NAG Library Routine Document
D01ASF calculates an approximation to the sine or the cosine transform of a function
(for a user-specified value of
|SUBROUTINE D01ASF (
||G, A, OMEGA, KEY, EPSABS, RESULT, ABSERR, LIMLST, LST, ERLST, RSLST, IERLST, W, LW, IW, LIW, IFAIL)
||KEY, LIMLST, LST, IERLST(LIMLST), LW, IW(LIW), LIW, IFAIL
||G, A, OMEGA, EPSABS, RESULT, ABSERR, ERLST(LIMLST), RSLST(LIMLST), W(LW)
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
over a semi-infinite interval, where
Over successive intervals
integration is performed by the same algorithm as is used by D01ANF
. The intervals
are of constant length
represents the largest integer less than or equal to
equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function
is positive and monotonically decreasing over
. The algorithm, described in Piessens et al. (1983)
, incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)
) together with the
-algorithm (see Wynn (1956)
) to perform extrapolation. The local error estimation is described by Piessens et al. (1983)
, the routine uses the same algorithm as D01AMF
In contrast to the other routines in Chapter D01
, D01ASF works only with an absolute
error tolerance (EPSABS
). Over the interval
it attempts to satisfy the absolute accuracy requirement
However, when difficulties occur during the integration over the
such that the error flag
is nonzero, the accuracy requirement over subsequent intervals is relaxed. See Piessens et al. (1983)
for more details.
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 transformation Math. Tables Aids Comput. 10 91–96
- 1: G – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
must return the value of the function
at a given point X
The specification of G
- 1: X – REAL (KIND=nag_wp)Input
On entry: the point at which the function must be evaluated.
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01ASF is called. Parameters denoted as Input
be changed by this procedure.
- 2: A – REAL (KIND=nag_wp)Input
On entry: , the lower limit of integration.
- 3: OMEGA – REAL (KIND=nag_wp)Input
On entry: the parameter in the weight function of the transform.
- 4: KEY – INTEGERInput
: indicates which integral is to be computed.
- 5: EPSABS – REAL (KIND=nag_wp)Input
: the absolute accuracy required. If EPSABS
is negative, the absolute value is used. See Section 7
- 6: RESULT – REAL (KIND=nag_wp)Output
On exit: the approximation to the integral .
- 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 .
- 8: LIMLST – INTEGERInput
On entry: an upper bound on the number of intervals needed for the integration.
is adequate for most problems.
- 9: LST – INTEGEROutput
On exit: the number of intervals actually used for the integration.
- 10: ERLST(LIMLST) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the error estimate corresponding to the integral contribution over the interval , for .
- 11: RSLST(LIMLST) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the integral contribution over the interval , for .
- 12: IERLST(LIMLST) – INTEGER arrayOutput
contains the error flag corresponding to
. See Section 6
- 13: W(LW) – REAL (KIND=nag_wp) arrayWorkspace
- 14: LW – INTEGERInput
: 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
may be divided by the routine. The number of sub-intervals cannot exceed
. The more difficult the integrand, the larger LW
a value in the range to is adequate for most problems.
- 15: IW(LIW) – INTEGER arrayOutput
On exit: contains the maximum number of sub-intervals actually used for integrating over any of the intervals . The rest of the array is used as workspace.
- 16: LIW – INTEGERInput
: 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
may be divided cannot exceed
- 17: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
on exit, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, 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:
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.
Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.
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 .
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 .
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
The integral is probably divergent, or slowly convergent. Please note that divergence can occur with any nonzero value of IFAIL
|On entry,|| or ,|
Bad integration behaviour occurs within one or more of the intervals
. Location and type of the difficulty involved can be determined from the vector IERLST
Maximum number of intervals
) allowed has been achieved. Increase the value of LIMLST
to allow more cycles.
The extrapolation table constructed for convergence acceleration of the series formed by the integral contribution over the intervals , does not converge to the required accuracy.
In the cases
, additional information about the cause of the error can be obtained from the array IERLST
, as follows:
- The maximum number of has been achieved on the th interval.
- Occurrence of round-off error is detected and prevents the tolerance imposed on the th interval from being achieved.
- Extremely bad integrand behaviour occurs at some points of the th interval.
- The integration procedure over the th 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.
- The integral over the th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of .
D01ASF cannot guarantee, but in practice usually achieves, the following accuracy:
is the user-specified absolute error tolerance. Moreover, it returns the quantity ABSERR
, which, in normal circumstances, satisfies
9.1 Program Text
Program Text (d01asfe.f90)
9.2 Program Data
9.3 Program Results
Program Results (d01asfe.r)