NAG Library Routine Document
D01ASF
1 Purpose
D01ASF calculates an approximation to the sine or the cosine transform of a function
g over
a∞:
(for a user-specified value of
ω).
2 Specification
| SUBROUTINE D01ASF ( | G, A, OMEGA, KEY, EPSABS, RESULT, ABSERR, LIMLST, LST, ERLST, RSLST, IERLST, W, LW, IW, LIW, IFAIL) |
| INTEGER | KEY, LIMLST, LST, IERLST(LIMLST), LW, IW(LIW), LIW, IFAIL |
| double precision | G, A, OMEGA, EPSABS, RESULT, ABSERR, ERLST(LIMLST), RSLST(LIMLST), W(LW) |
| EXTERNAL | G |
3 Description
D01ASF is based on the QUADPACK routine QAWFE (see
Piessens et al. (1983)). It is an adaptive routine, designed to integrate a function of the form
gxwx over a semi-infinite interval, where
wx is either
sinωx or
cosω x.
Over successive intervals
integration is performed by the same algorithm as is used by
D01ANF. The intervals
Ck are of constant length
where
ω represents the largest integer less than or equal to
ω. Since
c equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function
g is positive and monotonically decreasing over
a∞. The algorithm, described in
Piessens et al. (1983), 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 by
Piessens et al. (1983).
If
ω=0 and
KEY=1, the routine uses the same algorithm as
D01AMF (with
EPSREL=0.0).
In contrast to the other routines in
Chapter D01, D01ASF works only with an
absolute error tolerance (
EPSABS). Over the interval
Ck it attempts to satisfy the absolute accuracy requirement
where
Uk=1-ppk-1, for
k=1,2,… and
p=0.9.
However, when difficulties occur during the integration over the
kth sub-interval
Ck such that the error flag
IERLSTk is nonzero, the accuracy requirement over subsequent intervals is relaxed. See
Piessens et al. (1983) for more details.
4 References
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: G – double precision FUNCTION, supplied by the user.External Procedure
G must return the value of the function
g at a given point
X.
The specification of
G is:
| double precision FUNCTION G ( | X) |
| double precision | X |
- 1: X – double precisionInput
On entry: the point at which the function g must be evaluated.
G must be declared as EXTERNAL in the (sub)program from which D01ASF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 2: A – double precisionInput
On entry: a, the lower limit of integration.
- 3: OMEGA – double precisionInput
On entry: the parameter ω in the weight function of the transform.
- 4: KEY – INTEGERInput
On entry: indicates which integral is to be computed.
- KEY=1
- wx=cosωx.
- KEY=2
- wx=sinωx.
Constraint:
KEY=1 or 2.
- 5: EPSABS – double precisionInput
On entry: the absolute accuracy required. If
EPSABS 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: LIMLST – INTEGERInput
On entry:
an upper bound on the number of intervals Ck needed for the integration.
Suggested value:
LIMLST=50 is adequate for most problems.
Constraint:
LIMLST≥3.
- 9: LST – INTEGEROutput
On exit: the number of intervals Ck actually used for the integration.
- 10: ERLST(LIMLST) – double precision arrayOutput
On exit: ERLSTk contains the error estimate corresponding to the integral contribution over the interval Ck, for k=1,2,…,LST.
- 11: RSLST(LIMLST) – double precision arrayOutput
On exit: RSLSTk contains the integral contribution over the interval Ck, for k=1,2,…,LST.
- 12: IERLST(LIMLST) – INTEGER arrayOutput
On exit:
IERLSTk contains the error flag corresponding to
RSLSTk, for
k=1,2,…,LST. See
Section 6.
- 13: W(LW) – double precision arrayWorkspace
- 14: LW – INTEGERInput
On entry: the dimension of the array
W as declared in the (sub)program from which D01ASF is called. The value of
LW (together with that of
LIW) imposes a bound on the number of sub-intervals into which each interval
Ck 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:
a value in the range 800 to 2000 is adequate for most problems.
Constraint:
LW≥4.
- 15: IW(LIW) – INTEGER arrayOutput
On exit: IW1 contains the maximum number of sub-intervals actually used for integrating over any of the intervals Ck. The rest of the array is used as workspace.
- 16: LIW – INTEGERInput
On entry: the dimension of the array
IW as declared in the (sub)program from which D01ASF is called. The number of sub-intervals into which each interval
Ck may be divided cannot exceed
LIW/2.
Suggested value:
LIW=LW/2.
Constraint:
LIW≥2.
- 17: 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: D01ASF 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 D01ASF on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by
EPSABS 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.
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 D01ASF on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by
EPSABS or increasing the amount of workspace.
Please note that divergence can occur with any nonzero value of
IFAIL.
- 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, | KEY≠1 or 2, |
| or | LIMLST<3. |
- IFAIL=7
-
Bad integration behaviour occurs within one or more of the intervals
Ck. Location and type of the difficulty involved can be determined from the vector
IERLST.
- IFAIL=8
-
Maximum number of intervals
Ck (
=LIMLST) allowed has been achieved. Increase the value of
LIMLST to allow more cycles.
- IFAIL=9
-
The extrapolation table constructed for convergence acceleration of the series formed by the integral contribution over the intervals Ck, does not converge to the required accuracy.
- IFAIL=10
In the cases
IFAIL=7,
8 or
9, additional information about the cause of the error can be obtained from the array
IERLST, as follows:
- IERLSTk=1
- The maximum number of subdivisions=minLW/4,LIW/2 has been achieved on the kth interval.
- IERLSTk=2
- Occurrence of round-off error is detected and prevents the tolerance imposed on the kth interval from being achieved.
- IERLSTk=3
- Extremely bad integrand behaviour occurs at some points of the kth interval.
- IERLSTk=4
- The integration procedure over the kth interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.
- IERLSTk=5
- The integral over the kth interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of IERLSTk.
7 Accuracy
D01ASF cannot guarantee, but in practice usually achieves, the following accuracy:
where
EPSABS is the user-specified absolute error tolerance. Moreover, it returns the quantity
ABSERR, which, in normal circumstances, satisfies
8 Further Comments
None.
9 Example
9.1 Program Text
Program Text (d01asfe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d01asfe.r)