# 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()

nx = 40; ny = 40; x = linspace(-1,1,nx); y = linspace(-1,1,ny); [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]);

    // 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()

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") xtitle("A 3d grid !");

## See Also

- kron