NAG Library Routine Document
D01AQF
1 Purpose
D01AQF calculates an approximation to the Hilbert transform of a function
$g\left(x\right)$ over
$\left[a,b\right]$:
for userspecified values of
$a$,
$b$ and
$c$.
2 Specification
SUBROUTINE D01AQF ( 
G, A, B, C, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) 
INTEGER 
LW, IW(LIW), LIW, IFAIL 
REAL (KIND=nag_wp) 
G, A, B, C, EPSABS, EPSREL, RESULT, ABSERR, W(LW) 
EXTERNAL 
G 

3 Description
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
is that of the Hilbert transform. (If
$a<c<b$ 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 subinterval (see
Piessens et al. (1976)). On each subinterval
$\left({c}_{1},{c}_{2}\right)$ 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).
4 References
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
5 Arguments
 1: $\mathrm{G}$ – 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:
 1: $\mathrm{X}$ – 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.
 2: $\mathrm{A}$ – REAL (KIND=nag_wp)Input

On entry: $a$, the lower limit of integration.
 3: $\mathrm{B}$ – REAL (KIND=nag_wp)Input

On entry: $b$, the upper limit of integration. It is not necessary that $a<b$.
 4: $\mathrm{C}$ – REAL (KIND=nag_wp)Input

On entry: the argument $c$ in the weight function.
Constraint:
${\mathbf{C}}$ must not equal
A or
B.
 5: $\mathrm{EPSABS}$ – REAL (KIND=nag_wp)Input

On entry: the absolute accuracy required. If
EPSABS is negative, the absolute value is used. See
Section 7.
 6: $\mathrm{EPSREL}$ – REAL (KIND=nag_wp)Input

On entry: the relative accuracy required. If
EPSREL is negative, the absolute value is used. See
Section 7.
 7: $\mathrm{RESULT}$ – REAL (KIND=nag_wp)Output

On exit: the approximation to the integral $I$.
 8: $\mathrm{ABSERR}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\leftI{\mathbf{RESULT}}\right$.
 9: $\mathrm{W}\left({\mathbf{LW}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: details of the computation see
Section 9 for more information.
 10: $\mathrm{LW}$ – 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 subintervals into which the interval of integration may be divided by the routine. The number of subintervals 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: $\mathrm{IW}\left({\mathbf{LIW}}\right)$ – INTEGER arrayOutput

On exit: ${\mathbf{IW}}\left(1\right)$ contains the actual number of subintervals used. The rest of the array is used as workspace.
 12: $\mathrm{LIW}$ – INTEGERInput

On entry: the dimension of the array
IW as declared in the (sub)program from which D01AQF is called. The number of subintervals 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: $\mathrm{IFAIL}$ – 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
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.
 ${\mathbf{IFAIL}}=2$

Roundoff 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.
7 Accuracy
D01AQF cannot guarantee, but in practice usually achieves, the following accuracy:
where
and
EPSABS and
EPSREL are userspecified absolute and relative error tolerances. Moreover, it returns the quantity
ABSERR which, in normal circumstances satisfies:
8 Parallelism and Performance
D01AQF is not threaded in any implementation.
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 subintervals used by D01AQF along with the integral contributions and error estimates over these subintervals.
Specifically, for
$i=1,2,\dots ,n$, let
${r}_{i}$ denote the approximation to the value of the integral over the subinterval [
${a}_{i},{b}_{i}$] 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
${\mathbf{RESULT}}={\displaystyle \sum _{i=1}^{n}}{r}_{i}$. 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
 ${r}_{i}={\mathbf{W}}\left(3n+i\right)$.
10 Example
This example computes the Cauchy principal value of
10.1 Program Text
Program Text (d01aqfe.f90)
10.2 Program Data
None.
10.3 Program Results
Program Results (d01aqfe.r)