The routine may be called by the names d01aqf or nagf_quad_dim1_fin_wcauchy.
d01aqf is based on the QUADPACK routine QAWC (see Piessens et al. (1983)) and integrates a function of the form , where the weight function
is that of the Hilbert transform. (If the integral has to be interpreted in the sense of a Cauchy principal value.) It is an adaptive routine which employs a ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)). Special care is taken to ensure that is never the end point of a sub-interval (see Piessens et al. (1976)). On each sub-interval modified Clenshaw–Curtis integration of orders and is performed if where . Otherwise the Gauss
-point and Kronrod -point rules are used. The local error estimation is described by Piessens et al. (1983).
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software1 129–146
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Piessens R, van Roy–Branders M and Mertens I (1976) The automatic evaluation of Cauchy principal value integrals Angew. Inf.18 31–35
1: – real (Kind=nag_wp) Function, supplied by the user.External Procedure
g must return the value of the function at a given point x.
On entry: the point at which the function must be evaluated.
g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01aqf is called. Arguments denoted as Input must not be changed by this procedure.
Note:g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01aqf. If your code inadvertently does return any NaNs or infinities, d01aqf is likely to produce unexpected results.
2: – Real (Kind=nag_wp)Input
On entry: , the lower limit of integration.
3: – Real (Kind=nag_wp)Input
On entry: , the upper limit of integration. It is not necessary that .
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
6: – Real (Kind=nag_wp)Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
7: – Real (Kind=nag_wp)Output
On exit: the approximation to the integral .
8: – Real (Kind=nag_wp)Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for .
9: – Real (Kind=nag_wp) arrayOutput
On exit: details of the computation see Section 9 for more information.
10: – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which d01aqf 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 . The more difficult the integrand, the larger lw should be.
to is adequate for most problems.
11: – Integer arrayOutput
On exit: contains the actual number of sub-intervals used. The rest of the array is used as workspace.
12: – IntegerInput
On entry: the dimension of the array iw as declared in the (sub)program from which d01aqf is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed liw.
13: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended since useful values can be provided in some output arguments even when on exit. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , 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 d01aqf 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: and .
Extremely bad integrand behaviour occurs around the sub-interval . The same advice applies as in the case of .
On entry, , and .
Constraint: and .
On entry, .
On entry, .
An unexpected error has been triggered by this routine. Please
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.
d01aqf cannot guarantee, but in practice usually achieves, the following accuracy:
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances satisfies:
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d01aqf is not threaded in any implementation.
The time taken by d01aqf depends on the integrand and the accuracy required.
If 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 d01aqf along with the integral contributions and error estimates over these sub-intervals.
Specifically, for , let denote the approximation to the value of the integral over the sub-interval  in the partition of and be the corresponding absolute error estimate. Then, and . The value of is returned in
and the values , , and are stored consecutively in the
This example computes the Cauchy principal value of