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deriv.c

/* deriv/deriv.c
 * 
 * Copyright (C) 2004, 2007 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 3 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;

  /* The next term is due to finite precision in x+h = O (eps * x) */

  double dy = GSL_MAX (fabs (r3 / h), fabs (r5 / h)) *(fabs (x) / h) * 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;

  /* The next term is due to finite precision in x+h = O (eps * x) */

  double dy = GSL_MAX (fabs (r2 / h), fabs (r4 / h)) * fabs (x / h) * 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);
}

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