The routine may be called by the names d01apf or nagf_quad_dim1_fin_wsing.
3Description
d01apf is based on the QUADPACK routine QAWSE (see Piessens et al. (1983)) and integrates a function of the form , where the weight function may have algebraico-logarithmic singularities at the end points and/or . The strategy is a modification of that in d01rkf. We start by bisecting the original interval and applying modified Clenshaw–Curtis integration of orders and to both halves. Clenshaw–Curtis integration is then used on all sub-intervals which have or as one of their end points (see Piessens et al. (1974)). On the other sub-intervals Gauss–Kronrod (– point) integration is carried out.
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 d01apf 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 d01apf. If your code inadvertently does return any NaNs or infinities, d01apf 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.
Constraint:
.
4: – Real (Kind=nag_wp)Input
On entry: the argument in the weight function.
Constraint:
.
5: – Real (Kind=nag_wp)Input
On entry: the argument in the weight function.
Constraint:
.
6: – IntegerInput
On entry: indicates which weight function is to be used.
.
.
.
.
Constraint:
, , or .
7: – Real (Kind=nag_wp)Input
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
8: – Real (Kind=nag_wp)Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
9: – Real (Kind=nag_wp)Output
On exit: the approximation to the integral .
10: – Real (Kind=nag_wp)Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for .
11: – Real (Kind=nag_wp) arrayOutput
On exit: details of the computation see Section 9 for more information.
12: – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which d01apf 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.
Suggested value:
to is adequate for most problems.
Constraint:
.
13: – Integer arrayOutput
On exit: contains the actual number of sub-intervals used. The rest of the array is used as workspace.
14: – IntegerInput
On entry: the dimension of the array iw as declared in the (sub)program from which d01apf is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed liw.
Suggested value:
.
Constraint:
.
15: – 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 d01apf 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 the position of a discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called on the subranges. 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, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
An unexpected error has been triggered by this routine. Please
contact NAG.
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.
7Accuracy
d01apf 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
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d01apf is not threaded in any implementation.
9Further Comments
The time taken by d01apf 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 d01apf 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
array w,
that is: