.p>.CMCH4.DOC!INT_FCN;int_fcn
Integrates a function using a globally adaptive scheme based on Gauss-Kronrod rules.
Synopsis
#include <imsl.h>
float imsl_f_int_fcn (float fcn(), float a, float b, ¼, 0)
The type double function is imsl_d_int_fcn.
Required Arguments
float fcn
(float
x) (Input)
User-supplied function to be
integrated.
float a
(Input)
Lower limit of integration.
float b
(Input)
Upper limit of integration.
Return Value
The value of
is returned. If no value can be computed, then NaN is returned.
Synopsis with Optional Arguments
#include <imsl.h>
float
imsl_f_int_fcn (float
fcn(float x), float a, float b,
IMSL_RULE, int rule,
IMSL_ERR_ABS, float
err_abs,
IMSL_ERR_REL, float
err_rel,
IMSL_ERR_EST, float
*err_est,
IMSL_MAX_SUBINTER, int
max_subinter,
IMSL_N_SUBINTER, int
*n_subinter,
IMSL_N_EVALS, int
*n_evals,
IMSL_FCN_W_DATA,
float
fcn(),
void *data,
0)
Optional Arguments
IMSL_RULE, int rule
(Input)
Choice of quadrature rule.
|
rule |
Gauss-Kronrod Rule |
|
1 |
7-15 points |
|
2 |
10-21 points |
|
3 |
15-31 points |
|
4 |
20-41 points |
|
5 |
25-51 points |
|
6 |
30-61 points |
Default: rule = 1
IMSL_ERR_ABS, float err_abs
(Input)
Absolute accuracy desired.
Default: 
where ɛ is the machine precision
IMSL_ERR_REL, float err_rel
(Input)
Relative accuracy desired.
Default: 
where ɛ is the machine precision
IMSL_ERR_EST, float *err_est
(Output)
Address to store an estimate of the absolute value of the error.
IMSL_MAX_SUBINTER, int
max_subinter (Input)
Number of subintervals allowed.
Default:
max_subinter = 500
IMSL_N_SUBINTER, int
*n_subinter (Output)
Address to store the number of
subintervals generated.
IMSL_N_EVALS, int *n_evals
(Output)
Address to store the number of evaluations of fcn.
IMSL_FCN_W_DATA, float fcn
(float x, void *data), void *data (Input)
User
supplied function to be integrated, which also accepts a pointer to data that is
supplied by the user. data is a pointer to
the data to be passed to the user-supplied function. See the
Introduction, Passing Data to User-Supplied Functions at the beginning of
this manual for more details.
Description
The function imsl_f_int_fcn is a general-purpose integrator that uses a globally adaptive scheme to reduce the absolute error. It subdivides the interval [a, b] and uses a (2k + 1)-point Gauss-Kronrod rule to estimate the integral over each subinterval. The error for each subinterval is estimated by comparison with the k-point Gauss quadrature rule. The subinterval with the largest estimated error is then bisected, and the same procedure is applied to both halves. The bisection process is continued until either the error criterion is satisfied, roundoff error is detected, the subintervals become too small, or the maximum number of subintervals allowed is reached. The function imsl_f_int_fcn is based on the subroutine QAG by Piessens et al. (1983).
Should imsl_f_int_fcn fail to produce acceptable results, consider one of the more specialized functions documented in this chaptersection.
Examples
Example 1
The value of
is computed. Since the integrand is not oscillatory, all of the default values are used. The values of the actual and estimated error are machine dependent.
#include <math.h>
#include
<imsl.h>
float
fcn(float
x);
float
q;
float
exact;
main()
{
/* evaluate the integral */
q = imsl_f_int_fcn (fcn, 0.0,
2.0,
0);
/* print the result and the exact answer */
exact =
exp(2.0) + 1.0;
printf("integral =
%10.3f\nexact = %10.3f\n", q, exact);
}
float
fcn(float x)
{
float
y;
y = x * (exp(x));
return y;
}
Output
integral =
8.389
exact = 8.389
Example 2
The value of
is computed. Since the integrand is oscillatory, rule = 6 is used. The exact value is 0.50406706. The values of the actual and estimated error are machine dependent.
#include <math.h>
#include
<imsl.h>
float
fcn(float
x);
main()
{
float
q, err_est, err_abs= 0.0001, exact = 0.50406706,
error;
/* intergrate fcn(x) from 0 to 1 */
q = imsl_f_int_fcn
(fcn, 0.0,
1.0,
IMSL_ERR_ABS, err_abs,/* set abs error
value*/
IMSL_RULE, 6,
IMSL_ERR_EST, &err_est, /* pass in address
*/
0);
error = q -
exact;
/* print the result and the exact answer */
printf("
integral = %10.3f\n exact = %10.3f\n error =
%10.3f\n
",
q,
exact , error);
printf("
err_est = %g\n",
err_est);
}
float fcn(float
x)
{
/* compute sin(1/x), avoiding division by zero */
return ((x)>1.0e-5) ? sin(1.0/(x)) :
0.0;
}
Output
integral =
0.504
exact =
0.504
error =
0.000
err_est = 0.000170593
Warning Errors
IMSL_ROUNDOFF_CONTAMINATION Roundoff error, preventing the requested tolerance from being achieved, has been detected.
IMSL_PRECISION_DEGRADATION A degradation in precision has been detected.
Fatal Errors
IMSL_MAX_SUBINTERVALS The maximum number of subintervals allowed has been reached.
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