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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="lsq">
    <refnamediv>
        <refname>lsq</refname>
        <refpurpose>linear least square solution of A*X=B with minimal norm(X)</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            X = lsq(A, B)
            X = lsq(A, B, tol)
        </synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>Real or complex (m x n) matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>B</term>
                <listitem>
                    <para>real or complex (m x p) matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>tol</term>
                <listitem>
                    <para>positive scalar,  used to determine the effective rank of A
                        (defined as the order of the largest leading triangular
                        submatrix R11 in the QR factorization with pivoting of A,
                        whose estimated condition number &lt;= 1/tol). The tol default value is
                        set to <literal>sqrt(%eps)</literal>.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>X</term>
                <listitem>
                    <para>real or complex (n x p) matrix</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            <literal>X=lsq(A,B)</literal> computes the minimum norm least square solution of
            the equation <literal>A*X=B</literal>, while <literal>X=A \ B</literal> compute a least square
            solution with at most <literal>rank(A)</literal> nonzero components per column.
        </para>
    </refsection>
    <refsection>
        <title>References</title>
        <para>
            <literal>lsq</literal> function is  based on the LApack functions DGELSY for
            real matrices and ZGELSY for complex matrices.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
//Build the data
x=(1:10)';

y1=3*x+4.5+3*rand(x,'normal');
y2=1.8*x+0.5+2*rand(x,'normal');
plot2d(x,[y1,y2],[-2,-3])
//Find the linear regression
A=[x,ones(x)];B=[y1,y2];
X=lsq(A,B);

y1e=X(1,1)*x+X(2,1);
y2e=X(1,2)*x+X(2,2);
plot2d(x,[y1e,y2e],[2,3])

//Difference between lsq(A,b) and A\b
A=rand(4,2)*rand(2,3);//a rank 2 matrix
b=rand(4,1);
X1=lsq(A,b)
X2=A\b
[A*X1-b, A*X2-b] //the residuals are the same
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="backslash">backslash</link>
            </member>
            <member>
                <link linkend="inv">inv</link>
            </member>
            <member>
                <link linkend="pinv">pinv</link>
            </member>
            <member>
                <link linkend="rank">rank</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
