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)

3

Description

d01aqf is based on the QUADPACK routine QAWC (see Piessens et al. (1983)) and integrates a function of the form

g (x) w (x)

, where the weight function

w (x) = \frac{1}{x - c}

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 sub-interval (see Piessens et al. (1976)). On each sub-interval

(c_{1}, c_{2})

modified Clenshaw–Curtis integration of orders

12

and

24

is performed if

c_{1} - d \leq c \leq c_{2} + d

where

d = (c_{2} - c_{1}) / 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: $g$ – 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 (

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

C Header Interface

#include nagmk26.h

double

g ( const double *x)

1: $x$ – 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: $a$ – Real (Kind=nag_wp)Input

On entry:

a

, the lower limit of integration.

3: $b$ – Real (Kind=nag_wp)Input

On entry:

b

, the upper limit of integration. It is not necessary that

a < b

4: $c$ – Real (Kind=nag_wp)Input

On entry: the argument

c

in the weight function.

Constraint:

c

must not equal a or b.

5: $epsabs$ – Real (Kind=nag_wp)Input

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

6: $epsrel$ – Real (Kind=nag_wp)Input

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

7: $result$ – Real (Kind=nag_wp)Output

On exit: the approximation to the integral

I

8: $abserr$ – Real (Kind=nag_wp)Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for

|I - result|

9: $w (lw)$ – Real (Kind=nag_wp) arrayOutput

On exit: details of the computation see Section 9 for more information.

10: $lw$ – 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

lw / 4

. The more difficult the integrand, the larger lw should be.

Suggested value:

lw = 800

2000

is adequate for most problems.

Constraint:

lw \geq 4

11: $iw (liw)$ – Integer arrayOutput

On exit:

iw (1)

contains the actual number of sub-intervals used. The rest of the array is used as workspace.

12: $liw$ – 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:

liw = lw / 4

Constraint:

liw \geq 1

13: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ 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 ​ 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

ifail \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 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:

$ifail = 1$: 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.

$ifail = 2$: Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.

$ifail = 3$: Extremely bad local behaviour of $g (x)$ causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of $ifail = 1$ .

$ifail = 4$

On entry,

c = a

c = b

$ifail = 5$

On entry,	$lw < 4$ ,
or	$liw < 1$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: 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:

|I - result| \leq tol,

where

tol = \max \{|epsabs|, |epsrel| \times |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| \leq abserr \leq tol .

8

Parallelism and Performance

d01aqf is not threaded in any implementation.

9

Further Comments

The time taken by d01aqf depends on the integrand and the accuracy required.

ifail \neq 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

[a, b]

and

e_{i}

be the corresponding absolute error estimate. Then,

\int_{a_{i}}^{b_{i}} g (x) w (x) d x ≃ r_{i}

and

result = \sum_{i = 1}^{n} r_{i}

. The value of

n

is returned in

iw (1)

, and the values

a_{i}

b_{i}

e_{i}

and

r_{i}

are stored consecutively in the array w, that is:

$a_{i} = w (i)$ ,
$b_{i} = w (n + i)$ ,
$e_{i} = w (2 n + i)$ and
$r_{i} = w (3 n + i)$ .

10

Example

This example computes the Cauchy principal value of

\int_{- 1}^{1} \frac{d x}{(x^{2} + {0.01}^{2}) (x - \frac{1}{2})} .

10.1

Program Text

Program Text (d01aqfe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (d01aqfe.r)

d01aqf (PDF version)

D01 (quad) Chapter Contents

D01 (quad) Chapter Introduction

NAG Library Manual

NAG Library Routine Document

d01aqf (dim1_fin_wcauchy)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results