<?xml version="1.0" encoding="UTF-8"?>
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 * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
 * Copyright (C) 2008 - INRIA
 *
 * Copyright (C) 2012 - 2016 - Scilab Enterprises
 *
 * This file is hereby licensed under the terms of the GNU GPL v2.0,
 * pursuant to article 5.3.4 of the CeCILL v.2.1.
 * This file was originally licensed under the terms of the CeCILL v2.1,
 * and continues to be available under such terms.
 * For more information, see the COPYING file which you should have received
 * along with this program.
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<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="en" xml:id="linsolve">
    <refnamediv>
        <refname>linsolve</refname>
        <refpurpose>linear equation solver</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[x0,kerA]=linsolve(A,b [,x0])</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>
                        a <literal>na x ma</literal> real matrix (possibly sparse)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>b</term>
                <listitem>
                    <para>
                        a <literal>na x 1</literal> vector (same row dimension as <literal>A</literal>)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>x0</term>
                <listitem>
                    <para>a real vector</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>kerA</term>
                <listitem>
                    <para>
                        a <literal>ma x k</literal> real matrix
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            <literal>linsolve</literal>  computes all the solutions to <literal> A*x+b=0</literal>.
        </para>
        <para>
            <literal>x0</literal> is a particular solution (if any) and <literal> kerA= </literal>nullspace
            of <literal>A</literal>. Any <literal>x=x0+kerA*w</literal> with arbitrary <literal>w</literal> satisfies
            <literal> A*x+b=0</literal>.
        </para>
        <para>
            If compatible <literal>x0</literal> is given on entry, <literal>x0</literal> is returned. If not
            a compatible <literal>x0</literal>, if any, is returned.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
A=rand(5,3)*rand(3,8);
b=A*ones(8,1);[x,kerA]=linsolve(A,b);A*x+b   //compatible b
b=ones(5,1);[x,kerA]=linsolve(A,b);A*x+b   //uncompatible b
A=rand(5,5);[x,kerA]=linsolve(A,b), -inv(A)*b  //x is unique

// Benchmark with other linear sparse solver:
[A,descr,ref,mtype] = ReadHBSparse(SCI+"/modules/umfpack/demos/bcsstk24.rsa");

b = zeros(size(A,1),1);

tic();
res = umfpack(A,'\',b);
mprintf('\ntime needed to solve the system with umfpack: %.3f\n',toc());

tic();
res = linsolve(A,b);
mprintf('\ntime needed to solve the system with linsolve: %.3f\n',toc());

tic();
res = A\b;
mprintf('\ntime needed to solve the system with the backslash operator: %.3f\n',toc());
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="inv">inv</link>
            </member>
            <member>
                <link linkend="pinv">pinv</link>
            </member>
            <member>
                <link linkend="colcomp">colcomp</link>
            </member>
            <member>
                <link linkend="im_inv">im_inv</link>
            </member>
            <member>
                <link linkend="umfpack">umfpack</link>
            </member>
            <member>
                <link linkend="backslash">backslash</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
