hwpfilter/source/cspline.cxx

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  19
  20 // Natural, Clamped, or Periodic Cubic Splines
  21 //
  22 // Input:  A list of N+1 points (x_i,a_i), 0 <= i <= N, which are sampled
  23 // from a function, a_i = f(x_i).  The function f is unknown.  Boundary
  24 // conditions are
  25 //   (1) Natural splines:  f"(x_0) = f"(x_N) = 0
  26 //   (2) Clamped splines:  f'(x_0) and f'(x_N) are user-specified.
  27 //   (3) Periodic splines:  f(x_0) = f(x_N) [in which case a_N = a_0 is
  28 //       required in the input], f'(x_0) = f'(x_N), and f"(x_0) = f"(x_N).
  29 //
  30 // Output: b_i, c_i, d_i, 0 <= i <= N-1, which are coefficients for the cubic
  31 // spline S_i(x) = a_i + b_i(x-x_i) + c_i(x-x_i)^2 + d_i(x-x_i)^3 for
  32 // x_i <= x < x_{i+1}.
  33 //
  34 // The natural and clamped algorithms were implemented from
  35 //
  36 //    Numerical Analysis, 3rd edition
  37 //    Richard L. Burden and J. Douglas Faires
  38 //    Prindle, Weber & Schmidt
  39 //    Boston, 1985, pp. 122-124.
  40 //
  41 // The algorithm sets up a tridiagonal linear system of equations in the
  42 // c_i values.  This can be solved in O(N) time.
  43 //
  44 // The periodic spline algorithm was implemented from my own derivation.  The
  45 // linear system of equations is not tridiagonal.  For now I use a standard
  46 // linear solver that does not take advantage of the sparseness of the
  47 // matrix.  Therefore for very large N, you may have to worry about memory
  48 // usage.
  49
  50 #include <sal/config.h>
  51 #include <memory>
  52
  53 #include "cspline.h"
  54 #include "solver.h"
  55
  56 void NaturalSpline (int N, const double* x, const double* a, std::unique_ptr<double[]>& b, std::unique_ptr<double[]>& c,
  57     std::unique_ptr<double[]>& d)
  58 {
  59   const double oneThird = 1.0/3.0;
  60
  61   int i;
  62   std::unique_ptr<double[]> h(new double[N]);
  63   std::unique_ptr<double[]> hdiff(new double[N]);
  64   std::unique_ptr<double[]> alpha(new double[N]);
  65
  66   for (i = 0; i < N; i++){
  67     h[i] = x[i+1]-x[i];
  68   }
  69
  70   for (i = 1; i < N; i++)
  71     hdiff[i] = x[i+1]-x[i-1];
  72
  73   for (i = 1; i < N; i++)
  74   {
  75     double numer = 3.0*(a[i+1]*h[i-1]-a[i]*hdiff[i]+a[i-1]*h[i]);
  76     double denom = h[i-1]*h[i];
  77     alpha[i] = numer/denom;
  78   }
  79
  80   std::unique_ptr<double[]> ell(new double[N+1]);
  81   std::unique_ptr<double[]> mu(new double[N]);
  82   std::unique_ptr<double[]> z(new double[N+1]);
  83   double recip;
  84
  85   ell[0] = 1.0;
  86   mu[0] = 0.0;
  87   z[0] = 0.0;
  88
  89   for (i = 1; i < N; i++)
  90   {
  91     ell[i] = 2.0*hdiff[i]-h[i-1]*mu[i-1];
  92     recip = 1.0/ell[i];
  93     mu[i] = recip*h[i];
  94     z[i] = recip*(alpha[i]-h[i-1]*z[i-1]);
  95   }
  96   ell[N] = 1.0;
  97   z[N] = 0.0;
  98
  99   b.reset(new double[N]);
 100   c.reset(new double[N+1]);
 101   d.reset(new double[N]);
 102
 103   c[N] = 0.0;
 104
 105   for (i = N-1; i >= 0; i--)
 106   {
 107     c[i] = z[i]-mu[i]*c[i+1];
 108     recip = 1.0/h[i];
 109     b[i] = recip*(a[i+1]-a[i])-h[i]*(c[i+1]+2.0*c[i])*oneThird;
 110     d[i] = oneThird*recip*(c[i+1]-c[i]);
 111   }
 112 }
 113
 114 void PeriodicSpline (int N, const double* x, const double* a, std::unique_ptr<double[]>& b, std::unique_ptr<double[]>& c,
 115     std::unique_ptr<double[]>& d)
 116 {
 117   std::unique_ptr<double[]> h(new double[N]);
 118   int i;
 119   for (i = 0; i < N; i++)
 120     h[i] = x[i+1]-x[i];
 121
 122   std::unique_ptr<std::unique_ptr<double[]>[]> mat = mgcLinearSystemD::NewMatrix(N+1);  // guaranteed to be zeroed memory
 123   c = mgcLinearSystemD::NewVector(N+1);   // guaranteed to be zeroed memory
 124
 125   // c[0] - c[N] = 0
 126   mat[0][0] = +1.0f;
 127   mat[0][N] = -1.0f;
 128
 129   // h[i-1]*c[i-1]+2*(h[i-1]+h[i])*c[i]+h[i]*c[i+1] =
 130   //   3*{(a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]}
 131   for (i = 1; i <= N-1; i++)
 132   {
 133     mat[i][i-1] = h[i-1];
 134     mat[i][i  ] = 2.0f*(h[i-1]+h[i]);
 135     mat[i][i+1] = h[i];
 136     c[i] = 3.0f*((a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]);
 137   }
 138
 139   // "wrap around equation" for periodicity
 140   // h[N-1]*c[N-1]+2*(h[N-1]+h[0])*c[0]+h[0]*c[1] =
 141   //   3*{(a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]}
 142   mat[N][N-1] = h[N-1];
 143   mat[N][0  ] = 2.0f*(h[N-1]+h[0]);
 144   mat[N][1  ] = h[0];
 145   c[N] = 3.0f*((a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]);
 146
 147   // solve for c[0] through c[N]
 148   mgcLinearSystemD::Solve(N+1,mat,c.get());
 149
 150   const double oneThird = 1.0/3.0;
 151   b.reset(new double[N]);
 152   d.reset(new double[N]);
 153   for (i = 0; i < N; i++)
 154   {
 155     b[i] = (a[i+1]-a[i])/h[i] - oneThird*(c[i+1]+2.0f*c[i])*h[i];
 156     d[i] = oneThird*(c[i+1]-c[i])/h[i];
 157   }
 158 }
 159
 160 /* vim:set shiftwidth=4 softtabstop=4 expandtab: */