hwpfilter/source/cspline.cpp

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  27
  28 // Natural, Clamped, or Periodic Cubic Splines
  29 //
  30 // Input:  A list of N+1 points (x_i,a_i), 0 <= i <= N, which are sampled
  31 // from a function, a_i = f(x_i).  The function f is unknown.  Boundary
  32 // conditions are
  33 //   (1) Natural splines:  f"(x_0) = f"(x_N) = 0
  34 //   (2) Clamped splines:  f'(x_0) and f'(x_N) are user-specified.
  35 //   (3) Periodic splines:  f(x_0) = f(x_N) [in which case a_N = a_0 is
  36 //       required in the input], f'(x_0) = f'(x_N), and f"(x_0) = f"(x_N).
  37 //
  38 // Output: b_i, c_i, d_i, 0 <= i <= N-1, which are coefficients for the cubic
  39 // spline S_i(x) = a_i + b_i(x-x_i) + c_i(x-x_i)^2 + d_i(x-x_i)^3 for
  40 // x_i <= x < x_{i+1}.
  41 //
  42 // The natural and clamped algorithms were implemented from
  43 //
  44 //    Numerical Analysis, 3rd edition
  45 //    Richard L. Burden and J. Douglas Faires
  46 //    Prindle, Weber & Schmidt
  47 //    Boston, 1985, pp. 122-124.
  48 //
  49 // The algorithm sets up a tridiagonal linear system of equations in the
  50 // c_i values.  This can be solved in O(N) time.
  51 //
  52 // The periodic spline algorithm was implemented from my own derivation.  The
  53 // linear system of equations is not tridiagonal.  For now I use a standard
  54 // linear solver that does not take advantage of the sparseness of the
  55 // matrix.  Therefore for very large N, you may have to worry about memory
  56 // usage.
  57
  58 #include "solver.h"
  59 //-----------------------------------------------------------------------------
  60 void NaturalSpline (int N, double* x, double* a, double*& b, double*& c,
  61     double*& d)
  62 {
  63   const double oneThird = 1.0/3.0;
  64
  65   int i;
  66   double* h = new double[N];
  67   double* hdiff = new double[N];
  68   double* alpha = new double[N];
  69
  70   for (i = 0; i < N; i++){
  71     h[i] = x[i+1]-x[i];
  72   }
  73
  74   for (i = 1; i < N; i++)
  75     hdiff[i] = x[i+1]-x[i-1];
  76
  77   for (i = 1; i < N; i++)
  78   {
  79     double numer = 3.0*(a[i+1]*h[i-1]-a[i]*hdiff[i]+a[i-1]*h[i]);
  80     double denom = h[i-1]*h[i];
  81     alpha[i] = numer/denom;
  82   }
  83
  84   double* ell = new double[N+1];
  85   double* mu = new double[N];
  86   double* z = new double[N+1];
  87   double recip;
  88
  89   ell[0] = 1.0;
  90   mu[0] = 0.0;
  91   z[0] = 0.0;
  92
  93   for (i = 1; i < N; i++)
  94   {
  95     ell[i] = 2.0*hdiff[i]-h[i-1]*mu[i-1];
  96     recip = 1.0/ell[i];
  97     mu[i] = recip*h[i];
  98     z[i] = recip*(alpha[i]-h[i-1]*z[i-1]);
  99   }
 100   ell[N] = 1.0;
 101   z[N] = 0.0;
 102
 103   b = new double[N];
 104   c = new double[N+1];
 105   d = new double[N];
 106
 107   c[N] = 0.0;
 108
 109   for (i = N-1; i >= 0; i--)
 110   {
 111     c[i] = z[i]-mu[i]*c[i+1];
 112     recip = 1.0/h[i];
 113     b[i] = recip*(a[i+1]-a[i])-h[i]*(c[i+1]+2.0*c[i])*oneThird;
 114     d[i] = oneThird*recip*(c[i+1]-c[i]);
 115   }
 116
 117   delete[] h;
 118   delete[] hdiff;
 119   delete[] alpha;
 120   delete[] ell;
 121   delete[] mu;
 122   delete[] z;
 123 }
 124
 125 void PeriodicSpline (int N, double* x, double* a, double*& b, double*& c,
 126     double*& d)
 127 {
 128   double* h = new double[N];
 129   int i;
 130   for (i = 0; i < N; i++)
 131     h[i] = x[i+1]-x[i];
 132
 133   mgcLinearSystemD sys;
 134   double** mat = sys.NewMatrix(N+1);  // guaranteed to be zeroed memory
 135   c = sys.NewVector(N+1);   // guaranteed to be zeroed memory
 136
 137   // c[0] - c[N] = 0
 138   mat[0][0] = +1.0f;
 139   mat[0][N] = -1.0f;
 140
 141   // h[i-1]*c[i-1]+2*(h[i-1]+h[i])*c[i]+h[i]*c[i+1] =
 142   //   3*{(a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]}
 143   for (i = 1; i <= N-1; i++)
 144   {
 145     mat[i][i-1] = h[i-1];
 146     mat[i][i  ] = 2.0f*(h[i-1]+h[i]);
 147     mat[i][i+1] = h[i];
 148     c[i] = 3.0f*((a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]);
 149   }
 150
 151   // "wrap around equation" for periodicity
 152   // h[N-1]*c[N-1]+2*(h[N-1]+h[0])*c[0]+h[0]*c[1] =
 153   //   3*{(a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]}
 154   mat[N][N-1] = h[N-1];
 155   mat[N][0  ] = 2.0f*(h[N-1]+h[0]);
 156   mat[N][1  ] = h[0];
 157   c[N] = 3.0f*((a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]);
 158
 159   // solve for c[0] through c[N]
 160   sys.Solve(N+1,mat,c);
 161
 162   const double oneThird = 1.0/3.0;
 163   b = new double[N];
 164   d = new double[N];
 165   for (i = 0; i < N; i++)
 166   {
 167     b[i] = (a[i+1]-a[i])/h[i] - oneThird*(c[i+1]+2.0f*c[i])*h[i];
 168     d[i] = oneThird*(c[i+1]-c[i])/h[i];
 169   }
 170
 171   delete[] h;
 172   sys.DeleteMatrix(N+1,mat);
 173 }