicExtension">OddPeriodicExtension</link>
and
<link linkend="gel-function-EvenPeriodicExtension">EvenPeriodicExtension</link>.
</para>
<para>Version 1.0.7 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RightLimit"/>RightLimit</term>
<listitem>
<synopsis>RightLimit (f,x0)</synopsis>
<para>Calculate the right limit of a real-valued function at x0.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-TwoSidedFivePointFormula"/>TwoSidedFivePointFormula</term>
<listitem>
<synopsis>TwoSidedFivePointFormula (f,x0,h)</synopsis>
<para>Compute two-sided derivative using five-point formula.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-TwoSidedThreePointFormula"/>TwoSidedThreePointFormula</term>
<listitem>
<synopsis>TwoSidedThreePointFormula (f,x0,h)</synopsis>
<para>Compute two-sided derivative using three-point formula.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-functions">
<title>Functions</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-Argument"/>Argument</term>
<listitem>
<synopsis>Argument (z)</synopsis>
<para>Aliases: <function>Arg</function> <function>arg</function></para>
<para>argument (angle) of complex number.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BesselJ0"/>BesselJ0</term>
<listitem>
<synopsis>BesselJ0 (x)</synopsis>
<para>Bessel function of the first kind of order 0. Only implemented for real numbers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bessel_functions">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BesselJ1"/>BesselJ1</term>
<listitem>
<synopsis>BesselJ1 (x)</synopsis>
<para>Bessel function of the first kind of order 1. Only implemented for real numbers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bessel_functions">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BesselJn"/>BesselJn</term>
<listitem>
<synopsis>BesselJn (n,x)</synopsis>
<para>Bessel function of the first kind of order <varname>n</varname>. Only implemented for real numbers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bessel_functions">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BesselY0"/>BesselY0</term>
<listitem>
<synopsis>BesselY0 (x)</synopsis>
<para>Bessel function of the second kind of order 0. Only implemented for real numbers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bessel_functions">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BesselY1"/>BesselY1</term>
<listitem>
<synopsis>BesselY1 (x)</synopsis>
<para>Bessel function of the second kind of order 1. Only implemented for real numbers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bessel_functions">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-BesselYn"/>BesselYn</term>
<listitem>
<synopsis>BesselYn (n,x)</synopsis>
<para>Bessel function of the second kind of order <varname>n</varname>. Only implemented for real numbers.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Bessel_functions">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DirichletKernel"/>DirichletKernel</term>
<listitem>
<synopsis>DirichletKernel (n,t)</synopsis>
<para>Dirichlet kernel of order <varname>n</varname>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DiscreteDelta"/>DiscreteDelta</term>
<listitem>
<synopsis>DiscreteDelta (v)</synopsis>
<para>Returns 1 if and only if all elements are zero.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-ErrorFunction"/>ErrorFunction</term>
<listitem>
<synopsis>ErrorFunction (x)</synopsis>
<para>Aliases: <function>erf</function></para>
<para>The error function, 2/sqrt(pi) * int_0^x e^(-t^2) dt.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Error_function">Wikipedia</ulink> or
<ulink url="http://planetmath.org/ErrorFunction">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FejerKernel"/>FejerKernel</term>
<listitem>
<synopsis>FejerKernel (n,t)</synopsis>
<para>Fejer kernel of order <varname>n</varname> evaluated at
<varname>t</varname></para>
<para>
See
<ulink url="http://planetmath.org/FejerKernel">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GammaFunction"/>GammaFunction</term>
<listitem>
<synopsis>GammaFunction (x)</synopsis>
<para>Aliases: <function>Gamma</function></para>
<para>The Gamma function. Currently only implemented for real values.</para>
<para>
See
<ulink url="http://planetmath.org/GammaFunction">Planetmath</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Gamma_function">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-KroneckerDelta"/>KroneckerDelta</term>
<listitem>
<synopsis>KroneckerDelta (v)</synopsis>
<para>Returns 1 if and only if all elements are equal.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LambertW"/>LambertW</term>
<listitem>
<synopsis>LambertW (x)</synopsis>
<para>
The principal branch of Lambert W function computed for only
real values greater than or equal to <userinput>-1/e</userinput>.
That is, <function>LambertW</function> is the inverse of
the expression <userinput>x*e^x</userinput>. Even for
real <varname>x</varname> this expression is not one to one and
therefore has two branches over <userinput>[-1/e,0)</userinput>.
See <link linkend="gel-function-LambertWm1"><function>LambertWm1</function></link> for the other real branch.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Lambert_W_function">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LambertWm1"/>LambertWm1</term>
<listitem>
<synopsis>LambertWm1 (x)</synopsis>
<para>
The minus-one branch of Lambert W function computed for only
real values greater than or equal to <userinput>-1/e</userinput>
and less than 0.
That is, <function>LambertWm1</function> is the second
branch of the inverse of <userinput>x*e^x</userinput>.
See <link linkend="gel-function-LambertW"><function>LambertW</function></link> for the principal branch.
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Lambert_W_function">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MinimizeFunction"/>MinimizeFunction</term>
<listitem>
<synopsis>MinimizeFunction (func,x,incr)</synopsis>
<para>Find the first value where f(x)=0.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MoebiusDiskMapping"/>MoebiusDiskMapping</term>
<listitem>
<synopsis>MoebiusDiskMapping (a,z)</synopsis>
<para>Moebius mapping of the disk to itself mapping a to 0.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MobiusTransformation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MoebiusMapping"/>MoebiusMapping</term>
<listitem>
<synopsis>MoebiusMapping (z,z2,z3,z4)</synopsis>
<para>Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity respectively.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MobiusTransformation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MoebiusMappingInftyToInfty"/>MoebiusMappingInftyToInfty</term>
<listitem>
<synopsis>MoebiusMappingInftyToInfty (z,z2,z3)</synopsis>
<para>Moebius mapping using the cross ratio taking infinity to infinity and z2,z3 to 1 and 0 respectively.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MobiusTransformation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MoebiusMappingInftyToOne"/>MoebiusMappingInftyToOne</term>
<listitem>
<synopsis>MoebiusMappingInftyToOne (z,z3,z4)</synopsis>
<para>Moebius mapping using the cross ratio taking infinity to 1 and z3,z4 to 0 and infinity respectively.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MobiusTransformation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MoebiusMappingInftyToZero"/>MoebiusMappingInftyToZero</term>
<listitem>
<synopsis>MoebiusMappingInftyToZero (z,z2,z4)</synopsis>
<para>Moebius mapping using the cross ratio taking infinity to 0 and z2,z4 to 1 and infinity respectively.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation">Wikipedia</ulink> or
<ulink url="http://planetmath.org/MobiusTransformation">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PoissonKernel"/>PoissonKernel</term>
<listitem>
<synopsis>PoissonKernel (r,sigma)</synopsis>
<para>Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is 2pi).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PoissonKernelRadius"/>PoissonKernelRadius</term>
<listitem>
<synopsis>PoissonKernelRadius (r,sigma)</synopsis>
<para>Poisson kernel on D(0,R) (not normalized to 1).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RiemannZeta"/>RiemannZeta</term>
<listitem>
<synopsis>RiemannZeta (x)</synopsis>
<para>Aliases: <function>zeta</function></para>
<para>The Riemann zeta function. Currently only implemented for real values.</para>
<para>
See
<ulink url="http://planetmath.org/RiemannZetaFunction">Planetmath</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Riemann_zeta_function">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-UnitStep"/>UnitStep</term>
<listitem>
<synopsis>UnitStep (x)</synopsis>
<para>The unit step function is 0 for x<0, 1 otherwise. This is the integral of the Dirac Delta function. Also called the Heaviside function.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Unit_step">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-cis"/>cis</term>
<listitem>
<synopsis>cis (x)</synopsis>
<para>Функция <function>cis</function>, то же самое, что <userinput>cos(x)+1i*sin(x)</userinput></para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-deg2rad"/>deg2rad</term>
<listitem>
<synopsis>deg2rad (x)</synopsis>
<para>Преобразует градусы в радианы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-rad2deg"/>rad2deg</term>
<listitem>
<synopsis>rad2deg (x)</synopsis>
<para>Преобразует радианы в градусы.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-sinc"/>sinc</term>
<listitem>
<synopsis>sinc (x)</synopsis>
<para>Calculates the unnormalized sinc function, that is
<userinput>sin(x)/x</userinput>.
If you want the normalized function call <userinput>sinc(pi*x)</userinput>.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Sinc">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-equation-solving">
<title>Решение уравнений</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-CubicFormula"/>CubicFormula</term>
<listitem>
<synopsis>CubicFormula (p)</synopsis>
<para>
Compute roots of a cubic (degree 3) polynomial using the
cubic formula. The polynomial should be given as a
vector of coefficients. That is
<userinput>4*x^3 + 2*x + 1</userinput> corresponds to the vector
<userinput>[1,2,0,4]</userinput>.
Returns a column vector of the three solutions. The first solution is always
the real one as a cubic always has one real solution.
</para>
<para>
See
<ulink url="http://planetmath.org/CubicFormula">Planetmath</ulink>,
<ulink url="http://mathworld.wolfram.com/CubicFormula.html">Mathworld</ulink>, or
<ulink url="https://en.wikipedia.org/wiki/Cubic_equation">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-EulersMethod"/>EulersMethod</term>
<listitem>
<synopsis>EulersMethod (f,x0,y0,x1,n)</synopsis>
<para>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <varname>x0</varname>, <varname>y0</varname> going to
<varname>x1</varname> with <varname>n</varname> increments,
returns <varname>y</varname> at <varname>x1</varname>.
Unless you explicitly want to use Euler's method, you should really
think about using
<link linkend="gel-function-RungeKutta">RungeKutta</link>
for solving ODE.
</para>
<para>
Systems can be solved by just having <varname>y</varname> be a
(column) vector everywhere. That is, <varname>y0</varname> can
be a vector in which case <varname>f</varname> should take a number
<varname>x</varname> and a vector of the same size for the second
argument and should return a vector of the same size.
</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/EulerForwardMethod.html">Mathworld</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Eulers_method">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-EulersMethodFull"/>EulersMethodFull</term>
<listitem>
<synopsis>EulersMethodFull (f,x0,y0,x1,n)</synopsis>
<para>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <varname>x0</varname>, <varname>y0</varname> going to
<varname>x1</varname> with <varname>n</varname> increments,
returns an <userinput>n+1</userinput> by 2 matrix with the
<varname>x</varname> and <varname>y</varname> values.
Unless you explicitly want to use Euler's method, you should really
think about using
<link linkend="gel-function-RungeKuttaFull">RungeKuttaFull</link>
for solving ODE.
Suitable
for plugging into
<link linkend="gel-function-LinePlotDrawLine">LinePlotDrawLine</link> or
<link linkend="gel-function-LinePlotDrawPoints">LinePlotDrawPoints</link>.
</para>
<para>
Example:
<screen><prompt>genius></prompt> <userinput>LinePlotClear();</userinput>
<prompt>genius></prompt> <userinput>line = EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential growth");</userinput>
</screen>
</para>
<para>
Systems can be solved by just having <varname>y</varname> be a
(column) vector everywhere. That is, <varname>y0</varname> can
be a vector in which case <varname>f</varname> should take a number
<varname>x</varname> and a vector of the same size for the second
argument and should return a vector of the same size.
</para>
<para>
The output for a system is still a n by 2 matrix with the second
entry being a vector. If you wish to plot the line, make sure to use row vectors, and then flatten the matrix with
<link linkend="gel-function-ExpandMatrix">ExpandMatrix</link>,
and pick out the right columns. Example:
<screen><prompt>genius></prompt> <userinput>LinePlotClear();</userinput>
<prompt>genius></prompt> <userinput>lines = EulersMethodFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,500);</userinput>
<prompt>genius></prompt> <userinput>lines = ExpandMatrix(lines);</userinput>
<prompt>genius></prompt> <userinput>firstline = lines@(,[1,2]);</userinput>
<prompt>genius></prompt> <userinput>secondline = lines@(,[1,3]);</userinput>
<prompt>genius></prompt> <userinput>LinePlotWindow = [0,10,-2,2];</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine(firstline,"color","blue","legend","First");</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</userinput>
</screen>
</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/EulerForwardMethod.html">Mathworld</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Eulers_method">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FindRootBisection"/>FindRootBisection</term>
<listitem>
<synopsis>FindRootBisection (f,a,b,TOL,N)</synopsis>
<para>Find root of a function using the bisection method.
<varname>a</varname> and <varname>b</varname> are the initial guess interval,
<userinput>f(a)</userinput> and <userinput>f(b)</userinput> should have opposite signs.
<varname>TOL</varname> is the desired tolerance and
<varname>N</varname> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <userinput>[success,value,iteration]</userinput>, where <varname>success</varname> is a boolean indicating success, <varname>value</varname> is the last value computed, and <varname>iteration</varname> is the number of iterations done.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FindRootFalsePosition"/>FindRootFalsePosition</term>
<listitem>
<synopsis>FindRootFalsePosition (f,a,b,TOL,N)</synopsis>
<para>Find root of a function using the method of false position.
<varname>a</varname> and <varname>b</varname> are the initial guess interval,
<userinput>f(a)</userinput> and <userinput>f(b)</userinput> should have opposite signs.
<varname>TOL</varname> is the desired tolerance and
<varname>N</varname> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <userinput>[success,value,iteration]</userinput>, where <varname>success</varname> is a boolean indicating success, <varname>value</varname> is the last value computed, and <varname>iteration</varname> is the number of iterations done.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FindRootMullersMethod"/>FindRootMullersMethod</term>
<listitem>
<synopsis>FindRootMullersMethod (f,x0,x1,x2,TOL,N)</synopsis>
<para>Find root of a function using the Muller's method.
<varname>TOL</varname> is the desired tolerance and
<varname>N</varname> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <userinput>[success,value,iteration]</userinput>, where <varname>success</varname> is a boolean indicating success, <varname>value</varname> is the last value computed, and <varname>iteration</varname> is the number of iterations done.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-FindRootSecant"/>FindRootSecant</term>
<listitem>
<synopsis>FindRootSecant (f,a,b,TOL,N)</synopsis>
<para>Find root of a function using the secant method.
<varname>a</varname> and <varname>b</varname> are the initial guess interval,
<userinput>f(a)</userinput> and <userinput>f(b)</userinput> should have opposite signs.
<varname>TOL</varname> is the desired tolerance and
<varname>N</varname> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <userinput>[success,value,iteration]</userinput>, where <varname>success</varname> is a boolean indicating success, <varname>value</varname> is the last value computed, and <varname>iteration</varname> is the number of iterations done.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-HalleysMethod"/>HalleysMethod</term>
<listitem>
<synopsis>HalleysMethod (f,df,ddf,guess,epsilon,maxn)</synopsis>
<para>Find zeros using Halley's method. <varname>f</varname> is
the function, <varname>df</varname> is the derivative of
<varname>f</varname>, and <varname>ddf</varname> is the second derivative of
<varname>f</varname>. <varname>guess</varname> is the initial
guess. The function returns after two successive values are
within <varname>epsilon</varname> of each other, or after <varname>maxn</varname> tries, in which case the function returns <constant>null</constant> indicating failure.
</para>
<para>
See also <link linkend="gel-function-NewtonsMethod"><function>NewtonsMethod</function></link> and <link linkend="gel-function-SymbolicDerivative"><function>SymbolicDerivative</function></link>.
</para>
<para>
Example to find the square root of 10:
<screen><prompt>genius></prompt> <userinput>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</userinput>
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Halley%27s_method">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NewtonsMethod"/>NewtonsMethod</term>
<listitem>
<synopsis>NewtonsMethod (f,df,guess,epsilon,maxn)</synopsis>
<para>Find zeros using Newton's method. <varname>f</varname> is
the function and <varname>df</varname> is the derivative of
<varname>f</varname>. <varname>guess</varname> is the initial
guess. The function returns after two successive values are
within <varname>epsilon</varname> of each other, or after <varname>maxn</varname> tries, in which case the function returns <constant>null</constant> indicating failure.
</para>
<para>
See also <link linkend="gel-function-NewtonsMethodPoly"><function>NewtonsMethodPoly</function></link> and <link linkend="gel-function-SymbolicDerivative"><function>SymbolicDerivative</function></link>.
</para>
<para>
Example to find the square root of 10:
<screen><prompt>genius></prompt> <userinput>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</userinput>
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Newtons_method">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PolynomialRoots"/>PolynomialRoots</term>
<listitem>
<synopsis>PolynomialRoots (p)</synopsis>
<para>
Compute roots of a polynomial (degrees 1 through 4)
using one of the formulas for such polynomials.
The polynomial should be given as a
vector of coefficients. That is
<userinput>4*x^3 + 2*x + 1</userinput> corresponds to the vector
<userinput>[1,2,0,4]</userinput>.
Returns a column vector of the solutions.
</para>
<para>
The function calls
<link linkend="gel-function-QuadraticFormula">QuadraticFormula</link>,
<link linkend="gel-function-CubicFormula">CubicFormula</link>, and
<link linkend="gel-function-QuarticFormula">QuarticFormula</link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-QuadraticFormula"/>QuadraticFormula</term>
<listitem>
<synopsis>QuadraticFormula (p)</synopsis>
<para>
Compute roots of a quadratic (degree 2) polynomial using the
quadratic formula. The polynomial should be given as a
vector of coefficients. That is
<userinput>3*x^2 + 2*x + 1</userinput> corresponds to the vector
<userinput>[1,2,3]</userinput>.
Returns a column vector of the two solutions.
</para>
<para>
See
<ulink url="http://planetmath.org/QuadraticFormula">Planetmath</ulink>, or
<ulink url="http://mathworld.wolfram.com/QuadraticFormula.html">Mathworld</ulink>, or
<ulink url="https://en.wikipedia.org/wiki/Quadratic_formula">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-QuarticFormula"/>QuarticFormula</term>
<listitem>
<synopsis>QuarticFormula (p)</synopsis>
<para>
Compute roots of a quartic (degree 4) polynomial using the
quartic formula. The polynomial should be given as a
vector of coefficients. That is
<userinput>5*x^4 + 2*x + 1</userinput> corresponds to the vector
<userinput>[1,2,0,0,5]</userinput>.
Returns a column vector of the four solutions.
</para>
<para>
See
<ulink url="http://planetmath.org/QuarticFormula">Planetmath</ulink>,
<ulink url="http://mathworld.wolfram.com/QuarticEquation.html">Mathworld</ulink>, or
<ulink url="https://en.wikipedia.org/wiki/Quartic_equation">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RungeKutta"/>RungeKutta</term>
<listitem>
<synopsis>RungeKutta (f,x0,y0,x1,n)</synopsis>
<para>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <varname>x0</varname>, <varname>y0</varname>
going to <varname>x1</varname> with <varname>n</varname>
increments, returns <varname>y</varname> at <varname>x1</varname>.
</para>
<para>
Systems can be solved by just having <varname>y</varname> be a
(column) vector everywhere. That is, <varname>y0</varname> can
be a vector in which case <varname>f</varname> should take a number
<varname>x</varname> and a vector of the same size for the second
argument and should return a vector of the same size.
</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/Runge-KuttaMethod.html">Mathworld</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Runge-Kutta_methods">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RungeKuttaFull"/>RungeKuttaFull</term>
<listitem>
<synopsis>RungeKuttaFull (f,x0,y0,x1,n)</synopsis>
<para>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <varname>x0</varname>, <varname>y0</varname>
going to <varname>x1</varname> with <varname>n</varname>
increments,
returns an <userinput>n+1</userinput> by 2 matrix with the
<varname>x</varname> and <varname>y</varname> values. Suitable
for plugging into
<link linkend="gel-function-LinePlotDrawLine">LinePlotDrawLine</link> or
<link linkend="gel-function-LinePlotDrawPoints">LinePlotDrawPoints</link>.
</para>
<para>
Example:
<screen><prompt>genius></prompt> <userinput>LinePlotClear();</userinput>
<prompt>genius></prompt> <userinput>line = RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential growth");</userinput>
</screen>
</para>
<para>
Systems can be solved by just having <varname>y</varname> be a
(column) vector everywhere. That is, <varname>y0</varname> can
be a vector in which case <varname>f</varname> should take a number
<varname>x</varname> and a vector of the same size for the second
argument and should return a vector of the same size.
</para>
<para>
The output for a system is still a n by 2 matrix with the second
entry being a vector. If you wish to plot the line, make sure to use row vectors, and then flatten the matrix with
<link linkend="gel-function-ExpandMatrix">ExpandMatrix</link>,
and pick out the right columns. Example:
<screen><prompt>genius></prompt> <userinput>LinePlotClear();</userinput>
<prompt>genius></prompt> <userinput>lines = RungeKuttaFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,100);</userinput>
<prompt>genius></prompt> <userinput>lines = ExpandMatrix(lines);</userinput>
<prompt>genius></prompt> <userinput>firstline = lines@(,[1,2]);</userinput>
<prompt>genius></prompt> <userinput>secondline = lines@(,[1,3]);</userinput>
<prompt>genius></prompt> <userinput>LinePlotWindow = [0,10,-2,2];</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine(firstline,"color","blue","legend","First");</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</userinput>
</screen>
</para>
<para>
See
<ulink url="http://mathworld.wolfram.com/Runge-KuttaMethod.html">Mathworld</ulink> or
<ulink url="https://en.wikipedia.org/wiki/Runge-Kutta_methods">Wikipedia</ulink> for more information.
</para>
<para>Version 1.0.10 onwards.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-statistics">
<title>Статистика</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-Average"/>Average</term>
<listitem>
<synopsis>Average (m)</synopsis>
<para>Псевдонимы: <function>average</function><function>Mean</function><function>mean</function></para>
<para>Calculate average (the arithmetic mean) of an entire matrix.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Mean">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/ArithmeticMean.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GaussDistribution"/>GaussDistribution</term>
<listitem>
<synopsis>GaussDistribution (x,sigma)</synopsis>
<para>Integral of the GaussFunction from 0 to <varname>x</varname> (area under the normal curve).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Normal_distribution">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/NormalDistribution.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-GaussFunction"/>GaussFunction</term>
<listitem>
<synopsis>GaussFunction (x,sigma)</synopsis>
<para>The normalized Gauss distribution function (the normal curve).</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Normal_distribution">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/NormalDistribution.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Median"/>Median</term>
<listitem>
<synopsis>Median (m)</synopsis>
<para>Aliases: <function>median</function></para>
<para>Calculate median of an entire matrix.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Median">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/StatisticalMedian.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PopulationStandardDeviation"/>PopulationStandardDeviation</term>
<listitem>
<synopsis>PopulationStandardDeviation (m)</synopsis>
<para>Aliases: <function>stdevp</function></para>
<para>Calculate the population standard deviation of a whole matrix.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowAverage"/>RowAverage</term>
<listitem>
<synopsis>RowAverage (m)</synopsis>
<para>Aliases: <function>RowMean</function></para>
<para>Calculate average of each row in a matrix. That is, compute the
arithmetic mean.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Mean">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/ArithmeticMean.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowMedian"/>RowMedian</term>
<listitem>
<synopsis>RowMedian (m)</synopsis>
<para>Calculate median of each row in a matrix and return a column
vector of the medians.</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Median">Wikipedia</ulink> or
<ulink url="http://mathworld.wolfram.com/StatisticalMedian.html">Mathworld</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowPopulationStandardDeviation"/>RowPopulationStandardDeviation</term>
<listitem>
<synopsis>RowPopulationStandardDeviation (m)</synopsis>
<para>Aliases: <function>rowstdevp</function></para>
<para>Calculate the population standard deviations of rows of a matrix and return a vertical vector.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-RowStandardDeviation"/>RowStandardDeviation</term>
<listitem>
<synopsis>RowStandardDeviation (m)</synopsis>
<para>Aliases: <function>rowstdev</function></para>
<para>Calculate the standard deviations of rows of a matrix and return a vertical vector.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StandardDeviation"/>StandardDeviation</term>
<listitem>
<synopsis>StandardDeviation (m)</synopsis>
<para>Aliases: <function>stdev</function></para>
<para>Calculate the standard deviation of a whole matrix.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-polynomials">
<title>Многочлены</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-AddPoly"/>AddPoly</term>
<listitem>
<synopsis>AddPoly (p1,p2)</synopsis>
<para>Складывает два многочлена (в виде векторов).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-DividePoly"/>DividePoly</term>
<listitem>
<synopsis>DividePoly (p,q,&r)</synopsis>
<para>Divide two polynomials (as vectors) using long division.
Returns the quotient
of the two polynomials. The optional argument <varname>r</varname>
is used to return the remainder. The remainder will have lower
degree than <varname>q</varname>.</para>
<para>
See
<ulink url="http://planetmath.org/PolynomialLongDivision">Planetmath</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsPoly"/>IsPoly</term>
<listitem>
<synopsis>IsPoly (p)</synopsis>
<para>Проверяет, можно ли использовать вектор в качестве многочлена.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MultiplyPoly"/>MultiplyPoly</term>
<listitem>
<synopsis>MultiplyPoly (p1,p2)</synopsis>
<para>Умножает два многочлена (в виде векторов).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-NewtonsMethodPoly"/>NewtonsMethodPoly</term>
<listitem>
<synopsis>NewtonsMethodPoly (poly,guess,epsilon,maxn)</synopsis>
<para>Find a root of a polynomial using Newton's method. <varname>poly</varname> is
the polynomial as a vector and <varname>guess</varname> is the initial
guess. The function returns after two successive values are
within <varname>epsilon</varname> of each other, or after <varname>maxn</varname> tries, in which case the function returns <constant>null</constant> indicating failure.
</para>
<para>
See also <link linkend="gel-function-NewtonsMethod"><function>NewtonsMethod</function></link>.
</para>
<para>
Example to find the square root of 10:
<screen><prompt>genius></prompt> <userinput>NewtonsMethodPoly([-10,0,1],3,10^-10,100)</userinput>
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Newtons_method">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Poly2ndDerivative"/>Poly2ndDerivative</term>
<listitem>
<synopsis>Poly2ndDerivative (p)</synopsis>
<para>Находит вторую производную многочлена (как вектора).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PolyDerivative"/>PolyDerivative</term>
<listitem>
<synopsis>PolyDerivative (p)</synopsis>
<para>Находит производную многочлена (как вектора).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PolyToFunction"/>PolyToFunction</term>
<listitem>
<synopsis>PolyToFunction (p)</synopsis>
<para>Make function out of a polynomial (as vector).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PolyToString"/>PolyToString</term>
<listitem>
<synopsis>PolyToString (p,var...)</synopsis>
<para>Make string out of a polynomial (as vector).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SubtractPoly"/>SubtractPoly</term>
<listitem>
<synopsis>SubtractPoly (p1,p2)</synopsis>
<para>Subtract two polynomials (as vectors).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-TrimPoly"/>TrimPoly</term>
<listitem>
<synopsis>TrimPoly (p)</synopsis>
<para>Trim zeros from a polynomial (as vector).</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-set-theory">
<title>Теория множеств</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-Intersection"/>Intersection</term>
<listitem>
<synopsis>Intersection (X,Y)</synopsis>
<para>Возвращает пересечение множеств X и Y (X и Y — векторы, изображающие множества).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsIn"/>IsIn</term>
<listitem>
<synopsis>IsIn (x,X)</synopsis>
<para>Возвращает <constant>true</constant>, если элемент x присуствует в множестве X (где X — вектор, изображающий множество).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-IsSubset"/>IsSubset</term>
<listitem>
<synopsis>IsSubset (X, Y)</synopsis>
<para>Возвращает <constant>true</constant>, если X является подмножеством Y (X и Y — векторы, изображающие множество).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MakeSet"/>MakeSet</term>
<listitem>
<synopsis>MakeSet (X)</synopsis>
<para>Returns a vector where every element of X appears only once.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SetMinus"/>SetMinus</term>
<listitem>
<synopsis>SetMinus (X,Y)</synopsis>
<para>Returns a set theoretic difference X-Y (X and Y are vectors pretending to be sets).</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-Union"/>Union</term>
<listitem>
<synopsis>Union (X,Y)</synopsis>
<para>Возвращает объединение множеств X и Y (X и Y — векторы, изображающие множества).</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-commutative-algebra">
<title>Commutative Algebra</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-MacaulayBound"/>MacaulayBound</term>
<listitem>
<synopsis>MacaulayBound (c,d)</synopsis>
<para>For a Hilbert function that is c for degree d, given the Macaulay bound for the Hilbert function of degree d+1 (The c^<d> operator from Green's proof).</para>
<para>Version 1.0.15 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MacaulayLowerOperator"/>MacaulayLowerOperator</term>
<listitem>
<synopsis>MacaulayLowerOperator (c,d)</synopsis>
<para>The c_<d> operator from Green's proof of Macaulay's Theorem.</para>
<para>Version 1.0.15 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-MacaulayRep"/>MacaulayRep</term>
<listitem>
<synopsis>MacaulayRep (c,d)</synopsis>
<para>Return the dth Macaulay representation of a positive integer c.</para>
<para>Version 1.0.15 onwards.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-miscellaneous">
<title>Прочие</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-ASCIIToString"/>ASCIIToString</term>
<listitem>
<synopsis>ASCIIToString (vec)</synopsis>
<para>Convert a vector of ASCII values to a string.
See also
<link linkend="gel-function-StringToASCII"><function>StringToASCII</function></link>.
</para>
<para>
Example:
<screen><prompt>genius></prompt> <userinput>ASCIIToString([97,98,99])</userinput>
= "abc"
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/ASCII">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-AlphabetToString"/>AlphabetToString</term>
<listitem>
<synopsis>AlphabetToString (vec,alphabet)</synopsis>
<para>Convert a vector of 0-based alphabet values (positions in the alphabet string) to a string. A <constant>null</constant> vector results in an empty string.
See also
<link linkend="gel-function-StringToAlphabet"><function>StringToAlphabet</function></link>.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>AlphabetToString([1,2,3,0,0],"abcd")</userinput>
= "bcdaa"
<prompt>genius></prompt> <userinput>AlphabetToString(null,"abcd")</userinput>
= ""
</screen>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StringToASCII"/>StringToASCII</term>
<listitem>
<synopsis>StringToASCII (str)</synopsis>
<para>Convert a string to a (row) vector of ASCII values.
See also
<link linkend="gel-function-ASCIIToString"><function>ASCIIToString</function></link>.
</para>
<para>
Example:
<screen><prompt>genius></prompt> <userinput>StringToASCII("abc")</userinput>
= [97, 98, 99]
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/ASCII">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-StringToAlphabet"/>StringToAlphabet</term>
<listitem>
<synopsis>StringToAlphabet (str,alphabet)</synopsis>
<para>Convert a string to a (row) vector of 0-based alphabet values
(positions in the alphabet string), -1's for unknown letters.
An empty string results in a <constant>null</constant>.
See also
<link linkend="gel-function-AlphabetToString"><function>AlphabetToString</function></link>.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>StringToAlphabet("cca","abcd")</userinput>
= [2, 2, 0]
<prompt>genius></prompt> <userinput>StringToAlphabet("ccag","abcd")</userinput>
= [2, 2, 0, -1]
</screen>
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-symbolic">
<title>Symbolic Operations</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-SymbolicDerivative"/>SymbolicDerivative</term>
<listitem>
<synopsis>SymbolicDerivative (f)</synopsis>
<para>Attempt to symbolically differentiate the function f, where f is a function of one variable.</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SymbolicDerivative(sin)</userinput>
= (`(x)=cos(x))
<prompt>genius></prompt> <userinput>SymbolicDerivative(`(x)=7*x^2)</userinput>
= (`(x)=(7*(2*x)))
</screen>
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Derivative">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SymbolicDerivativeTry"/>SymbolicDerivativeTry</term>
<listitem>
<synopsis>SymbolicDerivativeTry (f)</synopsis>
<para>Attempt to symbolically differentiate the function f, where f is a function of one variable, returns <constant>null</constant> if unsuccessful but is silent.
(See <link linkend="gel-function-SymbolicDerivative"><function>SymbolicDerivative</function></link>)
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Derivative">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SymbolicNthDerivative"/>SymbolicNthDerivative</term>
<listitem>
<synopsis>SymbolicNthDerivative (f,n)</synopsis>
<para>Attempt to symbolically differentiate a function n times.
(See <link linkend="gel-function-SymbolicDerivative"><function>SymbolicDerivative</function></link>)
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Derivative">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SymbolicNthDerivativeTry"/>SymbolicNthDerivativeTry</term>
<listitem>
<synopsis>SymbolicNthDerivativeTry (f,n)</synopsis>
<para>Attempt to symbolically differentiate a function n times quietly and return <constant>null</constant> on failure
(See <link linkend="gel-function-SymbolicNthDerivative"><function>SymbolicNthDerivative</function></link>)
</para>
<para>
See
<ulink url="https://en.wikipedia.org/wiki/Derivative">Wikipedia</ulink> for more information.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SymbolicTaylorApproximationFunction"/>SymbolicTaylorApproximationFunction</term>
<listitem>
<synopsis>SymbolicTaylorApproximationFunction (f,x0,n)</synopsis>
<para>Attempt to construct the Taylor approximation function around x0 to the nth degree.
(See <link linkend="gel-function-SymbolicDerivative"><function>SymbolicDerivative</function></link>)
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-gel-function-list-plotting">
<title>Построение графиков</title>
<variablelist>
<varlistentry>
<term><anchor id="gel-function-ExportPlot"/>ExportPlot</term>
<listitem>
<synopsis>ExportPlot (file,type)</synopsis>
<synopsis>ExportPlot (file)</synopsis>
<para>
Export the contents of the plotting window to a file.
The type is a string that specifies the file type to
use, "png", "eps", or "ps". If the type is not
specified, then it is taken to be the extension, in
which case the extension must be ".png", ".eps", or ".ps".
</para>
<para>
Note that files are overwritten without asking.
</para>
<para>
On successful export, true is returned. Otherwise
error is printed and exception is raised.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>ExportPlot("file.png")</userinput>
<prompt>genius></prompt> <userinput>ExportPlot("/directory/file","eps")</userinput>
</screen>
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlot"/>LinePlot</term>
<listitem>
<synopsis>LinePlot (func1,func2,func3,...)</synopsis>
<synopsis>LinePlot (func1,func2,func3,x1,x2)</synopsis>
<synopsis>LinePlot (func1,func2,func3,x1,x2,y1,y2)</synopsis>
<synopsis>LinePlot (func1,func2,func3,[x1,x2])</synopsis>
<synopsis>LinePlot (func1,func2,func3,[x1,x2,y1,y2])</synopsis>
<para>
Plot a function (or several functions) with a line.
First (up to 10) arguments are functions, then optionally
you can specify the limits of the plotting window as
<varname>x1</varname>, <varname>x2</varname>,
<varname>y1</varname>, <varname>y2</varname>. If limits are not
specified, then the currently set limits apply
(See <link linkend="gel-function-LinePlotWindow"><function>LinePlotWindow</function></link>)
If the y limits are not specified, then the functions are computed and then the maxima and minima
are used.
</para>
<para>
The parameter
<link linkend="gel-function-LinePlotDrawLegends"><function>LinePlotDrawLegends</function></link>
controls the drawing of the legend.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>LinePlot(sin,cos)</userinput>
<prompt>genius></prompt> <userinput>LinePlot(`(x)=x^2,-1,1,0,1)</userinput>
</screen>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotClear"/>LinePlotClear</term>
<listitem>
<synopsis>LinePlotClear ()</synopsis>
<para>
Show the line plot window and clear out functions and any other
lines that were drawn.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotCParametric"/>LinePlotCParametric</term>
<listitem>
<synopsis>LinePlotCParametric (func,...)</synopsis>
<synopsis>LinePlotCParametric (func,t1,t2,tinc)</synopsis>
<synopsis>LinePlotCParametric (func,t1,t2,tinc,x1,x2,y1,y2)</synopsis>
<para>
Plot a parametric complex valued function with a line. First comes
the function that returns <computeroutput>x+iy</computeroutput>,
then optionally the <varname>t</varname> limits as <userinput>t1,t2,tinc</userinput>, then
optionally the limits as <userinput>x1,x2,y1,y2</userinput>.
</para>
<para>
If limits are not
specified, then the currently set limits apply
(See <link linkend="gel-function-LinePlotWindow"><function>LinePlotWindow</function></link>).
If instead the string "fit" is given for the x and y limits, then the limits are the maximum extent of
the graph
</para>
<para>
The parameter
<link linkend="gel-function-LinePlotDrawLegends"><function>LinePlotDrawLegends</function></link>
controls the drawing of the legend.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotDrawLine"/>LinePlotDrawLine</term>
<listitem>
<synopsis>LinePlotDrawLine (x1,y1,x2,y2,...)</synopsis>
<synopsis>LinePlotDrawLine (v,...)</synopsis>
<para>
Draw a line from <varname>x1</varname>,<varname>y1</varname> to
<varname>x2</varname>,<varname>y2</varname>.
<varname>x1</varname>,<varname>y1</varname>,
<varname>x2</varname>,<varname>y2</varname> can be replaced by an
<varname>n</varname> by 2 matrix for a longer polyline.
Alternatively the vector <varname>v</varname> may be a column vector of complex numbers,
that is an <varname>n</varname> by 1 matrix and each complex number is then
considered a point in the plane.
</para>
<para>
Extra parameters can be added to specify line color, thickness,
arrows, the plotting window, or legend.
You can do this by adding an argument string <userinput>"color"</userinput>,
<userinput>"thickness"</userinput>,
<userinput>"window"</userinput>,
<userinput>"arrow"</userinput>, or <userinput>"legend"</userinput>, and after it specify
the color, the thickness, the window
as 4-vector, type of arrow, or the legend. (Arrow and window are from version 1.0.6 onwards.)
</para>
<para>
If the line is to be treated as a filled polygon, filled with the given color, you
can specify the argument <userinput>"filled"</userinput>. Since version 1.0.22 onwards.
</para>
<para>
The color should be either a string indicating the common English word for the color
that GTK will recognize such as
<userinput>"red"</userinput>, <userinput>"blue"</userinput>, <userinput>"yellow"</userinput>, etc...
Alternatively the color can be specified in RGB format as
<userinput>"#rgb"</userinput>, <userinput>"#rrggbb"</userinput>, or
<userinput>"#rrrrggggbbbb"</userinput>, where the r, g, or b are hex digits of the red, green, and blue
components of the color. Finally, since version 1.0.18, the color
can also be specified as a real vector specifying the red green and
blue components where the components are between 0 and 1, e.g. <userinput>[1.0,0.5,0.1]</userinput>.
</para>
<para>
The window should be given as usual as <userinput>[x1,x2,y1,y2]</userinput>, or
alternatively can be given as a string
<userinput>"fit"</userinput> in which case,
the x range will be set precisely and the y range will be set with
five percent borders around the line.
</para>
<para>
Arrow specification should be
<userinput>"origin"</userinput>,
<userinput>"end"</userinput>,
<userinput>"both"</userinput>, or
<userinput>"none"</userinput>.
</para>
<para>
Finally, legend should be a string that can be used as the legend in the
graph. That is, if legends are being printed.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine([0,0;1,-1;-1,-1])</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine([0,0;1,1],"arrow","end")</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The Solution")</userinput>
<prompt>genius></prompt> <userinput>for r=0.0 to 1.0 by 0.1 do LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</userinput>
</screen>
</para>
<para>
Unlike many other functions that do not care if they take a
column or a row vector, if specifying points as a vector of
complex values, due to possible ambiguities, it must always
be given as a column vector.
</para>
<para>
Specifying <varname>v</varname> as a column vector of complex numbers is
implemented from version 1.0.22 and onwards.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotDrawPoints"/>LinePlotDrawPoints</term>
<listitem>
<synopsis>LinePlotDrawPoints (x,y,...)</synopsis>
<synopsis>LinePlotDrawPoints (v,...)</synopsis>
<para>
Draw a point at <varname>x</varname>,<varname>y</varname>.
The input can be an <varname>n</varname> by 2 matrix
for <varname>n</varname> different points. This function has essentially the same
input as <link linkend="gel-function-LinePlotDrawLine">LinePlotDrawLine</link>.
Alternatively the vector <varname>v</varname> may be a column vector of complex numbers,
that is an <varname>n</varname> by 1 matrix and each complex number is then
considered a point in the plane.
</para>
<para>
Extra parameters can be added to specify color, thickness,
the plotting window, or legend.
You can do this by adding an argument string <userinput>"color"</userinput>,
<userinput>"thickness"</userinput>,
<userinput>"window"</userinput>,
or <userinput>"legend"</userinput>, and after it specify
the color, the thickness, the window
as 4-vector, or the legend.
</para>
<para>
The color should be either a string indicating the common English word for the color
that GTK will recognize such as
<userinput>"red"</userinput>, <userinput>"blue"</userinput>, <userinput>"yellow"</userinput>, etc...
Alternatively the color can be specified in RGB format as
<userinput>"#rgb"</userinput>, <userinput>"#rrggbb"</userinput>, or
<userinput>"#rrrrggggbbbb"</userinput>, where the r, g, or b are hex digits of the red, green, and blue
components of the color. Finally the color can also be specified as a real vector specifying the red green
and blue components where the components are between 0 and 1.
</para>
<para>
The window should be given as usual as <userinput>[x1,x2,y1,y2]</userinput>, or
alternatively can be given as a string
<userinput>"fit"</userinput> in which case,
the x range will be set precisely and the y range will be set with
five percent borders around the line.
</para>
<para>
Finally, legend should be a string that can be used as the legend in the
graph. That is, if legends are being printed.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawPoints([0,0;1,-1;-1,-1])</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The Solution")</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</userinput>
<prompt>genius></prompt> <userinput>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The 7th roots of unity")</userinput>
</screen>
</para>
<para>
Unlike many other functions that do not care if they take a
column or a row vector, if specifying points as a vector of
complex values, due to possible ambiguities, it must always
be given as a column vector. Therefore, notice in the
last example the transpose of the vector <userinput>0:6</userinput>
to make it into a column vector.
</para>
<para>
Available from version 1.0.18 onwards. Specifying
<varname>v</varname> as a column vector of complex numbers is
implemented from version 1.0.22 and onwards.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotMouseLocation"/>LinePlotMouseLocation</term>
<listitem>
<synopsis>LinePlotMouseLocation ()</synopsis>
<para>
Returns a row vector of a point on the line plot corresponding to
the current mouse location. If the line plot is not visible,
then prints an error and returns <constant>null</constant>.
In this case you should run
<link linkend="gel-function-LinePlot"><function>LinePlot</function></link> or
<link linkend="gel-function-LinePlotClear"><function>LinePlotClear</function></link>
to put the graphing window into the line plot mode.
See also
<link linkend="gel-function-LinePlotWaitForClick"><function>LinePlotWaitForClick</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotParametric"/>LinePlotParametric</term>
<listitem>
<synopsis>LinePlotParametric (xfunc,yfunc,...)</synopsis>
<synopsis>LinePlotParametric (xfunc,yfunc,t1,t2,tinc)</synopsis>
<synopsis>LinePlotParametric (xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)</synopsis>
<synopsis>LinePlotParametric (xfunc,yfunc,t1,t2,tinc,[x1,x2,y1,y2])</synopsis>
<synopsis>LinePlotParametric (xfunc,yfunc,t1,t2,tinc,"fit")</synopsis>
<para>
Plot a parametric function with a line. First come the functions
for <varname>x</varname> and <varname>y</varname> then optionally the <varname>t</varname> limits as <userinput>t1,t2,tinc</userinput>, then optionally the
limits as <userinput>x1,x2,y1,y2</userinput>.
</para>
<para>
If x and y limits are not
specified, then the currently set limits apply
(See <link linkend="gel-function-LinePlotWindow"><function>LinePlotWindow</function></link>).
If instead the string "fit" is given for the x and y limits, then the limits are the maximum extent of
the graph
</para>
<para>
The parameter
<link linkend="gel-function-LinePlotDrawLegends"><function>LinePlotDrawLegends</function></link>
controls the drawing of the legend.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-LinePlotWaitForClick"/>LinePlotWaitForClick</term>
<listitem>
<synopsis>LinePlotWaitForClick ()</synopsis>
<para>
If in line plot mode, waits for a click on the line plot window
and returns the location of the click as a row vector.
If the window is closed
the function returns immediately with <constant>null</constant>.
If the window is not in line plot mode, it is put in it and shown
if not shown.
See also
<link linkend="gel-function-LinePlotMouseLocation"><function>LinePlotMouseLocation</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PlotCanvasFreeze"/>PlotCanvasFreeze</term>
<listitem>
<synopsis>PlotCanvasFreeze ()</synopsis>
<para>
Freeze drawing of the canvas plot temporarily. Useful if you need to draw a bunch of elements
and want to delay drawing everything to avoid flicker in an animation. After everything
has been drawn you should call <link linkend="gel-function-PlotCanvasThaw"><function>PlotCanvasThaw</function></link>.
</para>
<para>
The canvas is always thawed after end of any execution, so it will never remain frozen. The moment
a new command line is shown for example the plot canvas is thawed automatically. Also note that
calls to freeze and thaw may be safely nested.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PlotCanvasThaw"/>PlotCanvasThaw</term>
<listitem>
<synopsis>PlotCanvasThaw ()</synopsis>
<para>
Thaw the plot canvas frozen by
<link linkend="gel-function-PlotCanvasFreeze"><function>PlotCanvasFreeze</function></link>
and redraw the canvas immediately. The canvas is also always thawed after end of execution
of any program.
</para>
<para>Version 1.0.18 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-PlotWindowPresent"/>PlotWindowPresent</term>
<listitem>
<synopsis>PlotWindowPresent ()</synopsis>
<para>
Show and raise the plot window, creating it if necessary.
Normally the window is created when one of the plotting
functions is called, but it is not always raised if it
happens to be below other windows. So this function is
good to call in scripts where the plot window might have
been created before, and by now is hidden behind the
console or other windows.
</para>
<para>Version 1.0.19 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SlopefieldClearSolutions"/>SlopefieldClearSolutions</term>
<listitem>
<synopsis>SlopefieldClearSolutions ()</synopsis>
<para>
Clears the solutions drawn by the
<link linkend="gel-function-SlopefieldDrawSolution"><function>SlopefieldDrawSolution</function></link>
function.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SlopefieldDrawSolution"/>SlopefieldDrawSolution</term>
<listitem>
<synopsis>SlopefieldDrawSolution (x, y, dx)</synopsis>
<para>
When a slope field plot is active, draw a solution with
the specified initial condition. The standard
Runge-Kutta method is used with increment <varname>dx</varname>.
Solutions stay on the graph until a different plot is shown or until
you call
<link linkend="gel-function-SlopefieldClearSolutions"><function>SlopefieldClearSolutions</function></link>.
You can also use the graphical interface to draw solutions and specify
initial conditions with the mouse.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SlopefieldPlot"/>SlopefieldPlot</term>
<listitem>
<synopsis>SlopefieldPlot (func)</synopsis>
<synopsis>SlopefieldPlot (func,x1,x2,y1,y2)</synopsis>
<para>
Plot a slope field. The function <varname>func</varname>
should take two real numbers <varname>x</varname>
and <varname>y</varname>, or a single complex
number.
Optionally you can specify the limits of the plotting window as
<varname>x1</varname>, <varname>x2</varname>,
<varname>y1</varname>, <varname>y2</varname>. If limits are not
specified, then the currently set limits apply
(See <link linkend="gel-function-LinePlotWindow"><function>LinePlotWindow</function></link>).
</para>
<para>
The parameter
<link linkend="gel-function-LinePlotDrawLegends"><function>LinePlotDrawLegends</function></link>
controls the drawing of the legend.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</userinput>
</screen>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlot"/>SurfacePlot</term>
<listitem>
<synopsis>SurfacePlot (func)</synopsis>
<synopsis>SurfacePlot (func,x1,x2,y1,y2,z1,z2)</synopsis>
<synopsis>SurfacePlot (func,x1,x2,y1,y2)</synopsis>
<synopsis>SurfacePlot (func,[x1,x2,y1,y2,z1,z2])</synopsis>
<synopsis>SurfacePlot (func,[x1,x2,y1,y2])</synopsis>
<para>
Plot a surface function that takes either two arguments or a complex number. First comes the function then optionally limits as <varname>x1</varname>, <varname>x2</varname>,
<varname>y1</varname>, <varname>y2</varname>,
<varname>z1</varname>, <varname>z2</varname>. If limits are not
specified, then the currently set limits apply
(See <link linkend="gel-function-SurfacePlotWindow"><function>SurfacePlotWindow</function></link>).
Genius can only plot a single surface function at this time.
</para>
<para>
If the z limits are not specified then the maxima and minima of the function are used.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SurfacePlot(|sin|,-1,1,-1,1,0,1.5)</userinput>
<prompt>genius></prompt> <userinput>SurfacePlot(`(x,y)=x^2+y,-1,1,-1,1,-2,2)</userinput>
<prompt>genius></prompt> <userinput>SurfacePlot(`(z)=|z|^2,-1,1,-1,1,0,2)</userinput>
</screen>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotClear"/>SurfacePlotClear</term>
<listitem>
<synopsis>SurfacePlotClear ()</synopsis>
<para>
Show the surface plot window and clear out functions and any other
lines that were drawn.
</para>
<para>
Available in version 1.0.19 and onwards.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotData"/>SurfacePlotData</term>
<listitem>
<synopsis>SurfacePlotData (data)</synopsis>
<synopsis>SurfacePlotData (data,label)</synopsis>
<synopsis>SurfacePlotData (data,x1,x2,y1,y2,z1,z2)</synopsis>
<synopsis>SurfacePlotData (data,label,x1,x2,y1,y2,z1,z2)</synopsis>
<synopsis>SurfacePlotData (data,[x1,x2,y1,y2,z1,z2])</synopsis>
<synopsis>SurfacePlotData (data,label,[x1,x2,y1,y2,z1,z2])</synopsis>
<para>
Plot a surface from data. The data is an n by 3 matrix whose
rows are the x, y and z coordinates. The data can also be
simply a vector whose length is a multiple of 3 and so
contains the triples of x, y, z. The data should contain at
least 3 points.
</para>
<para>
Optionally we can give the label and also optionally the
limits. If limits are not given, they are computed from
the data, <link linkend="gel-function-SurfacePlotWindow"><function>SurfacePlotWindow</function></link>
is not used, if you want to use it, pass it in explicitly.
If label is not given then empty label is used.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SurfacePlotData([0,0,0;1,0,1;0,1,1;1,1,3])</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotData(data,"My data")</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotData(data,-1,1,-1,1,0,10)</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotData(data,SurfacePlotWindow)</userinput>
</screen>
</para>
<para>
Here's an example of how to plot in polar coordinates,
in particular how to plot the function
<userinput>-r^2 * theta</userinput>:
<screen><prompt>genius></prompt> <userinput>d:=null; for r=0 to 1 by 0.1 do for theta=0 to 2*pi by pi/5 do d=[d;[r*cos(theta),r*sin(theta),-r^2*theta]];</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotData(d)</userinput>
</screen>
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotDataGrid"/>SurfacePlotDataGrid</term>
<listitem>
<synopsis>SurfacePlotDataGrid (data,[x1,x2,y1,y2])</synopsis>
<synopsis>SurfacePlotDataGrid (data,[x1,x2,y1,y2,z1,z2])</synopsis>
<synopsis>SurfacePlotDataGrid (data,[x1,x2,y1,y2],label)</synopsis>
<synopsis>SurfacePlotDataGrid (data,[x1,x2,y1,y2,z1,z2],label)</synopsis>
<para>
Plot a surface from regular rectangular data.
The data is given in a n by m matrix where the rows are the
x coordinate and the columns are the y coordinate.
The x coordinate is divided into equal n-1 subintervals
and y coordinate is divided into equal m-1 subintervals.
The limits <varname>x1</varname> and <varname>x2</varname>
give the interval on the x-axis that we use, and
the limits <varname>y1</varname> and <varname>y2</varname>
give the interval on the y-axis that we use.
If the limits <varname>z1</varname> and <varname>z2</varname>
are not given they are computed from the data (to be
the extreme values from the data).
</para>
<para>
Optionally we can give the label, if label is not given then
empty label is used.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SurfacePlotDataGrid([1,2;3,4],[0,1,0,1])</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotDataGrid(data,[-1,1,-1,1],"My data")</userinput>
<prompt>genius></prompt> <userinput>d:=null; for i=1 to 20 do for j=1 to 10 do d@(i,j) = (0.1*i-1)^2-(0.1*j)^2;</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotDataGrid(d,[-1,1,0,1],"half a saddle")</userinput>
</screen>
</para>
<para>Version 1.0.16 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotDrawLine"/>SurfacePlotDrawLine</term>
<listitem>
<synopsis>SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</synopsis>
<synopsis>SurfacePlotDrawLine (v,...)</synopsis>
<para>
Draw a line from <varname>x1</varname>,<varname>y1</varname>,<varname>z1</varname> to
<varname>x2</varname>,<varname>y2</varname>,<varname>z2</varname>.
<varname>x1</varname>,<varname>y1</varname>,<varname>z1</varname>,
<varname>x2</varname>,<varname>y2</varname>,<varname>z2</varname> can be replaced by an
<varname>n</varname> by 3 matrix for a longer polyline.
</para>
<para>
Extra parameters can be added to specify line color, thickness,
the plotting window, or legend.
You can do this by adding an argument string <userinput>"color"</userinput>,
<userinput>"thickness"</userinput>,
<userinput>"window"</userinput>,
or <userinput>"legend"</userinput>, and after it specify
the color, the thickness, the window
as 6-vector, or the legend.
</para>
<para>
The color should be either a string indicating the common English word for the color
that GTK will recognize such as
<userinput>"red"</userinput>, <userinput>"blue"</userinput>, <userinput>"yellow"</userinput>, etc...
Alternatively the color can be specified in RGB format as
<userinput>"#rgb"</userinput>, <userinput>"#rrggbb"</userinput>, or
<userinput>"#rrrrggggbbbb"</userinput>, where the r, g, or b are hex digits of the red, green, and blue
components of the color. Finally, since version 1.0.18, the color
can also be specified as a real vector specifying the red green and
blue components where the components are between 0 and 1, e.g. <userinput>[1.0,0.5,0.1]</userinput>.
</para>
<para>
The window should be given as usual as <userinput>[x1,x2,y1,y2,z1,z2]</userinput>, or
alternatively can be given as a string
<userinput>"fit"</userinput> in which case,
the x range will be set precisely and the y range will be set with
five percent borders around the line.
</para>
<para>
Finally, legend should be a string that can be used as the legend in the
graph. That is, if legends are being printed.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotDrawLine([0,0,0;1,-1,2;-1,-1,-3])</userinput>
</screen>
</para>
<para>
Available from version 1.0.19 onwards.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-SurfacePlotDrawPoints"/>SurfacePlotDrawPoints</term>
<listitem>
<synopsis>SurfacePlotDrawPoints (x,y,z,...)</synopsis>
<synopsis>SurfacePlotDrawPoints (v,...)</synopsis>
<para>
Draw a point at <varname>x</varname>,<varname>y</varname>,<varname>z</varname>.
The input can be an <varname>n</varname> by 3 matrix
for <varname>n</varname> different points. This function has essentially the same
input as <link linkend="gel-function-SurfacePlotDrawLine">SurfacePlotDrawLine</link>.
</para>
<para>
Extra parameters can be added to specify line color, thickness,
the plotting window, or legend.
You can do this by adding an argument string <userinput>"color"</userinput>,
<userinput>"thickness"</userinput>,
<userinput>"window"</userinput>,
or <userinput>"legend"</userinput>, and after it specify
the color, the thickness, the window
as 6-vector, or the legend.
</para>
<para>
The color should be either a string indicating the common English word for the color
that GTK will recognize such as
<userinput>"red"</userinput>, <userinput>"blue"</userinput>, <userinput>"yellow"</userinput>, etc...
Alternatively the color can be specified in RGB format as
<userinput>"#rgb"</userinput>, <userinput>"#rrggbb"</userinput>, or
<userinput>"#rrrrggggbbbb"</userinput>, where the r, g, or b are hex digits of the red, green, and blue
components of the color. Finally the color can also be specified as a real vector specifying the red green
and blue components where the components are between 0 and 1.
</para>
<para>
The window should be given as usual as <userinput>[x1,x2,y1,y2,z1,z2]</userinput>, or
alternatively can be given as a string
<userinput>"fit"</userinput> in which case,
the x range will be set precisely and the y range will be set with
five percent borders around the line.
</para>
<para>
Finally, legend should be a string that can be used as the legend in the
graph. That is, if legends are being printed.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)</userinput>
<prompt>genius></prompt> <userinput>SurfacePlotDrawPoints([0,0,0;1,-1,2;-1,-1,1])</userinput>
</screen>
</para>
<para>
Available from version 1.0.19 onwards.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorfieldClearSolutions"/>VectorfieldClearSolutions</term>
<listitem>
<synopsis>VectorfieldClearSolutions ()</synopsis>
<para>
Clears the solutions drawn by the
<link linkend="gel-function-VectorfieldDrawSolution"><function>VectorfieldDrawSolution</function></link>
function.
</para>
<para>Version 1.0.6 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorfieldDrawSolution"/>VectorfieldDrawSolution</term>
<listitem>
<synopsis>VectorfieldDrawSolution (x, y, dt, tlen)</synopsis>
<para>
When a vector field plot is active, draw a solution with
the specified initial condition. The standard
Runge-Kutta method is used with increment <varname>dt</varname>
for an interval of length <varname>tlen</varname>.
Solutions stay on the graph until a different plot is shown or until
you call
<link linkend="gel-function-VectorfieldClearSolutions"><function>VectorfieldClearSolutions</function></link>.
You can also use the graphical interface to draw solutions and specify
initial conditions with the mouse.
</para>
<para>Version 1.0.6 onwards.</para>
</listitem>
</varlistentry>
<varlistentry>
<term><anchor id="gel-function-VectorfieldPlot"/>VectorfieldPlot</term>
<listitem>
<synopsis>VectorfieldPlot (funcx, funcy)</synopsis>
<synopsis>VectorfieldPlot (funcx, funcy, x1, x2, y1, y2)</synopsis>
<para>
Plot a two dimensional vector field. The function
<varname>funcx</varname>
should be the dx/dt of the vectorfield and the function
<varname>funcy</varname> should be the dy/dt of the vectorfield.
The functions
should take two real numbers <varname>x</varname>
and <varname>y</varname>, or a single complex
number. When the parameter
<link linkend="gel-function-VectorfieldNormalized"><function>VectorfieldNormalized</function></link>
is <constant>true</constant>, then the magnitude of the vectors is normalized. That is, only
the direction and not the magnitude is shown.
</para>
<para>
Optionally you can specify the limits of the plotting window as
<varname>x1</varname>, <varname>x2</varname>,
<varname>y1</varname>, <varname>y2</varname>. If limits are not
specified, then the currently set limits apply
(See <link linkend="gel-function-LinePlotWindow"><function>LinePlotWindow</function></link>).
</para>
<para>
The parameter
<link linkend="gel-function-LinePlotDrawLegends"><function>LinePlotDrawLegends</function></link>
controls the drawing of the legend.
</para>
<para>
Examples:
<screen><prompt>genius></prompt> <userinput>VectorfieldPlot(`(x,y)=x^2-y, `(x,y)=y^2-x, -1, 1, -1, 1)</userinput>
</screen>
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
</chapter>
<!-- ============= GEL examples ============================= -->
<chapter id="genius-gel-example-programs">
<title>Примеры программ на GEL</title>
<para>
Here is a function that calculates factorials:
<programlisting><![CDATA[function f(x) = if x <= 1 then 1 else (f(x-1)*x)
]]></programlisting>
</para>
<para>
With indentation it becomes:
<programlisting><![CDATA[function f(x) = (
if x <= 1 then
1
else
(f(x-1)*x)
)
]]></programlisting>
</para>
<para>
This is a direct port of the factorial function from the <application>bc</application> manpage. The syntax seems similar to <application>bc</application>, but different in that in GEL, the last expression is the one that is returned. Using the <literal>return</literal> function instead, it would be:
<programlisting><![CDATA[function f(x) = (
if (x <= 1) then return (1);
return (f(x-1) * x)
)
]]></programlisting>
</para>
<para>
By far the easiest way to define a factorial function would be using
the product loop as follows. This is not only the shortest and fastest,
but also probably the most readable version.
<programlisting>function f(x) = prod k=1 to x do k
</programlisting>
</para>
<para>
Here is a larger example, this basically redefines the internal
<link linkend="gel-function-ref"><function>ref</function></link> function to calculate the row echelon form of a
matrix. The function <function>ref</function> is built in and much faster,
but this example demonstrates some of the more complex features of GEL.
<programlisting><![CDATA[# Calculate the row-echelon form of a matrix
function MyOwnREF(m) = (
if not IsMatrix(m) or not IsValueOnly(m) then
(error("MyOwnREF: argument not a value only matrix");bailout);
s := min(rows(m), columns(m));
i := 1;
d := 1;
while d <= s and i <= columns(m) do (
# This just makes the anchor element non-zero if at
# all possible
if m@(d,i) == 0 then (
j := d+1;
while j <= rows(m) do (
if m@(j,i) == 0 then
(j=j+1;continue);
a := m@(j,);
m@(j,) := m@(d,);
m@(d,) := a;
j := j+1;
break
)
);
if m@(d,i) == 0 then
(i:=i+1;continue);
# Here comes the actual zeroing of all but the anchor
# element rows
j := d+1;
while j <= rows(m)) do (
if m@(j,i) != 0 then (
m@(j,) := m@(j,)-(m@(j,i)/m@(d,i))*m@(d,)
);
j := j+1
);
m@(d,) := m@(d,) * (1/m@(d,i));
d := d+1;
i := i+1
);
m
)
]]></programlisting>
</para>
</chapter>
<!-- ============= Customization ============================ -->
<chapter id="genius-prefs">
<title>Настройки</title>
<para>
To configure <application>Genius Mathematics Tool</application>, choose
<menuchoice><guimenu>Settings</guimenu>
<guimenuitem>Preferences</guimenuitem></menuchoice>.
There are several basic parameters provided by the calculator in addition
to the ones provided by the standard library. These control how the
calculator behaves.
</para>
<note>
<title>Changing Settings with GEL</title>
<para>
Many of the settings in Genius are simply global variables, and can
be evaluated and assigned to in the same way as normal variables. See
<xref linkend="genius-gel-variables"/> about evaluating and assigning
to variables, and <xref linkend="genius-gel-function-parameters"/> for
a list of settings that can be modified in this way.
</para>
<para>
As an example, you can set the maximum number of digits in a result to 12 by typing:
<programlisting>MaxDigits = 12
</programlisting>
</para>
</note>
<sect1 id="genius-prefs-output">
<title>Output</title>
<variablelist>
<varlistentry>
<term>
<guilabel>Maximum digits to output</guilabel>
</term>
<listitem>
<para>The maximum digits in a result (<link linkend="gel-function-MaxDigits"><function>MaxDigits</function></link>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Results as floats</guilabel>
</term>
<listitem>
<para>If the results should be always printed as floats (<link linkend="gel-function-ResultsAsFloats"><function>ResultsAsFloats</function></link>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Floats in scientific notation</guilabel>
</term>
<listitem>
<para>If floats should be in scientific notation (<link linkend="gel-function-ScientificNotation"><function>ScientificNotation</function></link>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Always print full expressions</guilabel>
</term>
<listitem>
<para>Should we print out full expressions for non-numeric return values (longer than a line) (<link linkend="gel-function-FullExpressions"><function>FullExpressions</function></link>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Use mixed fractions</guilabel>
</term>
<listitem>
<para>If fractions should be printed as mixed fractions such as "1 1/3" rather than "4/3". (<link linkend="gel-function-MixedFractions"><function>MixedFractions</function></link>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Display 0.0 when floating point number is less than 10^-x (0=never chop)</guilabel>
</term>
<listitem>
<para>How to chop output. But only when other numbers nearby are large.
See the documentation of the parameter
<link linkend="gel-function-OutputChopExponent"><function>OutputChopExponent</function></link>. </para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Only chop numbers when another number is greater than 10^-x</guilabel>
</term>
<listitem>
<para>When to chop output. This is set by the parameter <link linkend="gel-function-OutputChopWhenExponent"><function>OutputChopWhenExponent</function></link>.
See the documentation of the parameter
<link linkend="gel-function-OutputChopExponent"><function>OutputChopExponent</function></link>. </para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Remember output settings across sessions</guilabel>
</term>
<listitem>
<para>Should the output settings in the <guilabel>Number/Expression output options</guilabel> frame
be remembered for next session. Does not apply to the <guilabel>Error/Info output options</guilabel> frame.</para>
<para>
If unchecked,
either the default or any previously saved settings are used each time Genius starts
up. Note that
settings are saved at the end of the session, so if you wish to change the defaults
check this box, restart <application>Genius Mathematics Tool</application> and then uncheck it again.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Display errors in a dialog</guilabel>
</term>
<listitem>
<para>If set the errors will be displayed in a separate dialog, if
unset the errors will be printed on the console.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Display information messages in a dialog</guilabel>
</term>
<listitem>
<para>If set the information messages will be displayed in a separate
dialog, if unset the information messages will be printed on the
console.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Maximum errors to display</guilabel>
</term>
<listitem>
<para>
The maximum number of errors to return on one evaluation
(<link linkend="gel-function-MaxErrors"><function>MaxErrors</function></link>). If you set this to 0 then
all errors are always returned. Usually if some loop causes
many errors, then it is unlikely that you will be able to make
sense out of more than a few of these, so seeing a long list
of errors is usually not helpful.
</para>
</listitem>
</varlistentry>
</variablelist>
<para>
In addition to these preferences, there are some preferences that can
only be changed by setting them in the workspace console. For others
that may affect the output see <xref linkend="genius-gel-function-parameters"/>.
</para>
<variablelist>
<varlistentry>
<term>
<function>IntegerOutputBase</function>
</term>
<listitem>
<para>The base that will be used to output integers</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<function>OutputStyle</function>
</term>
<listitem>
<para>A string, can be <literal>"normal"</literal>,
<literal>"latex"</literal>, <literal>"mathml"</literal> or
<literal>"troff"</literal> and it will affect how matrices (and perhaps other
stuff) is printed, useful for pasting into documents. Normal style is the
default human readable printing style of <application>Genius Mathematics Tool</application>. The other styles are for
typesetting in LaTeX, MathML (XML), or in Troff.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-prefs-precision">
<title>Precision</title>
<variablelist>
<varlistentry>
<term>
<guilabel>Floating point precision</guilabel>
</term>
<listitem>
<para>
The floating point precision in bits
(<link linkend="gel-function-FloatPrecision"><function>FloatPrecision</function></link>).
Note that changing this only affects newly computed quantities.
Old values stored in variables are obviously still in the old
precision and if you want to have them more precise you will have
to recompute them. Exceptions to this are the system constants
such as <link linkend="gel-function-pi"><function>pi</function></link> or
<link linkend="gel-function-e"><function>e</function></link>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Remember precision setting across sessions</guilabel>
</term>
<listitem>
<para>
Should the precision setting be remembered for the next session. If unchecked,
either the default or any previously saved setting is used each time Genius starts
up. Note that
settings are saved at the end of the session, so if you wish to change the default
check this box, restart genius and then uncheck it again.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-prefs-terminal">
<title>Терминал</title>
<para>Терминалом называется консоль в рабочей области.</para>
<variablelist>
<varlistentry>
<term>
<guilabel>Scrollback lines</guilabel>
</term>
<listitem>
<para>Lines of scrollback in the terminal.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Шрифт</guilabel>
</term>
<listitem>
<para>Шрифт, используемый в терминале</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Чёрный на белом</guilabel>
</term>
<listitem>
<para>Использовать в терминале чёрный текст на белом фоне.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>
<guilabel>Мигающий курсор</guilabel>
</term>
<listitem>
<para>Должен ли курсор терминала мигать, если фокус ввода находится в терминале. Иногда это раздражает, кроме того это создаёт лишний трафик при удалённом доступе к Genius.</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
<sect1 id="genius-prefs-memory">
<title>Память</title>
<variablelist>
<varlistentry>
<term>
<guilabel>Maximum number of nodes to allocate</guilabel>
</term>
<listitem>
<para>
Internally all data is put onto small nodes in memory. This gives
a limit on the maximum number of nodes to allocate for
computations. This limit avoids the problem of running out of memory
if you do something by mistake that uses too much memory, such
as a recursion without end. This could slow your computer and make
it hard to even interrupt the program.
</para>
<para>
Once the limit is reached, <application>Genius Mathematics Tool</application> asks if you wish to interrupt
the computation or if you wish to continue. If you continue, no
limit is applied and it will be possible to run your computer
out of memory. The limit will be applied again next time you
execute a program or an expression on the Console regardless of how
you answered the question.
</para>
<para>
Setting the limit to zero means there is no limit to the amount of
memory that genius uses.
</para>
</listitem>
</varlistentry>
</variablelist>
</sect1>
</chapter>
<!-- ============= About ====================================== -->
<chapter id="genius-about">
<title>О <application>математическом инструменте Genius</application></title>
<para><application>Математический инструмент Genius</application> создан Jiří (George) Lebl (<email>jirka@5z.com</email>). История <application>математического инструмента Genius</application> началась в конце 1997 г. Это был первый калькулятор для GNOME, но с тех пор он вырос в нечто большее, чем простой настольный калькулятор. Чтобы узнать больше о <application>математическом инструменте Genius</application>, посетите <ulink url="http://www.jirka.org/genius.html" type="http">веб-страницу Genius</ulink>.</para>
<para>
To report a bug or make a suggestion regarding this application or
this manual, send email to me (the author) or post to the mailing
list (see the web page).
</para>
<para> This program is distributed 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. A copy of this license can be found at this
<ulink url="http://www.gnu.org/copyleft/gpl.html" type="http">link</ulink>, or in the file
COPYING included with the source code of this program. </para>
<para>Jiří Lebl was during various parts of the development
partially supported for the work by NSF grants DMS 0900885,
DMS 1362337,
the University of Illinois at Urbana-Champaign,
the University of California at San Diego,
the University of Wisconsin-Madison, and
Oklahoma State University. The software has
been used for both teaching and research.</para>
</chapter>
</book>
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