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// Copyright (c) 1998-2014
// Distributed under the Boost Software License, Version 1.0.
// http://www.boost.org/LICENSE_1_0.txt
// http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
//
// File Version: 5.0.1 (2010/10/01)
#ifndef WM5APPRPOLYFIT4_H
#define WM5APPRPOLYFIT4_H
#include "Wm5MathematicsLIB.h"
namespace Wm5
{
// The samples are (x[i],y[i],z[i],w[i]) for 0 <= i < S. Think of w as a
// function of x, y, and z, say w = f(x,y,z). The function fits the samples
// with a polynomial of degree d0 in x, degree d1 in y, and degree d2 in z,
// say
// w = sum_{i=0}^{d0} sum_{j=0}^{d1} sum_{k=0}^{d2} c[i][j][k]*x^i*y^j*z^k
// The method is a least-squares fitting algorithm. The return array stores
// the c[i][j][k] values according to
// returned[i+(d0+1)*(j+(d1+1)*k)] = c[i][j][k]
// for a total of (d0+1)*(d1+1)*(d2+1) coefficients. The caller is
// responsible for deleting the input arrays if they were dynamically
// allocated. The caller is also responsible for deleting the returned array.
//
// WARNING. The fitting algorithm for polynomial terms
// (1,x,x^2,...,x^d0), (1,y,y^2,...,y^d1), (1,z,z^2,...,z^d2)
// is known to be nonrobust for large degrees and for large magnitude data.
// One alternative is to use orthogonal polynomials
// (f[0](x),...,f[d0](x)), (g[0](y),...,g[d1](y)), (h[0](z),...,h[d2](z))
// and apply the least-squares algorithm to these. Another alternative is to
// transform
// (x',y',z',w') = ((x-xcen)/rng, (y-ycen)/rng, (z-zcen)/rng, w/rng)
// where xmin = min(x[i]), xmax = max(x[i]), xcen = (xmin+xmax)/2,
// ymin = min(y[i]), ymax = max(y[i]), ycen = (ymin+ymax)/2, zmin = min(z[i]),
// zmax = max(z[i]), zcen = (zmin+zmax)/2, and
// rng = max(xmax-xmin,ymax-ymin,zmax-zmin). Fit the (x',y',z',w') points,
// w' = sum_{i=0}^{d0} sum_{j=0}^{d1} sum_{k=0}^{d2} c'[i][j][k] *
// (x')^i*(y')^j*(z')^k
// The original polynomial is evaluated as
// w = rng * sum_{i=0}^{d0} sum_{j=0}^{d1} sum_{k=0}^{d2} c'[i][j][k] *
// ((x-xcen)/rng)^i * ((y-ycen)/rng)^j * ((z-zcen)/rng)^k
template <typename Real> WM5_MATHEMATICS_ITEM
Real* PolyFit4 (int numSamples, const Real* xSamples, const Real* ySamples,
const Real* zSamples, const Real* wSamples, int xDegree, int yDegree,
int zDegree);
}
#endif
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