NAG CL Interface
d01rkc (dim1_​fin_​osc_​fn)

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1 Purpose

d01rkc is an adaptive integrator, especially suited to oscillating, nonsingular integrands, which calculates an approximation to the integral of a function f(x) over a finite interval [a,b]:
I= ab f(x) dx .  

2 Specification

#include <nag.h>
void  d01rkc (
void (*f)(const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm),
double a, double b, Integer key, double epsabs, double epsrel, Integer maxsub, double *result, double *abserr, double rinfo[], Integer iinfo[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: d01rkc or nag_quad_dim1_fin_osc_fn.

3 Description

d01rkc is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive function, offering a choice of six Gauss–Kronrod rules. A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described in Piessens et al. (1983).
Because d01rkc is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.
d01rkc requires you to supply a function to evaluate the integrand at an array of points.

4 References

de Doncker E (1978) An adaptive extrapolation algorithm for automatic integration ACM SIGNUM Newsl. 13(2) 12–18
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Wynn P (1956) On a device for computing the em(Sn) transformation Math. Tables Aids Comput. 10 91–96

5 Arguments

1: f function, supplied by the user External Function
f must return the values of the integrand f at a set of points.
The specification of f is:
void  f (const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm)
1: x[dim] const double Input
On entry: the abscissae, xi, for i=1,2,,nx, at which function values are required.
2: nx Integer Input
On entry: the number of abscissae at which a function value is required. nx will be of size equal to the number of Kronrod points in the quadrature rule used, as determined by the choice of value for key.
3: fv[dim] double Output
On exit: fv must contain the values of the integrand f. fv[i-1]=f(xi) for all i=1,2,,nx.
4: iflag Integer * Input/Output
On entry: iflag=0.
On exit: set iflag<0 to force an immediate exit with fail.code= NE_USER_STOP.
5: comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
The type Pointer will be void *. Before calling d01rkc you may allocate memory and initialize these pointers with various quantities for use by f when called from d01rkc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01rkc. If your code inadvertently does return any NaNs or infinities, d01rkc is likely to produce unexpected results.
2: a double Input
On entry: a, the lower limit of integration.
3: b double Input
On entry: b, the upper limit of integration. It is not necessary that a<b.
Note: if a=b, the function will immediately return with result=0.0, abserr=0.0, rinfo=0.0 and iinfo=0.
4: key Integer Input
On entry: indicates which integration rule is to be used. The number of function evaluations required for an integral estimate over any segment will be the number of Kronrod points, nkron.
For the Gauss 7-point and Kronrod 15-point rule.
For the Gauss 10-point and Kronrod 21-point rule.
For the Gauss 15-point and Kronrod 31-point rule.
For the Gauss 20-point and Kronrod 41-point rule.
For the Gauss 25-point and Kronrod 51-point rule.
For the Gauss 30-point and Kronrod 61-point rule.
Suggested value: key=6.
Constraint: key=1, 2, 3, 4, 5 or 6.
5: epsabs double Input
On entry: εa, the absolute accuracy required. If epsabs is negative, εa=|epsabs|. See Section 7.
6: epsrel double Input
On entry: εr, the relative accuracy required. If epsrel is negative, εr=|epsrel|. See Section 7.
7: maxsub Integer Input
On entry: maxsdiv, the upper bound on the total number of subdivisions d01rkc may use to generate new segments. If maxsdiv=1, only the initial segment will be evaluated.
Suggested value: a value in the range 200 to 500 is adequate for most problems.
Constraint: maxsub1.
8: result double * Output
On exit: the approximation to the integral I.
9: abserr double * Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for |I-result|.
10: rinfo[4×maxsub] double Output
On exit: details of the computation. See Section 9 for more information.
11: iinfo[max(maxsub,4)] Integer Output
On exit: details of the computation. See Section 9 for more information.
12: comm Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
13: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, key=value.
Constraint: key=1, 2, 3, 4, 5 or 6.
On entry, maxsub=value.
Constraint: maxsub1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Extremely bad integrand behaviour occurs around the sub-interval (value,value). The same advice applies as in the case of fail.code= NE_QUAD_MAX_SUBDIV.
The maximum number of subdivisions allowed with the given workspace has been reached without the accuracy requirements being achieved. Look at the integrand in order to determine the integration difficulties. If necessary, another integrator, which is designed for handling the type of difficulty involved, must be used. Alternatively, consider relaxing the accuracy requirements specified by epsabs and epsrel, or increasing the amount of workspace.
Round-off error prevents the requested tolerance from being achieved: epsabs=value and epsrel=value.
Exit from f with iflag<0.

7 Accuracy

d01rkc cannot guarantee, but in practice usually achieves, the following accuracy:
tol= max{|epsabs|,|epsrel|×|I|} ,  
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances, satisfies

8 Parallelism and Performance

d01rkc is not threaded in any implementation.

9 Further Comments

The time taken by d01rkc depends on the integrand and the accuracy required.
If fail.code= NE_NOERROR, NE_QUAD_BAD_SUBDIV, NE_QUAD_MAX_SUBDIV or NE_QUAD_ROUNDOFF_TOL, or if fail.code= NE_USER_STOP and at least one complete vector evaluation of f was completed, result and abserr will contain computed results. If these results are unacceptable, or if otherwise required, then you may wish to examine the contents of the array rinfo, which contains the end points of the sub-intervals used by d01rkc along with the integral contributions and error estimates over the sub-intervals.
Specifically, for i=1,2,,n, let ri denote the approximation to the value of the integral over the sub-interval [ai,bi] in the partition of [a,b] and ei be the corresponding absolute error estimate. Then, ai bi f(x) dx ri and result = i=1 n ri . The value of n is returned in iinfo[0], and the values ai, bi, ei and ri are stored consecutively in the array rinfo, that is: The total number of abscissae at which the function was evaluated is returned in iinfo[1].

10 Example

This example computes
0 2π x sin(30x) cosx   dx .  

10.1 Program Text

Program Text (d01rkce.c)

10.2 Program Data


10.3 Program Results

Program Results (d01rkce.r)