NAG Library Routine Document

d01asf (dim1_inf_wtrig)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{g}$ – real (Kind=nag_wp) 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:

Fortran Interface

Function g ( x) Real (Kind=nag_wp) :: g Real (Kind=nag_wp), Intent (In) :: x

C Header Interface

#include <nagmk26.h> double  g (const double *x)

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the point at which the function $g$ must be evaluated.

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01asf 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 d01asf. If your code inadvertently does return any NaNs or infinities, d01asf is likely to produce unexpected results.

2:     $\mathbf{a}$ – Real (Kind=nag_wp)Input

On entry: $a$, the lower limit of integration.

3:     $\mathbf{omega}$ – Real (Kind=nag_wp)Input

On entry: the argument $\omega$ in the weight function of the transform.

4:     $\mathbf{key}$ – IntegerInput

On entry: indicates which integral is to be computed.

$\mathbf{key}=1$

$w\left(x\right)=\cos \left(\omega x\right)$.

$\mathbf{key}=2$

$w\left(x\right)=\sin \left(\omega x\right)$.

Constraint: $\mathbf{key}=1$ or $2$.

5:     $\mathbf{epsabs}$ – Real (Kind=nag_wp)Input

On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.

6:     $\mathbf{result}$ – Real (Kind=nag_wp)Output

On exit: the approximation to the integral $I$.

7:     $\mathbf{abserr}$ – Real (Kind=nag_wp)Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\left|I-\mathbf{result}\right|$.

8:     $\mathbf{limlst}$ – IntegerInput

On entry: an upper bound on the number of intervals $C_k$ needed for the integration.

Suggested value: $\mathbf{limlst}=50$ is adequate for most problems.

Constraint: $\mathbf{limlst}\ge 3$.

9:     $\mathbf{lst}$ – IntegerOutput

On exit: the number of intervals $C_k$ actually used for the integration.

10:   $\mathbf{erlst}\left(\mathbf{limlst}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{erlst}\left(\mathit{k}\right)$ contains the error estimate corresponding to the integral contribution over the interval $C_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,\mathbf{lst}$.

11:   $\mathbf{rslst}\left(\mathbf{limlst}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{rslst}\left(\mathit{k}\right)$ contains the integral contribution over the interval $C_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,\mathbf{lst}$.

12:   $\mathbf{ierlst}\left(\mathbf{limlst}\right)$ – Integer arrayOutput

On exit: $\mathbf{ierlst}\left(\mathit{k}\right)$ contains the error flag corresponding to $\mathbf{rslst}\left(\mathit{k}\right)$, for $\mathit{k}=1,2,\dots ,\mathbf{lst}$. See Section 6.

13:   $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) arrayWorkspace

14:   $\mathbf{lw}$ – 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 $C_k$ may be divided by the routine. The number of sub-intervals cannot exceed $\mathbf{lw}/4$. 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: $\mathbf{lw}\ge 4$.

15:   $\mathbf{iw}\left(\mathbf{liw}\right)$ – Integer arrayOutput

On exit: $\mathbf{iw}\left(1\right)$ contains the maximum number of sub-intervals actually used for integrating over any of the intervals $C_k$. The rest of the array is used as workspace.

16:   $\mathbf{liw}$ – 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 $C_k$ may be divided cannot exceed $\mathbf{liw}/2$.

Suggested value: $\mathbf{liw}=\mathbf{lw}/2$.

Constraint: $\mathbf{liw}\ge 2$.

17:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ 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 arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

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 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 or increasing the amount of workspace.

$\mathbf{ifail}=2$

Round-off error prevents the requested tolerance from being achieved: $\mathbf{epsabs}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=3$

Extremely bad integrand behaviour occurs around the sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$. The same advice applies as in the case of $\mathbf{ifail}=\mathbf{1}$.

$\mathbf{ifail}=4$

Round-off error is detected in the extrapolation table.
The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained. The same advice applies as in the case of $\mathbf{ifail}=\mathbf{1}$.
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.

$\mathbf{ifail}=5$

The integral is probably divergent or slowly convergent.

$\mathbf{ifail}=6$

On entry, $\mathbf{key}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{key}\le 2$.

On entry, $\mathbf{key}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{key}\ge 1$.

On entry, $\mathbf{limlst}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{limlst}\ge 3$.

$\mathbf{ifail}=7$

Bad integration behaviour occurs within interval $\mathit{K}$: $\mathit{K}=〈\mathit{\text{value}}〉$: $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$. Inspect ierlst for more details.

$\mathbf{ifail}=8$

The maximum number of intervals (limlst) has been reached: $\mathbf{limlst}=〈\mathit{\text{value}}〉$. Inspect ierlst for more details.

$\mathbf{ifail}=9$

Extrapolation does not converge to the requested accuracy. Inspect ierlst for more details.

$\mathbf{ifail}=10$

On entry, $\mathbf{liw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{liw}\ge 2$.

On entry, $\mathbf{lw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lw}\ge 4$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d01asfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (d01asfe.r)