<?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="pl_me_plt"><info><pubdate>02-May-2006</pubdate></info><refnamediv><refname>pl_me_plt</refname><refpurpose>Parametric Moreau Envelope, PME algorithm</refpurpose></refnamediv>
  
   
  
   
  
   
  
   
  
   

  
   <refsynopsisdiv><title>Calling Sequence</title><synopsis>[M, z] = pl_me_plt(X, f, df)</synopsis></refsynopsisdiv>

  
   <refsection><title>Parameters</title>
  
      <variablelist>

    
         <varlistentry>
    
            <term>X</term>
    
            <listitem>
       
               <para>column vector. A grid of points on which the function is sampled.
       </para>
    
            </listitem>
    
         </varlistentry>

    
         <varlistentry>
    
            <term>f</term>
    
            <listitem>
       
               <para>column vector. The value of the function on the grid X: usually Y(i)=f(X(i)) for some function f.
       </para>
    
            </listitem>
    
         </varlistentry>

    
         <varlistentry>
    
            <term>df</term>
    
            <listitem>
       
               <para>column vector. The value of the derivative of the function f evaluated on X, which is also the slope.
       </para>
    
            </listitem>
    
         </varlistentry>

  
      </variablelist>
  
   </refsection>
 
  
   <refsection><title>Description</title>
     
         <para>
	Compute numerically the discrete Moreau envelope of a set of planar points (X(i),f(i)) at slopes S(j), i.e. <!--<programlisting><![CDATA[
                                          2
       M(j) = min f(i) + || s(j) - x(i) ||.
               i
]]></programlisting>-->
			<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="pl_ImagesOfEquationsByLatex/pl_me_plt/pl_me_plt_1.png"/>
					</imageobject>
				</inlinemediaobject>
			</para>
	<para>It reduces computation to computing the Legendre conjugate through the formula</para> <!--<programlisting><![CDATA[
           2                                                      2
M(j) = s(j) - 2 g*(j) with g*(j) = max [ s(j) * x(i) - 1/2 * (x(i) + f(i)) ],
                                    i
where the conjugate is derived by the Parametric Legendre Transform.
]]></programlisting>-->
			<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="pl_ImagesOfEquationsByLatex/pl_me_plt/pl_me_plt_2.png"/>
					</imageobject>
				</inlinemediaobject>
			</para>
	<para>Here the conjugate is derived by the Parametric Legendre Transform.</para>
         </para>
     
   </refsection>

  
   <refsection><title>Examples</title><programlisting role="example"><![CDATA[
    function y = f(x), y=0.5 * x.^2, endfunction;
    function s = df(x), s= x, endfunction;
    b=%T;n=8;
    X=(-n/2:1:n/2)';S=df(X);Y=f(X);
    [M,z] = pl_me_plt(X, f, df);
  ]]></programlisting></refsection>

  
   <refsection><title>See Also</title><simplelist type="inline">
    
      <member> 
         <link linkend="pl_me_direct">pl_me_direct</link> 
      </member>
    
      <member> 
         <link linkend="pl_me_llt">pl_me_llt</link> 
      </member>
    
      <member> 
         <link linkend="pl_me_nep">pl_me_nep</link> 
      </member>
    
      <member> 
         <link linkend="pl_me_pe">pl_me_pe</link> 
      </member>
  
   </simplelist></refsection>

  
   <refsection><title>Authors</title><para>
	  
      <literal>Yves Lucet</literal>, University of British Columbia, BC, Canada
  </para></refsection>

</refentry>
