gaussQuad.c File Reference

Implements recursive Gaussian quadrature for numerical integration. More...

#include <math.h>
#include "mdb.h"

Go to the source code of this file.

Macros
#define	_CGQ   .57735026918962576451
#define	gauss_quad(fn, z, a, w)
#define	MAXSTACK   16384

Functions
long	gaussianQuadrature (double(*fn)(), double a, double b, long n, double err, double *result)
	Integrate a function using recursive Gaussian quadrature.

Detailed Description

Implements recursive Gaussian quadrature for numerical integration.

This file contains the implementation of the recursive Gaussian quadrature technique for integrating functions numerically. The primary function gaussianQuadrature divides the integration interval into panels and recursively refines the estimate to achieve the desired error tolerance.

Copyright

• (c) 2002 The University of Chicago, as Operator of Argonne National Laboratory.
• (c) 2002 The Regents of the University of California, as Operator of Los Alamos National Laboratory.

License

This file is distributed under the terms of the Software License Agreement found in the file LICENSE included with this distribution.

Author

M. Borland, R. Soliday

Definition in file gaussQuad.c.

Macro Definition Documentation

_CGQ

#define _CGQ   .57735026918962576451

Definition at line 24 of file gaussQuad.c.

gauss_quad

#define gauss_quad	(		fn,
			z,
			a,
			w )

Value:

(((*fn)(z + a) + (*fn)(a - z)) * w)

Definition at line 25 of file gaussQuad.c.

MAXSTACK

#define MAXSTACK   16384

Definition at line 27 of file gaussQuad.c.

Function Documentation

gaussianQuadrature()

long gaussianQuadrature	(	double(*	fn )(),
		double	a,
		double	b,
		long	n,
		double	err,
		double *	result )

Integrate a function using recursive Gaussian quadrature.

Performs numerical integration of the specified function over the interval [a, b] using a recursive Gaussian quadrature technique. The interval is initially divided into n panels, and each panel is recursively subdivided until the desired fractional error err is achieved.

Parameters

fn	Pointer to the function to be integrated.
a	Lower limit of integration.
b	Upper limit of integration.
n	Initial number of panels to divide the interval into.
err	Fractional error permissible on any quadrature.
result	Pointer to a double where the result of the integration will be stored.

Returns

The total number of function evaluations performed, or -1 if an error occurs.

Definition at line 45 of file gaussQuad.c.

                  {
  register long s; /* stack pointer for panel limits */
  double ave, i0, i1, i2, i3, idiv;
  long totalEvals;
  double d_ab, i_tot, d_ab2;
  double ab, z, width;
  /* stacks of upper,lower limits, and integral values from two
   * point quadratures on each interval */
  double a_stack[MAXSTACK];
  double b_stack[MAXSTACK];
  double i_stack[MAXSTACK];
 
  if (n > MAXSTACK)
    return -1;
 
  d_ab = (b - a) / n;
  d_ab2 = d_ab / 2;
  totalEvals = 0;
  for (s = 0; s < n; s++) {
    b = b_stack[s] = (a_stack[s] = a) + d_ab;
    ave = a + d_ab2;
    z = d_ab2 * _CGQ;
    i_stack[s] = gauss_quad(fn, z, ave, d_ab2);
    totalEvals += 2;
    a = b;
  }
 
  s = n - 1;
  i_tot = 0;
  while (s >= 0) {
    if (s == MAXSTACK)
      return -1;
    a = a_stack[s];
    b = b_stack[s];
    ab = (a + b) / 2;
    i0 = i_stack[s];
 
    ave = (a + ab) / 2;
    z = (width = (ab - a) / 2) * _CGQ;
    i1 = gauss_quad(fn, z, ave, width);
    totalEvals += 2;
 
    ave = (ab + b) / 2;
    z = (width = (b - ab) / 2) * _CGQ;
    i2 = gauss_quad(fn, z, ave, width);
    totalEvals += 2;
 
    i3 = idiv = i1 + i2;
    if (i3 == 0)
      idiv = i0;
    if (idiv != 0) {
      if (fabs((i3 - i0) / idiv) > err) {
        b_stack[s] = ab;
        i_stack[s] = i1;
        a_stack[++s] = ab;
        b_stack[s] = b;
        i_stack[s] = i2;
      } else {
        i_tot += i3;
        s--;
      }
    } else {
      i_tot += i3;
      s--;
    }
  }
  *result = i_tot;
  return totalEvals;
}