# NAG Library Routine Document

## 1Purpose

d01aqf calculates an approximation to the Hilbert transform of a function $g\left(x\right)$ over $\left[a,b\right]$:
 $I=∫abgx x-c dx$
for user-specified values of $a$, $b$ and $c$.

## 2Specification

Fortran Interface
 Subroutine d01aqf ( g, a, b, c, w, lw, iw, liw,
 Integer, Intent (In) :: lw, liw Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iw(liw) Real (Kind=nag_wp), External :: g Real (Kind=nag_wp), Intent (In) :: a, b, c, epsabs, epsrel Real (Kind=nag_wp), Intent (Out) :: result, abserr, w(lw)
C Header Interface
#include nagmk26.h
 void d01aqf_ (double (NAG_CALL *g)(const double *x),const double *a, const double *b, const double *c, const double *epsabs, const double *epsrel, double *result, double *abserr, double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)

## 3Description

d01aqf is based on the QUADPACK routine QAWC (see Piessens et al. (1983)) and integrates a function of the form $g\left(x\right)w\left(x\right)$, where the weight function
 $wx=1x-c$
is that of the Hilbert transform. (If $a 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 $c$ is never the end point of a sub-interval (see Piessens et al. (1976)). On each sub-interval $\left({c}_{1},{c}_{2}\right)$ modified Clenshaw–Curtis integration of orders $12$ and $24$ is performed if ${c}_{1}-d\le c\le {c}_{2}+d$ where $d=\left({c}_{2}-{c}_{1}\right)/20$. Otherwise the Gauss
$7$-point and Kronrod $15$-point rules are used. The local error estimation is described by
Piessens et al. (1983).

## 4References

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
Piessens R, van Roy–Branders M and Mertens I (1976) The automatic evaluation of Cauchy principal value integrals Angew. Inf. 18 31–35

## 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
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 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:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $a$, the lower limit of integration.
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $b$, the upper limit of integration. It is not necessary that $a.
4:     $\mathbf{c}$ – Real (Kind=nag_wp)Input
On entry: the argument $c$ in the weight function.
Constraint: ${\mathbf{c}}$ must not equal a or b.
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:     $\mathbf{epsrel}$ – Real (Kind=nag_wp)Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
7:     $\mathbf{result}$ – Real (Kind=nag_wp)Output
On exit: the approximation to the integral $I$.
8:     $\mathbf{abserr}$ – Real (Kind=nag_wp)Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\left|I-{\mathbf{result}}\right|$.
9:     $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: details of the computation see Section 9 for more information.
10:   $\mathbf{lw}$ – 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 ${\mathbf{lw}}/4$. The more difficult the integrand, the larger lw should be.
Suggested value: ${\mathbf{lw}}=800$ to $2000$ is adequate for most problems.
Constraint: ${\mathbf{lw}}\ge 4$.
11:   $\mathbf{iw}\left({\mathbf{liw}}\right)$ – Integer arrayOutput
On exit: ${\mathbf{iw}}\left(1\right)$ contains the actual number of sub-intervals used. The rest of the array is used as workspace.
12:   $\mathbf{liw}$ – 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.
Suggested value: ${\mathbf{liw}}={\mathbf{lw}}/4$.
Constraint: ${\mathbf{liw}}\ge 1$.
13:   $\mathbf{ifail}$ – 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).

## 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: d01aqf 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 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.
${\mathbf{ifail}}=2$
Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.
${\mathbf{ifail}}=3$
Extremely bad local behaviour of $g\left(x\right)$ causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of ${\mathbf{ifail}}={\mathbf{1}}$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{c}}={\mathbf{a}}$ or ${\mathbf{c}}={\mathbf{b}}$.
${\mathbf{ifail}}=5$
 On entry, ${\mathbf{lw}}<4$, or ${\mathbf{liw}}<1$.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

d01aqf cannot guarantee, but in practice usually achieves, the following accuracy:
 $I-result≤tol,$
where
 $tol=maxepsabs,epsrel×I ,$
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances satisfies:
 $I-result≤abserr≤tol.$

## 8Parallelism and Performance

d01aqf is not threaded in any implementation.

## 9Further Comments

The time taken by d01aqf depends on the integrand and the accuracy required.
If ${\mathbf{ifail}}\ne {\mathbf{0}}$ 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 $i=1,2,\dots ,n$, let ${r}_{i}$ denote the approximation to the value of the integral over the sub-interval [${a}_{i},{b}_{i}$] in the partition of $\left[a,b\right]$ and ${e}_{i}$ be the corresponding absolute error estimate. Then, $\underset{{a}_{i}}{\overset{{b}_{i}}{\int }}g\left(x\right)w\left(x\right)dx\simeq {r}_{i}$ and ${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$. The value of $n$ is returned in ${\mathbf{iw}}\left(1\right)$, and the values ${a}_{i}$, ${b}_{i}$, ${e}_{i}$ and ${r}_{i}$ are stored consecutively in the array w, that is:
• ${a}_{i}={\mathbf{w}}\left(i\right)$,
• ${b}_{i}={\mathbf{w}}\left(n+i\right)$,
• ${e}_{i}={\mathbf{w}}\left(2n+i\right)$ and
• ${r}_{i}={\mathbf{w}}\left(3n+i\right)$.

## 10Example

This example computes the Cauchy principal value of
 $∫ -1 1 dx x2 + 0.012 x - 12 .$

### 10.1Program Text

Program Text (d01aqfe.f90)

None.

### 10.3Program Results

Program Results (d01aqfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017