NAG Library Routine Document
D01AMF
1 Purpose
D01AMF calculates an approximation to the integral of a function
fx over an infinite or semi-infinite interval
a,b:
2 Specification
| SUBROUTINE D01AMF ( | F, BOUND, INF, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) |
| INTEGER | INF, LW, IW(LIW), LIW, IFAIL |
| double precision | F, BOUND, EPSABS, EPSREL, RESULT, ABSERR, W(LW) |
| EXTERNAL | F |
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:
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 Parameters
- 1: F – double precision 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:
| double precision FUNCTION F ( | X) |
| double precision | X |
- 1: X – double precisionInput
On entry: the point at which the integrand f must be evaluated.
F must be declared as EXTERNAL in the (sub)program from which D01AMF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 2: BOUND – double precisionInput
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 – double precisionInput
On entry: the absolute accuracy required. If
EPSABS is negative, the absolute value is used. See
Section 7.
- 5: EPSREL – double precisionInput
On entry: the relative accuracy required. If
EPSREL is negative, the absolute value is used. See
Section 7.
- 6: RESULT – double precisionOutput
On exit: the approximation to the integral I.
- 7: ABSERR – double precisionOutput
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for I-RESULT.
- 8: W(LW) – double precision arrayOutput
On exit: details of the computation, as described in
Section 8.
- 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:
LW≥4.
- 10: IW(LIW) – 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:
LIW≥1.
- 12: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
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 parameters may be useful even if
IFAIL≠0 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.
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. Consider requesting less accuracy.
- IFAIL=3
Extremely bad local integrand behaviour causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of IFAIL=1.
- IFAIL=4
The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best which can be obtained. The same advice applies as in the case of IFAIL=1.
- IFAIL=5
The integral is probably divergent, or slowly convergent. Please note that divergence can occur with any nonzero value of
IFAIL.
- IFAIL=6
-
| On entry, | LW<4, |
| or | LIW<1, |
| or | INF≠-1, 1 or 2. |
7 Accuracy
D01AMF cannot guarantee, but in practice usually achieves, the following accuracy:
where
and
EPSABS and
EPSREL are user-specified absolute and relative error tolerances. Moreover, it returns the quantity
ABSERR which, in normal circumstances, satisfies
8 Further Comments
The time taken by D01AMF depends on the integrand and the accuracy required.
If
IFAIL≠0 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:
- ai=Wi,
- bi=Wn+i,
- ei=W2n+i and
- ri=W3n+i.
Note: this information applies to the integral transformed to
0,1 as described in
Section 3, not to the original integral.
9 Example
This example computes
The exact answer is
π.
9.1 Program Text
Program Text (d01amfe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d01amfe.r)