ndgrid - arrays for multidimensional function evaluation on grid
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]
// 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); xbasc() plot3d(x,y,Z, flag=[2 6 4]); xselect() // 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 xbasc() plot3d(XF,YF,ZF, flag=[0 6 3], leg="X@Y@Z") xtitle("A 3d grid !"); xselect()
kron ,
B. Pincon