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<?xml version="1.0" encoding="UTF-8"?>
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns5="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="cshep2d" xml:lang="en">
<refnamediv>
<refname>cshep2d</refname>
<refpurpose>bidimensional cubic shepard (scattered)
interpolation
</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>tl_coef = cshep2d(xyz)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>xyz</term>
<listitem>
<para>a n x 3 matrix of the (no gridded) interpolation points (the i
th row given the (x,y) coordinates then the altitude z of the i th
interpolation point)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>tl_coef</term>
<listitem>
<para>a tlist scilab structure (of type cshep2d)</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>This function is useful to define a 2d interpolation function when
the interpolation points are not on a grid (you may use it in this case
but <link linkend="splin2d">splin2d</link> is better for that purpose).
The interpolant is a cubic shepard one and is a C2 (twice continuously
differentiable) bivariate function <emphasis>s(x,y)</emphasis> such that :
<emphasis>s(xi,yi)=zi</emphasis> for all <emphasis>i=1,..,n</emphasis>
(<emphasis>(xi,yi,zi)</emphasis> being the i th row of
<literal>xyz</literal>).
</para>
<para>
The evaluation of <emphasis>s</emphasis> at some points must be done
by the <link linkend="eval_cshep2d">eval_cshep2d</link> function.
</para>
</refsection>
<refsection>
<title>Remark</title>
<para>
The function works if <emphasis role="bold">n>= 10</emphasis>,
if the nodes are not all colinears (i.e. the <emphasis>(x,y)</emphasis>
coordinates of the interpolation points are not on the same straight
line), and if there is no duplicate nodes (i.e. 2 or more interpolation
points with the same <emphasis>(x,y)</emphasis> coordinates). An error is
issued if these conditions are not respected.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
// interpolation of cos(x)cos(y) with randomly chosen interpolation points
n = 150; // nb of interpolation points
xy = grand(n,2,"unf",0,2*%pi);
z = cos(xy(:,1)).*cos(xy(:,2));
xyz = [xy z];
tl_coef = cshep2d(xyz);
// evaluation on a grid
m = 30;
xx = linspace(0,2*%pi,m);
[X,Y] = ndgrid(xx,xx);
Z = eval_cshep2d(X,Y, tl_coef);
clf()
plot3d(xx,xx,Z,flag=[2 6 4])
param3d1(xy(:,1),xy(:,2),list(z,-9), flag=[0 0])
xtitle("Cubic Shepard Interpolation of cos(x)cos(y) with randomly chosen interpolation points")
legends("interpolation points",-9,1)
show_window()
]]></programlisting>
<scilab:image>
n = 150; // nb of interpolation points
xy = grand(n,2,"unf",0,2*%pi);
z = cos(xy(:,1)).*cos(xy(:,2));
xyz = [xy z];
tl_coef = cshep2d(xyz);
// evaluation on a grid
m = 30;
xx = linspace(0,2*%pi,m);
[X,Y] = ndgrid(xx,xx);
Z = eval_cshep2d(X,Y, tl_coef);
clf()
plot3d(xx,xx,Z,flag=[2 6 4])
param3d1(xy(:,1),xy(:,2),list(z,-9), flag=[0 0])
xtitle("Cubic Shepard Interpolation of cos(x)cos(y) with randomly chosen interpolation points")
legends("interpolation points",-9,1)
show_window()
</scilab:image>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="splin2d">splin2d</link>
</member>
<member>
<link linkend="eval_cshep2d">eval_cshep2d</link>
</member>
</simplelist>
</refsection>
<refsection>
<title>History</title>
<revhistory>
<revision>
<revnumber>5.4.0</revnumber>
<revremark>previously, imaginary part of input arguments were implicitly ignored.</revremark>
</revision>
</revhistory>
</refsection>
</refentry>
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