NAG Library Routine Document

D01ASF

INTEGER	KEY, LIMLST, LST, IERLST(LIMLST), LW, IW(LIW), LIW, IFAIL
REAL (KIND=nag_wp)	G, A, OMEGA, EPSABS, RESULT, ABSERR, ERLST(LIMLST), RSLST(LIMLST), W(LW)
EXTERNAL	G

3 Description

D01ASF is based on the QUADPACK routine QAWFE (see Piessens et al. (1983)). It is an adaptive routine, designed to integrate a function of the form

g (x) w (x)

over a semi-infinite interval, where

w (x)

is either

\sin (ω x)

\cos (ω x)

Over successive intervals

C_{k} = [a + (k - 1) c, a + k c], k = 1, 2, \dots, LST

integration is performed by the same algorithm as is used by D01ANF. The intervals

C_{k}

are of constant length

c = \{2 [|ω|] + 1\} π / |ω|, ω \neq 0,

where

[|ω|]

represents the largest integer less than or equal to

|ω|

. Since

c

equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function

g

is positive and monotonically decreasing over

[a, \infty)

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

ε

-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described by Piessens et al. (1983).

ω = 0

and

KEY = 1

, the routine uses the same algorithm as D01AMF (with

EPSREL = 0.0

In contrast to the other routines in Chapter D01, D01ASF works only with an absolute error tolerance (EPSABS). Over the interval

C_{k}

it attempts to satisfy the absolute accuracy requirement

{EPSA}_{k} = U_{k} \times EPSABS,

where

U_{k} = (1 - p) p^{k - 1}

, for

k = 1, 2, \dots

and

p = 0.9

However, when difficulties occur during the integration over the

k

th sub-interval

C_{k}

such that the error flag

IERLST (k)

is nonzero, the accuracy requirement over subsequent intervals is relaxed. See Piessens et al. (1983) for more details.

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

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 D01ASF 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: OMEGA – REAL (KIND=nag_wp)Input

On entry: the parameter

ω

in the weight function of the transform.

4: 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

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: RESULT – REAL (KIND=nag_wp)Output

On exit: the approximation to the integral

I

7: 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|

8: LIMLST – INTEGERInput

On entry: an upper bound on the number of intervals

C_{k}

needed for the integration.

Suggested value:

LIMLST = 50

is adequate for most problems.

Constraint:

LIMLST \geq 3

9: LST – INTEGEROutput

On exit: the number of intervals

C_{k}

actually used for the integration.

10: ERLST(LIMLST) – REAL (KIND=nag_wp) arrayOutput

On exit:

ERLST (k)

contains the error estimate corresponding to the integral contribution over the interval

C_{k}

, for

k = 1, 2, \dots, LST

11: RSLST(LIMLST) – REAL (KIND=nag_wp) arrayOutput

On exit:

RSLST (k)

contains the integral contribution over the interval

C_{k}

, for

k = 1, 2, \dots, LST

12: IERLST(LIMLST) – INTEGER arrayOutput

On exit:

IERLST (k)

contains the error flag corresponding to

RSLST (k)

, for

k = 1, 2, \dots, LST

. See Section 6.

13: W(LW) – REAL (KIND=nag_wp) arrayWorkspace

14: LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which D01ASF is called. The value of LW (together with that of LIW) imposes a bound on the number of sub-intervals into which each interval

C_{k}

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: a value in the range

800

2000

is adequate for most problems.

Constraint:

LW \geq 4

15: IW(LIW) – INTEGER arrayOutput

On exit:

IW (1)

contains the maximum number of sub-intervals actually used for integrating over any of the intervals

C_{k}

. The rest of the array is used as workspace.

16: LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which D01ASF is called. The number of sub-intervals into which each interval

C_{k}

may be divided cannot exceed

LIW / 2

Suggested value:

LIW = LW / 2

Constraint:

LIW \geq 2

17: 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: D01ASF 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 D01ASF on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by EPSABS 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 integrand behaviour 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 $IFAIL = 1$ .

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 D01ASF on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by EPSABS or increasing the amount of workspace.

Please note that divergence can occur with any nonzero value of IFAIL.

$IFAIL = 5$: 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 $2$ ,
or	$LIMLST < 3$ .

$IFAIL = 7$: Bad integration behaviour occurs within one or more of the intervals $C_{k}$ . Location and type of the difficulty involved can be determined from the vector IERLST.

$IFAIL = 8$: Maximum number of intervals $C_{k}$ ( $= LIMLST$ ) allowed has been achieved. Increase the value of LIMLST to allow more cycles.

$IFAIL = 9$: The extrapolation table constructed for convergence acceleration of the series formed by the integral contribution over the intervals $C_{k}$ , does not converge to the required accuracy.

$IFAIL = 10$

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

In the cases

IFAIL = 7

8

9

, additional information about the cause of the error can be obtained from the array IERLST, as follows:

$IERLST (k) = 1$: The maximum number of $subdivisions = \min (LW / 4, LIW / 2)$ has been achieved on the $k$ th interval.
$IERLST (k) = 2$: Occurrence of round-off error is detected and prevents the tolerance imposed on the $k$ th interval from being achieved.
$IERLST (k) = 3$: Extremely bad integrand behaviour occurs at some points of the $k$ th interval.
$IERLST (k) = 4$: The integration procedure over the $k$ th interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.
$IERLST (k) = 5$: The integral over the $k$ th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of $IERLST (k)$ .