<?xml version="1.0" encoding="ISO-8859-1"?><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" version="5.0-subset Scilab" xml:lang="en" xml:id="op_fitz"><info><pubdate>09-Jul-2008</pubdate></info><refnamediv><refname>op_fitz</refname><refpurpose>Evaluate the Fitzpatrick function of a given operator on a grid using the LLT2d algorithm.</refpurpose></refnamediv>
  
   
  
   
  
   
  
   
  
   

  
   <refsynopsisdiv><title>Calling Sequence</title><synopsis>F = op_fitz(A, n, x, xstar)</synopsis></refsynopsisdiv>

  
   <refsection><title>Parameters</title>
    
      <variablelist>
      
         <varlistentry>
        
            <term>A</term>
        
            <listitem>
          
               <para>matrix. An operator [a;a*] where a and a* are row vectors.
          </para>
        
            </listitem>
      
         </varlistentry>
      
         <varlistentry>
        
            <term>n</term>
        
            <listitem>
          
               <para>integer. The order of the Fitzpatrick function to compute.
          </para>
        
            </listitem>
      
         </varlistentry>
      
         <varlistentry>
        
            <term>x</term>
        
            <listitem>
          
               <para>column vector. The x values of the grid on which the Fitzpatrick function is evaluated.
          </para>
        
            </listitem>
      
         </varlistentry>
      
         <varlistentry>
        
            <term>xstar</term>
        
            <listitem>
          
               <para>column vector. The x* values of the grid on which the Fitzpatrick function is evaluated.
          </para>
        
            </listitem>
      
         </varlistentry>
    
      </variablelist>
  
   </refsection>

  
   <refsection><title>Description</title>
      
         <para>
        Evaluates the Fitzpatrick function with order n of an operator A on a grid (x,x*).
      </para>
      
         <para>
        This function uses a recursive algorithm that employs LLT2d to achieve a running time of O(n*m^2 + m*Nxstar + N).  When m==Nx==Nxstar, the running time is O(m^3).
<!--<programlisting><![CDATA[
    F(A, 1, x, xstar) = x * xstar'

    F(A, 2, x, xstar) =       max         [ x*astar + a*xstar - a*astar - I(a,astar*) ]
                         (a,astar) in A

    F(A, 3, x, xstar) =       max         [ x*astar2 + a1*xstar + f(a1,astar2) ]
                        (a1,astar2) in A
    f(a1, a2)         =       max         [ -a1*astar1 - a2*astar2 - I(a1,astar1) - I(a2,astar2) ]
                        (astar1,a2) in A

    where I is the indicator function for the set A (I(x)=0 if x is in A, +%inf otherwise).

    m = size(A, 2);
    Nx = size(x, 1);
    Nxstar = size(xstar, 1);
    N = Nx * Nxstar;
]]></programlisting>-->
				<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="op_ImagesOfEquationsByLatex/op_fitz/op_fitz_1.png"/>
					</imageobject>
				</inlinemediaobject>
				</para>
      
         </para>
    
   </refsection>

  
   <refsection><title>Examples</title><programlisting role="example"><![CDATA[
a = -2:2;
astar = 3*a - a + 3;
x = -3:3;
xstar = -3:3;
F1 = op_fitz([a; astar], 1, x, xstar);
F2 = op_fitz([a; astar], 2, x, xstar);
F3 = op_fitz([a; astar], 3, x, xstar);

clf(); alpha=60; theta=-60;
subplot(131); plot3d(x, xstar, F1, alpha=alpha, theta=theta);
subplot(132); plot3d(x, xstar, F2, alpha=alpha, theta=theta);
subplot(133); plot3d(x, xstar, F3, alpha=alpha, theta=theta);
  ]]></programlisting></refsection>

  
   <refsection><title>See Also</title><simplelist type="inline">
    
      <member>
         <link linkend="op_fitz_brute">op_fitz_brute</link>
      </member>
    
      <member>
         <link linkend="op_fitz_direct">op_fitz_direct</link>
      </member>
    
      <member>
         <link linkend="op_fitzinf">op_fitzinf</link>
      </member>
    
      <member>
         <link linkend="plq_fitzinf0">plq_fitzinf0</link>
      </member>
    
      <member>
         <link linkend="plq_fitzinf0_direct">plq_fitzinf0_direct</link>
      </member>
    
      <member>
         <link linkend="plq_rock">plq_rock</link>
      </member>
  
   </simplelist></refsection>

  
   <refsection><title>Authors</title><para>
    
      <literal>Bryan Gardiner</literal>, University of British Columbia, BC, Canada
  </para></refsection>

</refentry>