NAG FL Interface
d01rkf (dim1_​fin_​osc_​fn)

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1 Purpose

d01rkf 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

Fortran Interface
Subroutine d01rkf ( f, a, b, key, epsabs, epsrel, maxsub, result, abserr, rinfo, iinfo, iuser, ruser, cpuser, ifail)
Integer, Intent (In) :: key, maxsub
Integer, Intent (Inout) :: iuser(*), ifail
Integer, Intent (Out) :: iinfo(max(maxsub,4))
Real (Kind=nag_wp), Intent (In) :: a, b, epsabs, epsrel
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: result, abserr, rinfo(4*maxsub)
Type (c_ptr), Intent (In) :: cpuser
External :: f
C Header Interface
#include <nag.h>
void  d01rkf_ (
void (NAG_CALL *f)(const double x[], const Integer *nx, double fv[], Integer *iflag, Integer iuser[], double ruser[], void **cpuser),
const double *a, const double *b, const Integer *key, const double *epsabs, const double *epsrel, const Integer *maxsub, double *result, double *abserr, double rinfo[], Integer iinfo[], Integer iuser[], double ruser[], void **cpuser, Integer *ifail)
The routine may be called by the names d01rkf or nagf_quad_dim1_fin_osc_fn.

3 Description

d01rkf is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive routine, 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 d01rkf is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.
d01rkf requires you to supply a (sub)routine 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 Subroutine, supplied by the user. External Procedure
f must return the values of the integrand f at a set of points.
The specification of f is:
Fortran Interface
Subroutine f ( x, nx, fv, iflag, iuser, ruser, cpuser)
Integer, Intent (In) :: nx
Integer, Intent (Inout) :: iflag, iuser(*)
Real (Kind=nag_wp), Intent (In) :: x(nx)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fv(nx)
Type (c_ptr), Intent (In) :: cpuser
C Header Interface
void  f (const double x[], const Integer *nx, double fv[], Integer *iflag, Integer iuser[], double ruser[], void **cpuser)
1: x(nx) Real (Kind=nag_wp) array 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(nx) Real (Kind=nag_wp) array Output
On exit: fv must contain the values of the integrand f. fv(i)=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 ifail=-1.
5: iuser(*) Integer array User Workspace
6: ruser(*) Real (Kind=nag_wp) array User Workspace
7: cpuser Type (c_ptr) User Workspace
f is called with the arguments iuser, ruser and cpuser as supplied to d01rkf. You should use the arrays iuser and ruser, and the data handle cpuser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01rkf is called. Arguments denoted as Input must not be changed by this procedure.
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01rkf. If your code inadvertently does return any NaNs or infinities, d01rkf is likely to produce unexpected results.
2: a Real (Kind=nag_wp) Input
On entry: a, the lower limit of integration.
3: b Real (Kind=nag_wp) Input
On entry: b, the upper limit of integration. It is not necessary that a<b.
Note: if a=b, the routine 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 Real (Kind=nag_wp) Input
On entry: εa, the absolute accuracy required. If epsabs is negative, εa=|epsabs|. See Section 7.
6: epsrel Real (Kind=nag_wp) 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 d01rkf 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 Real (Kind=nag_wp) Output
On exit: the approximation to the integral I.
9: abserr Real (Kind=nag_wp) 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) Real (Kind=nag_wp) array Output
On exit: details of the computation. See Section 9 for more information.
11: iinfo(max(maxsub,4)) Integer array Output
On exit: details of the computation. See Section 9 for more information.
12: iuser(*) Integer array User Workspace
13: ruser(*) Real (Kind=nag_wp) array User Workspace
14: cpuser Type (c_ptr) User Workspace
iuser, ruser and cpuser are not used by d01rkf, but are passed directly to f and may be used to pass information to this routine. If you do not need to reference cpuser, it should be initialized to c_null_ptr.
15: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases d01rkf may return useful information.
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.
Extremely bad integrand behaviour occurs around the sub-interval (value,value). The same advice applies as in the case of ifail=1.
On entry, key=value.
Constraint: key=1, 2, 3, 4, 5 or 6.
On entry, maxsub=value.
Constraint: maxsub1.
Exit from f with iflag<0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

d01rkf 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

Background information to multithreading can be found in the Multithreading documentation.
d01rkf is not threaded in any implementation.

9 Further Comments

The time taken by d01rkf depends on the integrand and the accuracy required.
If ifail=0, 1, 2 or 3, or if ifail=-1 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 d01rkf 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(1), 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(2).

10 Example

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

10.1 Program Text

Program Text (d01rkfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (d01rkfe.r)