使用ALGLIB进行双三次样条插值时如何计算二阶导数？

Question

我使用这些代码进行双三次样条插值，并使用函数

`spline2ddiff(s, vx, vy, v, dx, dy, dxy);`

可以得到一阶导数和混合的二阶导数，如何得到二阶导数？

#include "stdafx.h"

#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
#define N 4

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled
    // at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
    //
    int i,j,k;
    real_1d_array x ;
    x.setlength(N);
    real_1d_array y ;
    y.setlength(N);
    real_1d_array f ;
    f.setlength(N*N);
    double h = 2.0/(N-1);
    for (i = 0; i<N; i++) {
        x[i] = -1.0+i*h;
        y[i] = -1.0+i*h;
    }
    //real_1d_array y = "[0.0, 1.0]";
    for (i=0; i<N; i++) {
        for (j=0; j<N; j++) {
            //f[i*N+j] = 4*pow((double)x[j],2)+3*pow((double)y[i],2);
            f[i*N+j] = 1+2*y[i]+3*y[i]*y[i]+4*y[i]*y[i]*y[i]+5*x[j]+6*x[j]*y[i]\
            +7*x[j]*y[i]*y[i]+8*x[j]*y[i]*y[i]*y[i]+9*x[j]*x[j]+10*x[j]*x[j]*y[i]+11*x[j]*x[j]*y[i]*y[i]+12*x[j]*x[j]*y[i]*y[i]*y[i]+13*x[j]*x[j]*x[j]+14*x[j]*x[j]*x[j]*y[i]+15*x[j]*x[j]*x[j]*y[i]*y[i]+16*x[j]*x[j]*x[j]*y[i]*y[i]*y[i];
        }
    }
    real_2d_array c;
    real_1d_array c1;
    c1.setlength(20);
    ae_int_t m;
    ae_int_t n;
    ae_int_t d;
    double vx = 0.25;
    double vy = 0.50;
    double v;
    double dx;
    double dy;
    double dxy;
    spline2dinterpolant s;
    // build spline
    spline2dbuildbicubicv(x, N, y, N, f, 1, s);
    // unpack and test
    spline2dunpackv(s, m, n, d, c);
    for (i=0; i<20; i++) {
        c1[i] = c[0][i];
    }
    printf("%s\n", c.tostring(0).c_str()); // EXPECTED: [[0, 1, 0, 1, 0,2,0,0, 1,3,0,0, 0,0,0,0, 0,0,0,0 ]]
    // calculate S(0.25,0.50)
    v = spline2dcalc(s, vx, vy);
    printf("%.10f\n", double(v)); // EXPECTED: 1.0625

    // calculate derivatives
    spline2ddiff(s, vx, vy, v, dx, dy, dxy);
    printf("%.10f\n", double(v)); // EXPECTED: 1.0625
    printf("%.10f\n", double(dx)); // EXPECTED: 0.5000
    printf("%.10f\n", double(dy)); // EXPECTED: 2.0000
    printf("%.10f\n", double(dxy));
    return 0;
}

我不知道是否有某些功能可以直接执行此操作？而且我不知道返回的系数均值，双三次样条插值应该有16个系数，但是有20个。