ndgrid
======

arrays for multidimensional function evaluation on grid



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [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)




Arguments
~~~~~~~~~

:x, y, z, ... vectors
: :X, Y, Z, ... matrices in case of 2 input arguments, or else
  hypermatrices
:



Description
~~~~~~~~~~~

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, `x` and `y` of length nx and ny, and
you want to evaluate a function (says *f*) on all the grid points,
that is on all the points of coordinates *(x(i),y(j))* with
*i=1,..,nx* and *j=1,..,ny*. In this case, this function can compute
the two matrices `X,Y` of size *nx x ny* such that :


::

    X(i,j) = x(i)   for all i in [1,nx]
    Y(i,j) = y(j)       `and`_ j in [1,ny]


and the evaluation may be done with `Z=f(X,Y)` (at the condition that
you have coded `f` for evaluation on vector arguments, which is done
(in general) by using the element-wise operators `.*`, `./` and `.^`
in place of `*`, `/` and `^`).

In the 3d case, considering 3 vectors `x,y,z` of length nx, ny and nz,
`X,Y,Z` are 3 hypermatrices of size *nx x ny x nz* such that :


::

    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)


In the general case of m input arguments `x1, x2, .., xm` ,then the m
output arguments `X1, X2, .., Xm` are hypermatrices of size *nx1 x nx2
x ... x nxm* and :


::

    Xj(i1,i2,...,ij,...,im) = xj(ij)   
    for all (i1,i2,...,im) in [1,nx1]x[1,nx2]x...x[1,nxm]




Examples
~~~~~~~~


::

    // 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`_()
    
    // 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`_()




See Also
~~~~~~~~


+ `kron`_ Kronecker product (.*.)


.. _kron: kron.html