text: // Geometric Tools, LLC
  // Copyright (c) 1998-2011
  // 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 WM5APPRPOLYFIT3_H
  #define WM5APPRPOLYFIT3_H
  
  #include "Wm5MathematicsLIB.h"
  
  namespace Wm5
  {
  
  // The samples are (x[i],y[i],w[i]) for 0 <= i < S.  Think of w as a function
  // of x and y, say w = f(x,y).  The function fits the samples with a
  // polynomial of degree d0 in x and degree d1 in y, say
  //   w = sum_{i=0}^{d0} sum_{j=0}^{d1} c[i][j]*x^i*y^j
  // The method is a least-squares fitting algorithm.  The returned array
  // stores the c[i][j] values according to returned[i+(d0+1)*j] = c[i][j]
  // for a total of (d0+1)*(d1+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)
  // 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))
  // and apply the least-squares algorithm to these.  Another alternative is to
  // transform
  //   (x',y',w') = ((x-xcen)/rng, (y-ycen)/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, and
  // rng = max(xmax-xmin,ymax-ymin).  Fit the (x',y',w') points,
  //   w' = sum_{i=0}^{d0} sum_{j=0}^{d1} c'[i][j]*(x')^i*(y')^j
  // The original polynomial is evaluated as
  //   w = rng * sum_{i=0}^{d0} sum_{j=0}^{d1} c'[i][j] *
  //         ((x-xcen)/rng)^i * ((y-ycen)/rng)^j
  
  template <typename Real> WM5_MATHEMATICS_ITEM
  Real* PolyFit3 (int numSamples, const Real* xSamples, const Real* ySamples,
      const Real* wSamples, int xDegree, int yDegree);
  
  }
  
  #endif
  