int_fcn_alg_log
Integrates a function with algebraic-logarithmic singularities.
Synopsis
#include <imsl.h>
float imsl_f_int_fcn_alg_log (float fcn(), float a, float b, Imsl_quad weight, float alpha, float beta, …, 0)
The type double function is imsl_d_int_fcn_alg_log.
Required Arguments
float fcn (float x) (Input)
User-supplied function to be integrated.
User-supplied function to be integrated.
float a (Input)
Lower limit of integration.
Lower limit of integration.
float b (Input)
Upper limit of integration.
Upper limit of integration.
Imsl_quad weight, float alpha, float beta (Input)
These three parameters are used to describe the weight function that may have algebraic or logarithmic singularities at the endpoints. The parameter weight can take on four values as described below. The parameters alpha = α and beta = β specify the strength of the singularities at a or b and hence, must be greater than −1.
These three parameters are used to describe the weight function that may have algebraic or logarithmic singularities at the endpoints. The parameter weight can take on four values as described below. The parameters alpha = α and beta = β specify the strength of the singularities at a or b and hence, must be greater than −1.
Weight | Integration Weight |
|---|---|
IMSL_ALG | (x - a)a (b - x)b |
IMSL_ALG_LEFT_LOG | (x - a)a (b - x)blog (x - a) |
IMSL_ALG_RIGHT_LOG | (x - a)a (b - x)blog (b - x) |
IMSL_ALG_LOG | (x - a)a (b - x)blog (x - a) log (b - x) |
Return Value
The value of
is returned where w(x) is one of the four weights above. If no value can be computed, then NaN is returned.
Synopsis with Optional Arguments
#include <imsl.h>
float imsl_f_int_fcn_alg_log (float fcn(float x), float a, float b, Imsl_quad weight, float alpha, float beta,
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_ERR_ABS, float err_abs (Input)
Absolute accuracy desired.
Default: where ɛ is the machine precision
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
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.
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
Number of subintervals allowed.
Default: max_subinter = 500
IMSL_N_SUBINTER, int *n_subinter (Output)
Address to store the number of subintervals generated.
Address to store the number of subintervals generated.
IMSL_N_EVALS, int *n_evals (Output)
Address to store the number of evaluations of fcn.
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 Passing Data to User-Supplied Functions in the introduction to this manual for more details.
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 Passing Data to User-Supplied Functions in the introduction to this manual for more details.
Description
The function imsl_f_int_fcn_alg_log is a special-purpose integrator that uses a globally adaptive scheme to reduce the absolute error. It computes integrals whose integrands have the special form w(x)f(x) where w(x) is a weight function described above. A combination of modified Clenshaw-Curtis and Gauss-Kronrod formulas is employed. This function is based on the subroutine QAWS, which is fully documented by Piessens et al. (1983).
On some platforms,imsl_f_int_fcn_alg_log can evaluate the user-supplied function fcn in parallel. This is done only if the function imsl_omp_options is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables.
Examples
Example 1
The value of
is computed.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
float fcn(float x);
int main()
{
float q, exact;
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_alg_log (fcn, 0.0, 1.0,
IMSL_ALG_LEFT_LOG, 1.0, 0.5,
0);
/* Print the result and the */
/* exact answer */
exact = (3.*log(2.)-4.)/9.;
printf("integral = %10.3f\nexact = %10.3f\n", q, exact);
}
float fcn(float x)
{
return sqrt(1+x);
}
Output
integral = -0.213
exact = -0.213
Example 2
The value of
is again computed. The values of the actual and estimated error are printed as well. Note that these numbers are machine dependent. Furthermore, the error estimate is usually pessimistic. That is, the actual error is usually smaller than the error estimate, as in this example. The number of function evaluations also are printed.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
float fcn(float x);
int main()
{
int n_evals;
float q, exact, err_est, exact_err;
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_alg_log (fcn, 0.0, 1.0,
IMSL_ALG_LEFT_LOG, 1.0, 0.5,
IMSL_ERR_EST, &err_est,
IMSL_N_EVALS, &n_evals,
0);
/* Print the result and the */
/* exact answer */
exact = (3.*log(2.)-4.)/9.;
exact_err = fabs(exact - q);
printf("integral = %10.3f\nexact = %10.3f\n", q, exact);
printf("error estimate = %e\nexact error = %e\n", err_est,
exact_err);
printf("The number of function evaluations = %d\n", n_evals);
}
float fcn(float x)
{
return sqrt(1+x);
}
Output
integral = -0.213
exact = -0.213
error estimate = 3.725290e-09
exact error = 1.490116e-08
The number of function evaluations = 50
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. |
IMSL_STOP_USER_FCN | Request from user supplied function to stop algorithm. User flag = "#". |
