1 files changed, 174 insertions, 0 deletions
diff --git a/gsl-1.9/deriv/deriv.c b/gsl-1.9/deriv/deriv.c
new file mode 100644
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+++ b/gsl-1.9/deriv/deriv.c
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+/* deriv/deriv.c
+ * 
+ * Copyright (C) 2004 Brian Gough
+ * 
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or (at
+ * your option) any later version.
+ * 
+ * This program is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+ * General Public License for more details.
+ * 
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ */
+
+#include <config.h>
+#include <stdlib.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_deriv.h>
+
+static void
+central_deriv (const gsl_function * f, double x, double h,
+               double *result, double *abserr_round, double *abserr_trunc)
+{
+  /* Compute the derivative using the 5-point rule (x-h, x-h/2, x,
+     x+h/2, x+h). Note that the central point is not used.  
+
+     Compute the error using the difference between the 5-point and
+     the 3-point rule (x-h,x,x+h). Again the central point is not
+     used. */
+
+  double fm1 = GSL_FN_EVAL (f, x - h);
+  double fp1 = GSL_FN_EVAL (f, x + h);
+
+  double fmh = GSL_FN_EVAL (f, x - h / 2);
+  double fph = GSL_FN_EVAL (f, x + h / 2);
+
+  double r3 = 0.5 * (fp1 - fm1);
+  double r5 = (4.0 / 3.0) * (fph - fmh) - (1.0 / 3.0) * r3;
+
+  double e3 = (fabs (fp1) + fabs (fm1)) * GSL_DBL_EPSILON;
+  double e5 = 2.0 * (fabs (fph) + fabs (fmh)) * GSL_DBL_EPSILON + e3;
+
+  double dy = GSL_MAX (fabs (r3), fabs (r5)) * fabs (x) * GSL_DBL_EPSILON;
+
+  /* The truncation error in the r5 approximation itself is O(h^4).
+     However, for safety, we estimate the error from r5-r3, which is
+     O(h^2).  By scaling h we will minimise this estimated error, not
+     the actual truncation error in r5. */
+
+  *result = r5 / h;
+  *abserr_trunc = fabs ((r5 - r3) / h); /* Estimated truncation error O(h^2) */
+  *abserr_round = fabs (e5 / h) + dy;   /* Rounding error (cancellations) */
+}
+
+int
+gsl_deriv_central (const gsl_function * f, double x, double h,
+                   double *result, double *abserr)
+{
+  double r_0, round, trunc, error;
+  central_deriv (f, x, h, &r_0, &round, &trunc);
+  error = round + trunc;
+
+  if (round < trunc && (round > 0 && trunc > 0))
+    {
+      double r_opt, round_opt, trunc_opt, error_opt;
+
+      /* Compute an optimised stepsize to minimize the total error,
+         using the scaling of the truncation error (O(h^2)) and
+         rounding error (O(1/h)). */
+
+      double h_opt = h * pow (round / (2.0 * trunc), 1.0 / 3.0);
+      central_deriv (f, x, h_opt, &r_opt, &round_opt, &trunc_opt);
+      error_opt = round_opt + trunc_opt;
+
+      /* Check that the new error is smaller, and that the new derivative 
+         is consistent with the error bounds of the original estimate. */
+
+      if (error_opt < error && fabs (r_opt - r_0) < 4.0 * error)
+        {
+          r_0 = r_opt;
+          error = error_opt;
+        }
+    }
+
+  *result = r_0;
+  *abserr = error;
+
+  return GSL_SUCCESS;
+}
+
+
+static void
+forward_deriv (const gsl_function * f, double x, double h,
+               double *result, double *abserr_round, double *abserr_trunc)
+{
+  /* Compute the derivative using the 4-point rule (x+h/4, x+h/2,
+     x+3h/4, x+h).
+
+     Compute the error using the difference between the 4-point and
+     the 2-point rule (x+h/2,x+h).  */
+
+  double f1 = GSL_FN_EVAL (f, x + h / 4.0);
+  double f2 = GSL_FN_EVAL (f, x + h / 2.0);
+  double f3 = GSL_FN_EVAL (f, x + (3.0 / 4.0) * h);
+  double f4 = GSL_FN_EVAL (f, x + h);
+
+  double r2 = 2.0*(f4 - f2);
+  double r4 = (22.0 / 3.0) * (f4 - f3) - (62.0 / 3.0) * (f3 - f2) +
+    (52.0 / 3.0) * (f2 - f1);
+
+  /* Estimate the rounding error for r4 */
+
+  double e4 = 2 * 20.67 * (fabs (f4) + fabs (f3) + fabs (f2) + fabs (f1)) * GSL_DBL_EPSILON;
+
+  double dy = GSL_MAX (fabs (r2), fabs (r4)) * fabs (x) * GSL_DBL_EPSILON;
+
+  /* The truncation error in the r4 approximation itself is O(h^3).
+     However, for safety, we estimate the error from r4-r2, which is
+     O(h).  By scaling h we will minimise this estimated error, not
+     the actual truncation error in r4. */
+
+  *result = r4 / h;
+  *abserr_trunc = fabs ((r4 - r2) / h); /* Estimated truncation error O(h) */
+  *abserr_round = fabs (e4 / h) + dy;
+}
+
+int
+gsl_deriv_forward (const gsl_function * f, double x, double h,
+                   double *result, double *abserr)
+{
+  double r_0, round, trunc, error;
+  forward_deriv (f, x, h, &r_0, &round, &trunc);
+  error = round + trunc;
+
+  if (round < trunc && (round > 0 && trunc > 0))
+    {
+      double r_opt, round_opt, trunc_opt, error_opt;
+
+      /* Compute an optimised stepsize to minimize the total error,
+         using the scaling of the estimated truncation error (O(h)) and
+         rounding error (O(1/h)). */
+
+      double h_opt = h * pow (round / (trunc), 1.0 / 2.0);
+      forward_deriv (f, x, h_opt, &r_opt, &round_opt, &trunc_opt);
+      error_opt = round_opt + trunc_opt;
+
+      /* Check that the new error is smaller, and that the new derivative 
+         is consistent with the error bounds of the original estimate. */
+
+      if (error_opt < error && fabs (r_opt - r_0) < 4.0 * error)
+        {
+          r_0 = r_opt;
+          error = error_opt;
+        }
+    }
+
+  *result = r_0;
+  *abserr = error;
+
+  return GSL_SUCCESS;
+}
+
+int
+gsl_deriv_backward (const gsl_function * f, double x, double h,
+                    double *result, double *abserr)
+{
+  return gsl_deriv_forward (f, x, -h, result, abserr);
+}