NAG Library Routine Document

D01ALF

D01ALF is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to D01AJF, but allows you to supply ‘break points’, points at which the integrand is known to be difficult. It employs an adaptive algorithm, using the Gauss

10

-point and Kronrod

21

-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 user-supplied ‘break points’ always occur as the end points of some sub-interval during the adaptive process. The local error estimation is described in Piessens et al. (1983).

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 D01ALF 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: NPTS – INTEGERInput

On entry: the number of user-supplied break points within the integration interval.

Constraint:

NPTS \geq 0

and

NPTS < \min ((LW - 2 \times NPTS - 4) / 4, (LIW - NPTS - 2) / 2)

5: POINTS( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array POINTS must be at least

\max (1, NPTS)

On entry: the user-specified break points.

Constraint: the break points must all lie within the interval of integration (but may be supplied in any order).

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 D01ALF 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 - 2 \times NPTS - 4) / 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 2 \times NPTS + 8

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 D01ALF is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed

(LIW - NPTS - 2) / 2

Suggested value:

LIW = LW / 2

Constraint:

LIW \geq NPTS + 4

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: D01ALF 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.) it should be supplied to the routine as an element of the vector POINTS. 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.

$IFAIL = 6$: The input is invalid: break points are specified outside the integration range, $NPTS > \min ((LW - 2 \times NPTS - 4) / 4, (LIW - NPTS - 2) / 2)$ or $NPTS < 0$ . RESULT and ABSERR are set to zero.

$IFAIL = 7$

On entry,	$LW < 2 \times NPTS + 8$ ,
or	$LIW < NPTS + 4$ .

7 Accuracy

D01ALF 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 D01ALF 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 D01ALF 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}} f (x) d x ≃ r_{i}

and

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

unless D01ALF 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
$r_{i} = W (3 n + i)$ .

9 Example

This example computes

\int_{0}^{1} \frac{1}{\sqrt{|x - 1 / 7|}} d x .

A break point is specified at

x = 1 / 7

, at which point the integrand is infinite. (For definiteness the function FST returns the value

0.0

at this point.)

9.1 Program Text

Program Text (d01alfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (d01alfe.r)

D01ALF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentD01ALF

+− 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

NAG Library Routine Document

D01ALF