<?xml version="1.0" encoding="UTF-8"?>
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 * Scilab ( https://www.scilab.org/ ) - This file is part of Scilab
 * Copyright (C) 2008 - INRIA
 * Copyright (C) 2015 - Samuel GOUGEON
 *
 * Copyright (C) 2012 - 2016 - Scilab Enterprises
 *
 * This file is hereby licensed under the terms of the GNU GPL v2.0,
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 * 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="householder">
    <refnamediv>
        <refname>householder</refname>
        <refpurpose>Householder orthogonal reflexion matrix. Symetrical wrt a plane</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            householder() // demo
            u = householder(v [,w])
            [u, H] = householder(v [,w])
        </synopsis>
    </refsynopsisdiv>
    <refsection role="arguments">
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>v</term>
                <listitem>
                    <para>real or complex column vector</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>w</term>
                <listitem>
                    <para>
                        real or complex column vector with same size as <literal>v</literal>.
                        Default value is <literal>eye(v)</literal> ((Ox) axis).
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>u</term>
                <listitem>
                    <para>
                        unit vector lying in the <literal>(v,w)</literal> plane and orthogonal
                        to the bisectrix of <literal>(v,w)</literal>.
                        Column of size(v) of real or complex numbers.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>H</term>
                <listitem>
                    <para>
                        Orthogonal Householder reflexion matrix: <literal>H= eye() - 2*u*u'</literal>.
                        <varname>H</varname> is such that <literal>inv(H)==H</literal>,
                        <literal>H'==H</literal>, and <literal>det(H)==-1</literal>.
                    </para>
                    <para>
                        If <varname>v</varname> and <varname>w</varname> are real,
                        <literal>H*v</literal> is proportional to <varname>w</varname>.
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection role="description">
        <title>Description</title>
        <para>
            <literal>householder(..)</literal> computes the unit vector <varname>u</varname>
            lying in the <literal>(v,w)</literal> plane and orthogonal to the bisectrix of
            <literal>(v,w)</literal>.
        </para>
        <para>
            If <varname>v</varname> and <varname>w</varname> are proportional:
            <itemizedlist>
                <listitem>
                    <para>
                        If they are opposite, <literal>u= v/|v|</literal> is returned.
                    </para>
                </listitem>
                <listitem>
                    If they are real and have the same direction, <varname>u</varname> is set
                    in the (xOy) plane with a priori <literal>u(1)>0</literal>, and orthogonal to
                    <varname>v</varname> (<literal>u'*v==0</literal>). However,
                    <itemizedlist>
                        <listitem>
                            If they are along (Ox), <literal>u = (Oy+)</literal> is returned instead.
                        </listitem>
                        <listitem>
                            If <varname>v</varname> and <varname>w</varname> are scalars with same
                            signs, the orthogonal sub-space is restricted to <literal>{0}</literal>
                            that can't be normalized:
                            <varname>u</varname> and <varname>H</varname> are then set to
                            <literal>%nan</literal>.
                        </listitem>
                    </itemizedlist>
                </listitem>
            </itemizedlist>
        </para>
        <note>
            If the related reflexion matrix <varname>H</varname> is computed, for any point A
            of column coordinates <literal>a</literal>, <literal>H*a</literal> are the coordinates of
            the symetrical image of A with respect to the <literal>(v,w)</literal> plane
            (see the example below).
        </note>
        <note>
            If <varname>v</varname> or/and <varname>w</varname> are in row, they are priorly
            transposed into columns.
        </note>
        <warning>
            If <varname>v</varname> or/and <varname>w</varname> are <literal>[]</literal>,
            <literal>[]</literal> is returned for <varname>u</varname> and <varname>H</varname>.
        </warning>
    </refsection>
    <refsection role="examples">
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
a = [ rand(1,1) 0  0 ]';
[ra hm] = householder(a);
[a ra hm*a ]
norm(ra)

b = rand(3,1);
[rb, hm] = householder(b);
[b rb eye(b) clean(hm*b) ]
norm(rb)

[rb2b, hm] = householder(b, 2*b);
[b rb2b clean(hm*b ./ b) ]  // last column must be uniform
norm(rb2b)                  // must be 1

c = rand(3,1);
[rbc, hm] = householder(b,c);
norm(rbc)          // must be 1
hm*b ./c           // must be uniform

d = b + %i*c;
e = rand(3,1) + %i*rand(3,1);
[rde, hm] = householder(d,e);
norm(rbc)               // must be 1
clean(inv(hm) - hm)     // must be zeros(3,3)
clean(hm' - hm)         // must be zeros(3,3)
clean(det(hm))          // must be -1
 ]]></programlisting>
        <para>
            <emphasis role="bold"> Application:</emphasis> Symetrical image of an object w.r.t. a given plane
        </para>
        <programlisting role="example"><![CDATA[
// (OA) = [0 0 1] is reflected in O into (OB) = [ 1 1 0.3 ]:
[n, H] = householder([0 0 1]', [ 1 1 0.3 ]');
// "n" is the unit vector orthogonal to the reflecting plane

// Emitting object (feature from shell demo):
u = linspace(0,2*%pi,40);
v = linspace(0,2*%pi,20);
Xe = (cos(u).*u)'*(1+cos(v)/2)+10;
Ye = (u/2)'*sin(v);
Ze = (sin(u).*u)'*(1+cos(v)/2);

// Symetrical object:
Pe = [ Xe(:)' ; Ye(:)' ; Ze(:)'];
Pr = H*Pe;
Xr = matrix(Pr(1,:),40,-1);
Yr = matrix(Pr(2,:),40,-1);
Zr = matrix(Pr(3,:),40,-1);

// Reflecting plane containing O: n(1).x + n(2).y + n(3).z = 0
//   Sampling space:
x = linspace(min([Xe(:);Xr(:)]), max([Xe(:);Xr(:)]),20);
y = linspace(min([Ye(:);Yr(:)]), max([Ye(:);Yr(:)]),20);
[X, Y] = meshgrid(x,y);
//   Generating the mirror:
deff("z = mirror(x,y,n)","z = -n(1)/n(3)*x - n(2)/n(3)*y")
Zm = mirror(X,Y,n);

// Plotting:
clf()
isoview()
drawlater()
f = gcf();
f.color_map = [ 0.8 0.8 0.8 ; jet(100)];
surf(Xe,Ye,Ze)
surf(X,Y,Zm)
surf(Xr,Yr,Zr)
a = gca();
a.rotation_angles = [74 123];
a.children.color_flag = 0;
a.children.color_mode = 0;
a.children(1).foreground = color("red");
a.children(2).foreground = 1;
a.children(3).foreground = color("green");
drawnow()
 ]]></programlisting>
        <scilab:image>
            // (OA) = [0 0 1] is reflected in O into (OB) = [ 1 1 0.3 ]:
            [n, H] = householder([0 0 1]', [ 1 1 0.3 ]');
            // "n" is the unit vector orthogonal to the reflecting plane

            // Emitting object (feature from shell demo):
            u = linspace(0,2*%pi,40);
            v = linspace(0,2*%pi,20);
            Xe = (cos(u).*u)'*(1+cos(v)/2)+10;
            Ye = (u/2)'*sin(v);
            Ze = (sin(u).*u)'*(1+cos(v)/2);

            // Symetrical object:
            Pe = [ Xe(:)' ; Ye(:)' ; Ze(:)'];
            Pr = H*Pe;
            Xr = matrix(Pr(1,:),40,-1);
            Yr = matrix(Pr(2,:),40,-1);
            Zr = matrix(Pr(3,:),40,-1);

            // Reflecting plane containing O: n(1).x + n(2).y + n(3).z = 0
            //   Sampling space:
            x = linspace(min([Xe(:);Xr(:)]), max([Xe(:);Xr(:)]),20);
            y = linspace(min([Ye(:);Yr(:)]), max([Ye(:);Yr(:)]),20);
            [X, Y] = meshgrid(x,y);
            //   Generating the mirror:
            deff("z = mirror(x,y,n)","z = -n(1)/n(3)*x - n(2)/n(3)*y")
            Zm = mirror(X,Y,n);

            // Plotting:
            clf
            drawlater
            f = gcf();
            f.color_map = [ 0.8 0.8 0.8 ; jet(100)];
            surf(Xe,Ye,Ze)
            surf(X,Y,Zm)
            surf(Xr,Yr,Zr)
            a = gca();
            a.isoview = "on";
            a.rotation_angles = [74 123];
            a.children.color_flag = 0;
            a.children.color_mode = 0;
            a.children(1).foreground = color("red");
            a.children(2).foreground = 1;
            a.children(3).foreground = color("green");
            drawnow
        </scilab:image>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="proj">proj</link>
            </member>
            <member>
                <link linkend="orthProj">orthProj</link>
            </member>
            <member>
                <link linkend="scaling">scaling</link>
            </member>
            <member>
                <link linkend="qr">qr</link>
            </member>
            <member>
                <link linkend="givens">givens</link>
            </member>
        </simplelist>
    </refsection>
    <refsection role="history">
        <title>History</title>
        <revhistory>
            <revision>
                <revnumber>6.0</revnumber>
                <revdescription>
                    <para>Householder reflexion matrix added as second output parameter.
                        Demo householder() added. Help page reviewed.
                    </para>
                </revdescription>
            </revision>
        </revhistory>
    </refsection>
</refentry>
