<?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="ndgrid" xml:lang="en">
+
<refnamediv>
+
<refname>ndgrid</refname>
- <refpurpose>arrays for multidimensional function evaluation on
- grid
+
+ <refpurpose>build matrices or N-D arrays by replicating some template vectors
+
</refpurpose>
+
</refnamediv>
+
<refsynopsisdiv>
+
<title>Calling Sequence</title>
+
<synopsis>[X, Y] = ndgrid(x,y)
+
[X, Y, Z] = ndgrid(x,y,z)
+
[X, Y, Z, T] = ndgrid(x,y,z,t)
+
[X1, X2, ..., Xm] = ndgrid(x1,x2,...,xm)
+
</synopsis>
+
</refsynopsisdiv>
- <refsection>
+
+ <refsection role="arguments">
+
<title>Arguments</title>
+
<variablelist>
+
<varlistentry>
+
<term>x, y, z, ...</term>
+
<listitem>
- <para>vectors</para>
+
+ <para>vectors of any data types. They may have distinct data types.</para>
+
</listitem>
+
</varlistentry>
+
<varlistentry>
+
<term>X, Y, Z, ...</term>
+
<listitem>
- <para>matrices in case of 2 input arguments, or else
- hypermatrices
+
+ <para>matrices in case of 2 input arguments, or hypermatrices otherwise.
+
+ They all have the same sizes: size(X,"*") rows, size(Y,"*") columns,
+
+ size(Z,"*") layers, etc.
+
+ They have the datatypes of respective input vectors:
+
+ <literal>typeof(X)==typeof(x)</literal>,
+
+ <literal>typeof(Y)==typeof(y)</literal>, etc.
+
</para>
+
</listitem>
+
</varlistentry>
+
</variablelist>
+
</refsection>
- <refsection>
+
+ <refsection role="description">
+
<title>Description</title>
- <para>This is an utility routine useful to create arrays for function
- evaluation on 2, 3, ..., n dimensional grids. For instance in 2d, a grid
- is defined by two vectors, <literal>x</literal> and <literal> y</literal>
- of length nx and ny, and you want to evaluate a function (says
- <emphasis>f</emphasis>) on all the grid points, that is on all the points
- of coordinates <emphasis>(x(i),y(j))</emphasis> with
- <emphasis>i=1,..,nx</emphasis> and <emphasis>j=1,..,ny</emphasis>. In this
- case, this function can compute the two matrices <literal>X,Y</literal> of
- size <emphasis>nx x ny</emphasis> such that :
+
+ <para>
+ The first application of <function>ndgrid</function> is to build
+
+ a grid of nodes meshing the 2D or 3D or N-D space according to 2, 3,
+
+ or more sets
+
+ <literal>x</literal>, <literal> y</literal>, etc.. of
+
+ "template" coordinates sampled along each direction/dimension of the
+
+ space that you want to mesh.
+
</para>
- <programlisting role="no-scilab-exec"><![CDATA[
-X(i,j) = x(i) for all i in [1,nx]
-Y(i,j) = y(j) and j in [1,ny]
- ]]></programlisting>
+
<para>
- and the evaluation may be done with <literal>Z=f(X,Y)</literal> (at
- the condition that you have coded <literal>f</literal> for evaluation on
- vector arguments, which is done (in general) by using the element-wise
- operators <literal>.*</literal>, <literal>./</literal> and
- <literal>.^</literal> in place of <literal>*</literal>,
- <literal>/</literal> and <literal>^</literal>).
+ Hence, the matrix or hypermatrix <literal>X</literal> is made
+
+ by replicating the vector <literal>x</literal> as all its columns ;
+
+ the matrix or hypermatrix <literal>Y</literal> is made
+
+ by replicating the vector <literal>y</literal> as all its rows ;
+
+ <literal>Z</literal> is made of replicating the vector
+
+ <literal>z</literal> along all its local thicknesses (3rd dimension);
+
+ etc
+
</para>
+
+ <screen>
+
+ <![CDATA[--> [X, Y] = ndgrid([1 3 4], [0 2 4 6])\r
+ X = \r
+ 1. 1. 1. 1.\r
+ 3. 3. 3. 3.\r
+ 4. 4. 4. 4.\r
+\r
+ Y = \r
+ 0. 2. 4. 6.\r
+ 0. 2. 4. 6.\r
+ 0. 2. 4. 6.\r
+]]>
+ </screen>
+
+ <para>
+
+ Then, the coordinates of the node(i,j) in the 2D space
+
+ will be
+
+ simply <literal>[x(i), y(j)]</literal> now given by
+
+ <literal>[X(i,j), Y(i,j)]</literal>. As well, the coordinates of a
+
+ <literal>node(i,j,k)</literal> of a 3D grid will be
+
+ <literal>[x(i), y(j), z(k)]</literal> now given by
+
+ <literal>[X(i,j,k), Y(i,j,k), Z(i,j,k)]</literal>.
+
+ </para>
+
<para>
- In the 3d case, considering 3 vectors <literal>x,y,z</literal> of
- length nx, ny and nz, <literal>X,Y,Z</literal> are 3 hypermatrices of size
- <emphasis>nx x ny x nz</emphasis> such that :
+
+ This replication scheme can be generalized to any number of dimensions,
+
+ as well to any type of uniform data. Let's for instance consider 2
+
+ attributes:
+
+ <orderedlist>
+
+ <listitem>The first is a number, to be chosen from the vector say
+
+ <literal>n = [ 3 7 ]</literal>
+
+ </listitem>
+
+ <listitem>The second is a letter, to be chosen from the vector
+
+ say <literal>c = ["a" "e" "i" "o" "u" "y"]</literal>
+
+ </listitem>
+
+ </orderedlist>
+
+ Then we want to build the set of all {n,c} possible pairs. It will
+
+ just be the 2D grid:
+
</para>
- <programlisting role="no-scilab-exec"><![CDATA[
-X(i,j,k) = x(i)
-Y(i,j,k) = y(j) for all (i,j,k) in [1,nx]x[1,ny]x[1,nz]
-Z(i,j,k) = z(k)
- ]]></programlisting>
+
+ <screen>
+
+ <![CDATA[--> [N, C] = ndgrid([3 7],["a" "e" "i" "o" "u" "y"])\r
+ C = \r
+!a e i o u y !\r
+!a e i o u y !\r
+\r
+ N = \r
+ 3. 3. 3. 3. 3. 3.\r
+ 7. 7. 7. 7. 7. 7.\r
+]]>
+ </screen>
+
+ <para>Then, the object(i,j) will have the properties
+
+ <literal>{n(i) c(j)}</literal> that now can be addressed with
+
+ <literal>{N(i,j) C(i,j)}</literal>.
+
+ This kind of grid may be useful to initialize an array of structures.
+
+ </para>
+
<para>
- In the general case of m input arguments <literal>x1, x2, ..,
- xm
- </literal>
- ,then the m output arguments <literal>X1, X2, ..,
- Xm
- </literal>
- are hypermatrices of size <emphasis>nx1 x nx2 x ... x
- nxm
- </emphasis>
- and :
+ Following examples show how to use <varname>X, Y, Z</varname> in
+
+ most frequent applications.
+
</para>
- <programlisting role="no-scilab-exec"><![CDATA[
-Xj(i1,i2,...,ij,...,im) = xj(ij)
-for all (i1,i2,...,im) in [1,nx1]x[1,nx2]x...x[1,nxm]
- ]]></programlisting>
+
+
</refsection>
- <refsection>
+
+ <refsection role="examples">
+
<title>Examples</title>
- <programlisting role="example"><![CDATA[
-// create a simple 2d grid
-nx = 40; ny = 40;
-x = linspace(-1,1,nx);
-y = linspace(-1,1,ny);
-[X,Y] = ndgrid(x,y);
-
-// compute a function on the grid and plot it
-//deff("z=f(x,y)","z=128*x.^2 .*(1-x).^2 .*y.^2 .*(1-y).^2");
-deff("z=f(x,y)","z=x.^2 + y.^3")
-Z = f(X,Y);
-clf()
-plot3d(x,y,Z, flag=[2 6 4]); show_window()
+
+ <para>
+ <emphasis role="bold">Example #1:</emphasis>
+ </para>
+
+ <programlisting role="example"><![CDATA[ \r
+// Create a simple 2d grid\r
+x = linspace(-10,2,40);\r
+y = linspace(-5,5,40);\r
+[X,Y] = ndgrid(x,y);\r
+\r
+// Compute ordinates Z(X,Y) on the {X, Y} grid and plot Z(X,Y)\r
+Z = X - 3*X.*sin(X).*cos(Y-4) ;\r
+clf()\r
+plot3d(x,y,Z, flag=[color("green") 2 4], alpha=7, theta=60); show_window()\r
]]></programlisting>
+
<scilab:image>
- nx = 40; ny = 40;
- x = linspace(-1,1,nx);
- y = linspace(-1,1,ny);
+
+ x = linspace(-10,2,40);
+
+ y = linspace(-5,5,40);
+
[X,Y] = ndgrid(x,y);
- deff("z=f(x,y)","z=x.^2 + y.^3")
- Z = f(X,Y);
- plot3d(x,y,Z, flag=[2 6 4]);
+
+ Z = X - 3*X.*sin(X).*cos(Y-4) ;
+
+ clf()
+
+ plot3d(x,y,Z, flag=[color("green") 2 4], alpha=7, theta=60); show_window()
+
</scilab:image>
- <programlisting role="example"><![CDATA[
-// create a simple 3d grid
-nx = 10; ny = 6; nz = 4;
-x = linspace(0,2,nx);
-y = linspace(0,1,ny);
-z = linspace(0,0.5,nz);
-[X,Y,Z] = ndgrid(x,y,z);
-
-// try to display this 3d grid ...
-XF=[]; YF=[]; ZF=[];
-
-for k=1:nz
- [xf,yf,zf] = nf3d(X(:,:,k),Y(:,:,k),Z(:,:,k));
- XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];
-end
-
-for j=1:ny
- [xf,yf,zf] = nf3d(matrix(X(:,j,:),[nx,nz]),...
- matrix(Y(:,j,:),[nx,nz]),...
- matrix(Z(:,j,:),[nx,nz]));
- XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];
-end
-
-clf()
-plot3d(XF,YF,ZF, flag=[0 6 3], leg="X@Y@Z")
-xtitle("A 3d grid !"); show_window()
+
+ <para>
+ <emphasis role="bold">Example #2:</emphasis>
+ </para>
+
+ <programlisting role="example"><![CDATA[ \r
+// Create a simple 3d grid\r
+nx = 10; ny = 6; nz = 4;\r
+x = linspace(0,2,nx);\r
+y = linspace(0,1,ny);\r
+z = linspace(0,0.5,nz);\r
+[X,Y,Z] = ndgrid(x,y,z);\r
+\r
+// Try to display this 3d grid ...\r
+XF=[]; YF=[]; ZF=[];\r
+\r
+for k=1:nz\r
+ [xf,yf,zf] = nf3d(X(:,:,k),Y(:,:,k),Z(:,:,k));\r
+ XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];\r
+end\r
+\r
+for j=1:ny\r
+ [xf,yf,zf] = nf3d(matrix(X(:,j,:),[nx,nz]),...\r
+ matrix(Y(:,j,:),[nx,nz]),...\r
+ matrix(Z(:,j,:),[nx,nz]));\r
+ XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];\r
+end\r
+\r
+clf()\r
+plot3d(XF,YF,ZF, flag=[0 6 3], 66, 61,leg="X@Y@Z")\r
+xtitle("A 3d grid !"); show_window()\r
]]></programlisting>
+
<scilab:image>
+
nx = 10; ny = 6; nz = 4;
+
x = linspace(0,2,nx);
+
y = linspace(0,1,ny);
+
z = linspace(0,0.5,nz);
+
[X,Y,Z] = ndgrid(x,y,z);
+
+
XF=[]; YF=[]; ZF=[];
+
+
for k=1:nz
+
[xf,yf,zf] = nf3d(X(:,:,k),Y(:,:,k),Z(:,:,k));
+
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];
+
end
+
+
for j=1:ny
+
[xf,yf,zf] = nf3d(matrix(X(:,j,:),[nx,nz]),...
+
matrix(Y(:,j,:),[nx,nz]),...
+
matrix(Z(:,j,:),[nx,nz]));
+
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];
+
end
- plot3d(XF,YF,ZF, flag=[0 6 3], leg="X@Y@Z")
+
+ plot3d(XF,YF,ZF, flag=[0 6 3], 66, 61, leg="X@Y@Z")
+
xtitle("A 3d grid !");
+
</scilab:image>
+
+
+
+ <para>
+ <emphasis role="bold">Example #3: Creates a table of digrams:</emphasis>
+ </para>
+
+ <programlisting role="example"><![CDATA[ \r
+[c1, c2] = ndgrid(["a" "b" "c"], ["a" "b" "c" "d" "e" "f" "g" "h"])\r
+c1+c2\r
+ ]]></programlisting>
+
+ <screen>
+
+ <![CDATA[--> [c1, c2] = ndgrid(["a" "b" "c"], ["a" "b" "c" "d" "e" "f" "g" "h"])\r
+ c2 = \r
+!a b c d e f g h !\r
+!a b c d e f g h !\r
+!a b c d e f g h !\r
+\r
+ c1 = \r
+!a a a a a a a a !\r
+!b b b b b b b b !\r
+!c c c c c c c c !\r
+\r
+--> c1+c2\r
+ ans =\r
+!aa ab ac ad ae af ag ah !\r
+!ba bb bc bd be bf bg bh !\r
+!ca cb cc cd ce cf cg ch !\r
+]]>
+ </screen>
+
</refsection>
+
<refsection role="see also">
+
<title>See Also</title>
+
<simplelist type="inline">
+
+ <member>
+
+ <link linkend="meshgrid">meshgrid</link>
+
+ </member>
+
<member>
+
<link linkend="kron">kron</link>
+
</member>
+
+ <member>
+
+ <link linkend="feval">feval</link>
+
+ </member>
+
+ <member>
+
+ <link linkend="eval3d">eval3d</link>
+
+ </member>
+
+ <member>
+
+ <link linkend="nf3d">nf3d</link>
+
+ </member>
+
</simplelist>
+
</refsection>
+
+ <refsection role="history">
+
+ <title>History</title>
+
+ <revhistory>
+
+ <revision>
+
+ <revnumber>6.0</revnumber>
+
+ <revdescription>Extension to all homogeneous datatypes ([],
+
+ booleans, encoded integers, polynomials, rationals, strings).
+
+ Revision of the help page.
+
+ </revdescription>
+
+ </revision>
+
+ </revhistory>
+
+ </refsection>
+
</refentry>
+
projects / scilab.git / blobdiff