eval_cshep2d bidimensional cubic shepard interpolation evaluation

## Syntax

            zp                                            = eval_cshep2d(xp, yp, tl_coef)
            [zp, dzpdx, dzpdy]                            = eval_cshep2d(xp, yp, tl_coef)
            [zp, dzpdx, dzpdy, d2zpdxx, d2zpdyy, d2zpdxy] = eval_cshep2d(xp, yp, tl_coef)

## Arguments

xp, yp

  two real vectors (or matrices) of the same size

tl_coef

  a tlist scilab structure (of type cshep2d) defining a cubic Shepard interpolation function (named `S` in the following)

zp

  vector (or matrix) of the same size than `xp` and `yp`, evaluation of the interpolant `S` at these points

dzpdx,dzpdy

  vectors (or matrices) of the same size than `xp` and `yp`, evaluation of the first derivatives of `S` at these points

d2zpdxx,d2zpdxy,d2zpdyy

  vectors (or matrices) of the same size than `xp` and `yp`, evaluation of the second derivatives of `S` at these points

## Description

This is the evaluation routine for cubic Shepard interpolation function computed with cshep2d, that is :

## Remark

The interpolant S is C2 (twice continuously differentiable) but is also extended by zero for (x,y) far enough the interpolation points. This leads to a discontinuity in a region far outside the interpolation points, and so, is not cumbersome in practice (in a general manner, evaluation outside interpolation points (i.e. extrapolation) leads to very inaccurate results).

## Examples

    // see example section of cshep2d

    // this example shows the behavior far from the interpolation points ...
    deff("z=f(x,y)","z = 1+ 50*(x.*(1-x).*y.*(1-y)).^2")
    x = linspace(0,1,10);
    [X,Y] = ndgrid(x,x);
    X = X(:); Y = Y(:); Z = f(X,Y);
    S = cshep2d([X Y Z]);
    // evaluation inside and outside the square [0,1]x[0,1]
    m = 40;
    xx = linspace(-1.5,0.5,m);
    [xp,yp] = ndgrid(xx,xx);
    zp = eval_cshep2d(xp,yp,S);
    // compute facet (to draw one color for extrapolation region
    // and another one for the interpolation region)
    [xf,yf,zf] = genfac3d(xx,xx,zp);
    colors = 2*ones(1,size(zf,2));
    // indices corresponding to facet in the interpolation region
    ind=find( mean(xf,"r")>0 & mean(xf,"r")<1 & mean(yf,"r")>0 & mean(yf,"r")<1 );
    colors(ind)=3;
    clf();
    plot3d(xf,yf,list(zf,colors), flag=[2 6 4])
    legends(["extrapolation region","interpolation region"],[2 3],1)
    show_window()

deff("z=f(x,y)","z = 1+ 50*(x.*(1-x).*y.*(1-y)).^2") x = linspace(0,1,10); [X,Y] = ndgrid(x,x); X = X(:); Y = Y(:); Z = f(X,Y); S = cshep2d([X Y Z]); // evaluation inside and outside the square [0,1]x[0,1] m = 40; xx = linspace(-1.5,0.5,m); [xp,yp] = ndgrid(xx,xx); zp = eval_cshep2d(xp,yp,S); // compute facet (to draw one color for extrapolation region // and another one for the interpolation region) [xf,yf,zf] = genfac3d(xx,xx,zp); colors = 2*ones(1,size(zf,2)); // indices corresponding to facet in the interpolation region ind=find( mean(xf,"r")>0 & mean(xf,"r")<1 & mean(yf,"r")>0 & mean(yf,"r")<1 ); colors(ind)=3; clf(); plot3d(xf,yf,list(zf,colors), flag=[2 6 4]) legends(["extrapolation region","interpolation region"],[2 3],1) show_window()

## See also

- cshep2d

## History

5.4.0 previously, imaginary part of input arguments were implicitly ignored.