NAG Library Routine Document

D01ANF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

D01ANF calculates an approximation to the sine or the cosine transform of a function

g

over

[a, b]

I = \int_{a}^{b} g (x) \sin (ω x) d x or I = \int_{a}^{b} g (x) \cos (ω x) d x

(for a user-specified value of

ω

2 Specification

SUBROUTINE D01ANF (

G, A, B, OMEGA, KEY, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL)

INTEGER	KEY, LW, IW(LIW), LIW, IFAIL
REAL (KIND=nag_wp)	G, A, B, OMEGA, EPSABS, EPSREL, RESULT, ABSERR, W(LW)
EXTERNAL	G

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 Parameters

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:

FUNCTION G (

REAL (KIND=nag_wp) G

REAL (KIND=nag_wp)

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. Parameters denoted as Input must not be changed by this procedure.

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 parameter

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 8 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 parameter you should refer to Section 3.3 in the Essential Introduction 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 parameters 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. 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 $IFAIL = 1$ .

$IFAIL = 4$: The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best which can be obtained. The same advice applies as in the case of $IFAIL = 1$ .

$IFAIL = 5$: The integral is probably divergent, or slowly convergent. Please note that divergence can occur with any nonzero value of IFAIL.

$IFAIL = 6$: On entry, $KEY \neq 1$ or $2$ .

$IFAIL = 7$

On entry,	$LW < 4$ ,
or	$LIW < 2$ .

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 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