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 "solver.h"
  51 //-----------------------------------------------------------------------------
  52 void NaturalSpline (int N, double* x, double* a, double*& b, double*& c,
  53     double*& d)
  54 {
  55   const double oneThird = 1.0/3.0;
  56
  57   int i;
  58   double* h = new double[N];
  59   double* hdiff = new double[N];
  60   double* alpha = new double[N];
  61
  62   for (i = 0; i < N; i++){
  63     h[i] = x[i+1]-x[i];
  64   }
  65
  66   for (i = 1; i < N; i++)
  67     hdiff[i] = x[i+1]-x[i-1];
  68
  69   for (i = 1; i < N; i++)
  70   {
  71     double numer = 3.0*(a[i+1]*h[i-1]-a[i]*hdiff[i]+a[i-1]*h[i]);
  72     double denom = h[i-1]*h[i];
  73     alpha[i] = numer/denom;
  74   }
  75
  76   double* ell = new double[N+1];
  77   double* mu = new double[N];
  78   double* z = new double[N+1];
  79   double recip;
  80
  81   ell[0] = 1.0;
  82   mu[0] = 0.0;
  83   z[0] = 0.0;
  84
  85   for (i = 1; i < N; i++)
  86   {
  87     ell[i] = 2.0*hdiff[i]-h[i-1]*mu[i-1];
  88     recip = 1.0/ell[i];
  89     mu[i] = recip*h[i];
  90     z[i] = recip*(alpha[i]-h[i-1]*z[i-1]);
  91   }
  92   ell[N] = 1.0;
  93   z[N] = 0.0;
  94
  95   b = new double[N];
  96   c = new double[N+1];
  97   d = new double[N];
  98
  99   c[N] = 0.0;
 100
 101   for (i = N-1; i >= 0; i--)
 102   {
 103     c[i] = z[i]-mu[i]*c[i+1];
 104     recip = 1.0/h[i];
 105     b[i] = recip*(a[i+1]-a[i])-h[i]*(c[i+1]+2.0*c[i])*oneThird;
 106     d[i] = oneThird*recip*(c[i+1]-c[i]);
 107   }
 108
 109   delete[] h;
 110   delete[] hdiff;
 111   delete[] alpha;
 112   delete[] ell;
 113   delete[] mu;
 114   delete[] z;
 115 }
 116
 117 void PeriodicSpline (int N, double* x, double* a, double*& b, double*& c,
 118     double*& d)
 119 {
 120   double* h = new double[N];
 121   int i;
 122   for (i = 0; i < N; i++)
 123     h[i] = x[i+1]-x[i];
 124
 125   mgcLinearSystemD sys;
 126   double** mat = sys.NewMatrix(N+1);  // guaranteed to be zeroed memory
 127   c = sys.NewVector(N+1);   // guaranteed to be zeroed memory
 128
 129   // c[0] - c[N] = 0
 130   mat[0][0] = +1.0f;
 131   mat[0][N] = -1.0f;
 132
 133   // h[i-1]*c[i-1]+2*(h[i-1]+h[i])*c[i]+h[i]*c[i+1] =
 134   //   3*{(a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]}
 135   for (i = 1; i <= N-1; i++)
 136   {
 137     mat[i][i-1] = h[i-1];
 138     mat[i][i  ] = 2.0f*(h[i-1]+h[i]);
 139     mat[i][i+1] = h[i];
 140     c[i] = 3.0f*((a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]);
 141   }
 142
 143   // "wrap around equation" for periodicity
 144   // h[N-1]*c[N-1]+2*(h[N-1]+h[0])*c[0]+h[0]*c[1] =
 145   //   3*{(a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]}
 146   mat[N][N-1] = h[N-1];
 147   mat[N][0  ] = 2.0f*(h[N-1]+h[0]);
 148   mat[N][1  ] = h[0];
 149   c[N] = 3.0f*((a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]);
 150
 151   // solve for c[0] through c[N]
 152   sys.Solve(N+1,mat,c);
 153
 154   const double oneThird = 1.0/3.0;
 155   b = new double[N];
 156   d = new double[N];
 157   for (i = 0; i < N; i++)
 158   {
 159     b[i] = (a[i+1]-a[i])/h[i] - oneThird*(c[i+1]+2.0f*c[i])*h[i];
 160     d[i] = oneThird*(c[i+1]-c[i])/h[i];
 161   }
 162
 163   delete[] h;
 164   sys.DeleteMatrix(N+1,mat);
 165 }
 166
 167 /* vim:set shiftwidth=4 softtabstop=4 expandtab: */