NAG Library Routine Document

D01AJF

-point rules. The algorithm, described in de Doncker (1978), 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 in Piessens et al. (1983).

The routine is suitable as a general purpose integrator, and can be used when the integrand has singularities, especially when these are of algebraic or logarithmic type.

D01AJF requires you to supply a function to evaluate the integrand at a single point.

The routine D01ATF uses an identical algorithm but requires you to supply a subroutine to evaluate the integrand at an array of points. Therefore D01ATF may be more efficient for some problem types and some machine architectures.

4 References

de Doncker E (1978) An adaptive extrapolation algorithm for automatic integration ACM SIGNUM Newsl. 13(2) 12–18

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: F – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

F must return the value of the integrand

f

at a given point.

The specification of F is:

FUNCTION F (

REAL (KIND=nag_wp) F

REAL (KIND=nag_wp)

1: X – REAL (KIND=nag_wp)Input: On entry: the point at which the integrand $f$ must be evaluated.

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01AJF 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: EPSABS – REAL (KIND=nag_wp)Input

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

5: EPSREL – REAL (KIND=nag_wp)Input

On entry: the relative accuracy required. If EPSREL 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: W(LW) – REAL (KIND=nag_wp) arrayOutput

On exit: details of the computation see Section 8 for more information.

9: LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which D01AJF 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

10: 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.

11: LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which D01AJF 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

12: 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: D01AJF 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 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 $IFAIL = 1$ .

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

$IFAIL = 6$

On entry,	$LW < 4$ ,
or	$LIW < 1$ .

7 Accuracy

D01AJF 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 Further Comments

The time taken by D01AJF 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 D01AJF along with the integral contributions and error estimates over the 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}} f (x) d x ≃ r_{i}

and

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

, unless D01AJF 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