lib/splines.c

   1 #include "xmath.h"
   2
   3 /* natural cubic spline interpolation based on Numerical Recipes in C */
   4
   5 /* calculate second derivatives; assumes strictly increasing x values */
   6 static void
   7 find_spline_derivs(double *x, double *y, int n,
   8                    double *y2, double *u)
   9 {
  10   int i, k;
  11   double p, sig;
  12
  13   y2[0] = u[0] = 0.0;  /* lower boundary condition for natural spline */
  14
  15   /* decomposition loop for the tridiagonal algorithm */
  16   for (i = 1; i < n - 1; i++) {
  17     y2[i] = u[i] = 0.0; /* set in case a zero test is failed */
  18     if (x[i - 1] < x[i] && x[i] < x[i + 1]) {
  19       sig = (x[i] - x[i - 1]) / (x[i + 1] - x[i - 1]);
  20       p = sig * y2[i - 1] + 2.0;
  21       if (p != 0.0) {
  22         y2[i] = (sig - 1.0) / p;
  23         u[i] = (y[i + 1] - y[i]) / (x[i + 1] - x[i])
  24              - (y[i] - y[i - 1]) / (x[i] - x[i - 1]);
  25         u[i] = (6.0 * u[i] / (x[i + 1] - x[i - 1]) - sig * u[i - 1]) / p;
  26       }
  27     }
  28   }
  29
  30   /* upper boundary condition for natural spline */
  31   y2[n - 1] = 0.0;
  32
  33   /* backsubstitution loop of the tridiagonal algorithm */
  34   for (k = n - 2; k >= 0; k--)
  35     y2[k] = y2[k] * y2[k + 1] + u[k];
  36 }
  37
  38 /* interpolate or extrapolate value at x using results of find_spline_derivs */
  39 static void
  40 spline_interp(double *xa, double *ya, double *y2a,
  41               int n, double x, double *y)
  42 {
  43   int klo, khi, k;
  44   double h, b, a;
  45
  46   if (x <= xa[0]) {
  47     h = xa[1] - xa[0];
  48     b = (h > 0.0) ? (ya[1] - ya[0]) / h - h * y2a[1] / 6.0 : 0.0;
  49     *y = ya[0] + b * (x - xa[0]);
  50   }
  51   else if (x >= xa[n - 1]) {
  52     h = xa[n - 1] - xa[n - 2];
  53     b = (h > 0.0) ? (ya[n - 1] - ya[n - 2]) / h + h * y2a[n - 2] / 6.0 : 0.0;
  54     *y = ya[n - 1] + b * (x - xa[n - 1]);
  55   }
  56   else {
  57     /* try a linear interpolation for equally spaced x values */
  58     k = (n - 1) * (x - xa[0]) / (xa[n - 1] - xa[0]);
  59
  60     /* make sure the range is right */
  61     if (k < 0) k = 0;
  62     if (k > n - 2) k = n - 2;
  63
  64     /* bisect if necessary until the bracketing interval is found */
  65     klo = (x >= xa[k]) ? k : 0;
  66     khi = (x < xa[k + 1]) ? k + 1 : n - 1;
  67     while (khi - klo > 1) {
  68       k = (khi + klo) / 2;
  69       if (xa[k] > x) khi = k;
  70       else klo = k;
  71     }
  72
  73     /* interpolate */
  74     h = xa[khi] - xa[klo];
  75     if (h > 0.0) {
  76       a = (xa[khi] - x) / h;
  77       b = (x - xa[klo]) / h;
  78       *y = a * ya[klo] + b * ya[khi]
  79          + ((a * a * a - a) * y2a[klo] + (b * b * b - b) * y2a[khi]) * (h * h) / 6.0;
  80     }
  81     else *y = (ya[klo] + ya[khi]) / 2.0; /* should not be needed */
  82   }
  83 }
  84
  85 int
  86 fit_spline(int n, double *x, double *y,
  87            int ns, double *xs, double *ys, double *work)
  88 {
  89   int i;
  90   double *y2, *u;
  91
  92   y2 = work; u = work + n;
  93
  94   if (n < 2 || ns < 1) return (1); /* signal an error */
  95   for (i = 1; i < n; i++)
  96     if (x[i - 1] >= x[i]) return(1); /* signal an error */
  97
  98   find_spline_derivs(x, y, n, y2, u);
  99
 100   for (i = 0; i < ns; i++)
 101     spline_interp(x, y, y2, n, xs[i], &ys[i]);
 102
 103   return(0);
 104 }