NAG Library Routine Document

Integer, Intent (In)	::	key, 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, omega, epsabs, epsrel
Real (Kind=nag_wp), Intent (Out)	::	result, abserr, w(lw)

C Header Interface

#include <nagmk26.h>

void	d01anf_ ( double (NAG_CALL g)(const double x), const double a, const double b, const double omega, const Integer key, const double epsabs, const double epsrel, double result, double abserr, double w[], const Integer lw, Integer iw[], const Integer liw, Integer *ifail)

3

Description

d01anf is based on the QUADPACK routine QFOUR (see Piessens et al. (1983)). It is an adaptive routine, designed to integrate a function of the form

g (x) w (x)

, where

w (x)

is either

\sin (ω x)

\cos (ω x)

. If a sub-interval has length

L = |b - a| 2^{- l}

then the integration over this sub-interval is performed by means of a modified Clenshaw–Curtis procedure (see Piessens and Branders (1975)) if

L ω > 4

and

l \leq 20 .

In this case a Chebyshev series approximation of degree

24

is used to approximate

g (x)

, while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree

12

. If the above conditions do not hold then Gauss

7

-point and Kronrod

15

-point rules are used. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined in Malcolm and Simpson (1976)) together with the

ε

-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described in
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 and Branders M (1975) Algorithm 002: computation of oscillating integrals J. Comput. Appl. Math. 1 153–164

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} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

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 d01anf 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 d01anf. If your code inadvertently does return any NaNs or infinities, d01anf 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: $omega$ – Real (Kind=nag_wp)Input

On entry: the argument

ω

in the weight function of the transform.

5: $key$ – IntegerInput

On entry: indicates which integral is to be computed.

$key = 1$: $w (x) = \cos (ω x)$ .
$key = 2$: $w (x) = \sin (ω x)$ .

Constraint:

key = 1

2

6: $epsabs$ – Real (Kind=nag_wp)Input

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

7: $epsrel$ – Real (Kind=nag_wp)Input

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

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

On exit: the approximation to the integral

I

9: $abserr$ – 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|

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

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

11: $lw$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which d01anf 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

12: $iw (liw)$ – Integer arrayOutput

On exit:

iw (1)

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

13: $liw$ – IntegerInput

On entry: the dimension of the array iw as declared in the (sub)program from which d01anf is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed

liw / 2

Suggested value:

liw = lw / 2

Constraint:

liw \geq 2

14: $ifail$ – 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: d01anf 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 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 the integrator 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 and epsrel, or increasing the amount of workspace.

$ifail = 2$: Round-off error prevents the requested tolerance from being achieved: $epsabs = 〈value〉$ and $epsrel = 〈value〉$ .

$ifail = 3$: Extremely bad integrand behaviour occurs around the sub-interval $(〈value〉, 〈value〉)$ . The same advice applies as in the case of $ifail = 1$ .

$ifail = 4$: 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 $ifail = 1$ .

$ifail = 5$: The integral is probably divergent or slowly convergent.

$ifail = 6$: On entry, $key = 〈value〉$ .
Constraint: $key \geq 1$ and $key \leq 2$ .

$ifail = 7$: On entry, $liw = 〈value〉$ .
Constraint: $liw \geq 2$ .

On entry, $lw = 〈value〉$ .
Constraint: $lw \geq 4$ .

$ifail = - 99$: 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

d01anf 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 tolerances. Moreover, it returns the quantity abserr which in normal circumstances, satisfies

|I - result| \leq abserr \leq tol .

8

Parallelism and Performance

d01anf is not threaded in any implementation.

9

Further Comments

The time taken by d01anf 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 d01anf 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}

unless d01anf terminates while testing for divergence of the integral (see Section 3.4.3 of Piessens et al. (1983)). In this case, result (and abserr) are taken to be the values returned from the extrapolation process. 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

\int_{0}^{1} \ln x \sin (10 π x) d x .

10.1

Program Text

Program Text (d01anfe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (d01anfe.r)

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

D01 (quad) Chapter Contents

D01 (quad) Chapter Introduction

NAG Library Routine Document

d01anf (dim1_fin_wtrig)

▸▿ 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