NAG Library Routine Document

d01amf (dim1_inf)

1
Purpose

d01amf calculates an approximation to the integral of a function fx over an infinite or semi-infinite interval a,b:
I= ab fx dx .  

2
Specification

Fortran Interface
Subroutine d01amf ( f, bound, inf, epsabs, epsrel, result, abserr, w, lw, iw, liw, ifail)
Integer, Intent (In):: inf, lw, liw
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: iw(liw)
Real (Kind=nag_wp), External:: f
Real (Kind=nag_wp), Intent (In):: bound, epsabs, epsrel
Real (Kind=nag_wp), Intent (Out):: result, abserr, w(lw)
C Header Interface
#include <nagmk26.h>
void  d01amf_ (
double (NAG_CALL *f)(const double *x),
const double *bound, const Integer *inf, const double *epsabs, const double *epsrel, double *result, double *abserr, double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)

3
Description

d01amf is based on the QUADPACK routine QAGI (see Piessens et al. (1983)). The entire infinite integration range is first transformed to 0,1  using one of the identities:
- a fx dx = 01 f a - 1-tt 1t2 dt  
a fx dx = 01 f a+ 1-tt 1t2 dt  
- fx dx = 0 fx + f-x dx = 01 ​ ​ f 1-tt + f -1+t t 1t2 dt  
where a represents a finite integration limit. An adaptive procedure, based on the Gauss 7-point and Kronrod 15-point rules, is then employed on the transformed integral. The algorithm, described in de Doncker (1978), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the ε-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described in Piessens et al. (1983).

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 emSn transformation Math. Tables Aids Comput. 10 91–96

5
Arguments

1:     f – real (Kind=nag_wp) Function, supplied by the user.External Procedure
f must return the value of the integrand f at a given point.
The specification of f is:
Fortran Interface
Function f ( x)
Real (Kind=nag_wp):: f
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include <nagmk26.h>
double  f (const double *x)
1:     x – Real (Kind=nag_wp)Input
On entry: the point at which the integrand f must be evaluated.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01amf 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 d01amf. If your code inadvertently does return any NaNs or infinities, d01amf is likely to produce unexpected results.
2:     bound – Real (Kind=nag_wp)Input
On entry: the finite limit of the integration range (if present). bound is not used if the interval is doubly infinite.
3:     inf – IntegerInput
On entry: indicates the kind of integration range.
inf=1
The range is bound,+.
inf=-1
The range is -,bound.
inf=2
The range is -,+.
Constraint: inf=-1, 1 or 2.
4:     epsabs – Real (Kind=nag_wp)Input
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
5:     epsrel – Real (Kind=nag_wp)Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
6:     result – Real (Kind=nag_wp)Output
On exit: the approximation to the integral I.
7:     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.
8:     wlw – Real (Kind=nag_wp) arrayOutput
On exit: details of the computation see Section 9 for more information.
9:     lw – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which d01amf is called. The value of lw (together with that of liw) imposes a bound on the number of sub-intervals into which the interval of integration may be divided by the routine. The number of sub-intervals cannot exceed lw/4. The more difficult the integrand, the larger lw should be.
Suggested value: lw=800 to 2000 is adequate for most problems.
Constraint: lw4.
10:   iwliw – Integer arrayOutput
On exit: iw1 contains the actual number of sub-intervals used. The rest of the array is used as workspace.
11:   liw – IntegerInput
On entry: the dimension of the array iw as declared in the (sub)program from which d01amf is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed liw.
Suggested value: liw=lw/4.
Constraint: liw1.
12:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ifail0 on exit, the recommended value is -1. 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).
Note: d01amf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
ifail=1
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 the position of a local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity, etc.) you will probably gain from splitting up the interval at this point and calling d01amf on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs and epsrel, or increasing the amount of workspace.
ifail=2
Round-off error prevents the requested tolerance from being achieved: epsabs=value and epsrel=value.
ifail=3
Extremely bad integrand behaviour occurs around one of the sub-intervals value,value or value,value. The same advice applies as in the case of ifail=1.
Extremely bad integrand behaviour occurs around the sub-interval value,value. The same advice applies as in the case of ifail=1.
ifail=4
Round-off error is detected in the extrapolation table. The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained. The same advice applies as in the case of ifail=1.
ifail=5
The integral is probably divergent or slowly convergent.
ifail=6
On entry, inf=value.
Constraint: inf=-1, 1 or 2.
On entry, liw=value.
Constraint: liw1.
On entry, lw=value.
Constraint: lw4.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

d01amf cannot guarantee, but in practice usually achieves, the following accuracy:
I-resulttol,  
where
tol=maxepsabs,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
I-resultabserrtol.  

8
Parallelism and Performance

d01amf is not threaded in any implementation.

9
Further Comments

The time taken by d01amf depends on the integrand and the accuracy required.
If ifail0 on exit, then you may wish to examine the contents of the array w, which contains the end points of the sub-intervals used by d01amf along with the integral contributions and error estimates over these 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 fx dx ri  and result = i=1 n ri , unless d01amf terminates while testing for divergence of the integral (see Section 3.4.3 of Piessens et al. (1983)). In this case, result (and abserr) are taken to be the values returned from the extrapolation process. The value of n is returned in iw1, and the values ai, bi, ei and ri are stored consecutively in the array w, that is: Note:  this information applies to the integral transformed to 0,1 as described in Section 3, not to the original integral.

10
Example

This example computes
0 1 x+1 x dx .  
The exact answer is π.

10.1
Program Text

Program Text (d01amfe.f90)

10.2
Program Data

None.

10.3
Program Results

Program Results (d01amfe.r)