/**
* This method straightforwardly applies the standard definition of the numerical estimates of the
* second order partial derivatives. See for example Section 5.7 of Numerical Recipes in C.
*/
public double secondPartialDerivative(FittingFunction f, int i, int j, double[] p, double delt) {
double[] arg = new double[p.length];
System.arraycopy(p, 0, arg, 0, p.length);
double center = f.evaluate(arg);
arg[i] += delt;
arg[j] += delt;
double ff1 = f.evaluate(arg);
arg[j] -= 2 * delt;
double ff2 = f.evaluate(arg);
arg[i] -= 2 * delt;
arg[j] += 2 * delt;
double ff3 = f.evaluate(arg);
arg[j] -= 2 * delt;
double ff4 = f.evaluate(arg);
if (Double.isNaN(ff1)) {
ff1 = center;
}
if (Double.isNaN(ff2)) {
ff2 = center;
}
if (Double.isNaN(ff3)) {
ff3 = center;
}
if (Double.isNaN(ff4)) {
ff4 = center;
}
double fsSum = ff1 - ff2 - ff3 + ff4;
return fsSum / (4.0 * delt * delt);
}
/**
* This method implements Ridder's algorithm for computing the second order partial derivatives.
* It is a translation of the C program in section 5.7 of Numerical Recipes in C. It is more
* robust than the above method in that it searches for a perferred value of delt. But based on
* our experience to date with SEM fitting functions, the above method seems to be adequately
* accurate and faster that this one.
*/
public double secondPartialDerivativeRidr(
FittingFunction f, int i, int j, double[] args, double delt) {
double[] arg = new double[args.length];
double[][] a = new double[NTAB][NTAB];
double hh = delt;
double errt;
double ans = 0.0;
double fac;
System.arraycopy(args, 0, arg, 0, args.length);
double center = f.evaluate(arg);
arg[i] += delt;
arg[j] += delt;
double ff1 = f.evaluate(arg);
arg[j] -= 2 * delt;
double ff2 = f.evaluate(arg);
arg[i] -= 2 * delt;
arg[j] += 2 * delt;
double ff3 = f.evaluate(arg);
arg[j] -= 2 * delt;
double ff4 = f.evaluate(arg);
if (Double.isNaN(ff1)) {
ff1 = center;
}
if (Double.isNaN(ff2)) {
ff2 = center;
}
if (Double.isNaN(ff3)) {
ff3 = center;
}
if (Double.isNaN(ff4)) {
ff4 = center;
}
a[0][0] = (ff1 - ff2 - ff3 + ff4) / (4.0 * delt * delt);
double err = BIG;
for (int ii = 1; ii < NTAB; ii++) {
hh /= CON;
System.arraycopy(args, 0, arg, 0, args.length);
arg[i] += hh;
arg[j] += hh;
ff1 = f.evaluate(arg);
arg[j] -= 2 * hh;
ff2 = f.evaluate(arg);
arg[i] -= 2 * hh;
arg[j] += 2 * hh;
ff3 = f.evaluate(arg);
arg[j] -= 2 * hh;
ff4 = f.evaluate(arg);
if (Double.isNaN(ff1)) {
ff1 = center;
}
if (Double.isNaN(ff2)) {
ff2 = center;
}
if (Double.isNaN(ff3)) {
ff3 = center;
}
if (Double.isNaN(ff4)) {
ff4 = center;
}
a[0][ii] = (ff1 - ff2 - ff3 + ff4) / (4.0 * hh * hh);
fac = CON2;
for (int jj = 1; jj < ii; jj++) {
a[jj][ii] = (a[jj - 1][ii] * fac - a[jj - 1][ii - 1]) / (fac - 1.0);
fac = CON2 * fac;
errt =
Math.max(Math.abs(a[jj][ii] - a[jj - 1][ii]), Math.abs(a[jj][ii] - a[jj - 1][ii - 1]));
if (errt < err) {
err = errt;
ans = a[jj][ii];
}
}
if (Math.abs(a[ii][ii] - a[ii - 1][ii - 1]) >= SAFE * err) {
break;
}
}
return ans;
}
