# NAG Library Routine Document

## 1Purpose

d01asf calculates an approximation to the sine or the cosine transform of a function $g$ over $\left[a,\infty \right)$:
 $I = ∫a∞ gx sinωx dx or I = ∫a∞ gx cosωx dx$
(for a user-specified value of $\omega$).

## 2Specification

Fortran Interface
 Subroutine d01asf ( g, a, key, lst, w, lw, iw, liw,
 Integer, Intent (In) :: key, limlst, lw, liw Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: lst, ierlst(limlst), iw(liw) Real (Kind=nag_wp), External :: g Real (Kind=nag_wp), Intent (In) :: a, omega, epsabs Real (Kind=nag_wp), Intent (Out) :: result, abserr, erlst(limlst), rslst(limlst), w(lw)
#include <nagmk26.h>
 void d01asf_ (double (NAG_CALL *g)(const double *x),const double *a, const double *omega, const Integer *key, const double *epsabs, double *result, double *abserr, const Integer *limlst, Integer *lst, double erlst[], double rslst[], Integer ierlst[], double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)

## 3Description

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 $g\left(x\right)w\left(x\right)$ over a semi-infinite interval, where $w\left(x\right)$ is either $\mathrm{sin}\left(\omega x\right)$ or $\mathrm{cos}\left(\omega x\right)$.
Over successive intervals
 $Ck = a+k-1c,a+kc , k=1,2,…,lst$
integration is performed by the same algorithm as is used by d01anf. The intervals ${C}_{k}$ are of constant length
 $c = 2ω +1 π/ω , ω≠0 ,$
where $\left[\left|\omega \right|\right]$ represents the largest integer less than or equal to $\left|\omega \right|$. 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 $\left[a,\infty \right)$. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the $\epsilon$-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described by Piessens et al. (1983).
If $\omega =0$ and ${\mathbf{key}}=1$, the routine uses the same algorithm as d01amf (with ${\mathbf{epsrel}}=0.0$).
In contrast to the other routines in Chapter D01, d01asf works only with an absolute error tolerance (epsabs). Over the interval ${C}_{k}$ it attempts to satisfy the absolute accuracy requirement
 $EPSAk = Uk×epsabs ,$
where ${U}_{\mathit{k}}=\left(1-p\right){p}^{\mathit{k}-1}$, for $\mathit{k}=1,2,\dots$ and $p=0.9$.
However, when difficulties occur during the integration over the $k$th sub-interval ${C}_{k}$ such that the error flag ${\mathbf{ierlst}}\left(k\right)$ 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 ${e}_{m}\left({S}_{n}\right)$ transformation Math. Tables Aids Comput. 10 91–96

## 5Arguments

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
#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.
${\mathbf{key}}=1$
$w\left(x\right)=\mathrm{cos}\left(\omega x\right)$.
${\mathbf{key}}=2$
$w\left(x\right)=\mathrm{sin}\left(\omega x\right)$.
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 $0$, . 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  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  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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ 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:
${\mathbf{ifail}}=1$
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: ${\mathbf{lw}}\ge 4$.
${\mathbf{ifail}}=-99$
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.
In the cases ${\mathbf{ifail}}={\mathbf{7}}$, ${\mathbf{8}}$ or ${\mathbf{9}}$, additional information about the cause of the error can be obtained from the array ierlst, as follows:
${\mathbf{ierlst}}\left(k\right)=1$
The maximum number of $\text{subdivisions}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lw}}/4,{\mathbf{liw}}/2\right)$ has been achieved on the $k$th interval.
${\mathbf{ierlst}}\left(k\right)=2$
Occurrence of round-off error is detected and prevents the tolerance imposed on the $k$th interval from being achieved.
${\mathbf{ierlst}}\left(k\right)=3$
Extremely bad integrand behaviour occurs at some points of the $k$th interval.
${\mathbf{ierlst}}\left(k\right)=4$
The integration procedure over the $k$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.
${\mathbf{ierlst}}\left(k\right)=5$
The integral over the $k$th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of ${\mathbf{ierlst}}\left(k\right)$.

## 7Accuracy

d01asf cannot guarantee, but in practice usually achieves, the following accuracy:
 $I-result≤epsabs,$
where epsabs is the user-specified absolute error tolerance. Moreover, it returns the quantity abserr, which, in normal circumstances, satisfies
 $I-result≤abserr≤epsabs.$

## 8Parallelism and Performance

d01asf is not threaded in any implementation.

None.

## 10Example

This example computes
 $∫0∞ 1x cosπx/2 dx .$

### 10.1Program Text

Program Text (d01asfe.f90)

None.

### 10.3Program Results

Program Results (d01asfe.r)