NAG Library Routine Document

D01AKF

+− 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

D01AKF is an adaptive integrator, especially suited to oscillating, nonsingular integrands, which calculates an approximation to the integral of a function

f (x)

over a finite interval

[a, b]

I = \int_{a}^{b} f (x) d x .

2 Specification

SUBROUTINE D01AKF (

F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL)

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

3 Description

D01AKF is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive routine, using the Gauss

30

-point and Kronrod

61

-point rules. A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described in Piessens et al. (1983).

Because D01AKF is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.

D01AKF requires you to supply a function to evaluate the integrand at a single point.

The routine D01AUF uses an identical algorithm but requires you to supply a subroutine to evaluate the integrand at an array of points. Therefore D01AUF will be more efficient if the evaluation can be performed in vector mode on a vector-processing machine.

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 (1973) An algorithm for automatic integration Angew. Inf. 15 399–401

Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

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 D01AKF 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 D01AKF 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 D01AKF 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, if you are not familiar with this parameter, the recommended value is

0

. 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).

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 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 $IFAIL = 1$ .

$IFAIL = 4$

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

7 Accuracy

D01AKF 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 D01AKF 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 D01AKF 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}

. 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}^{2 π} x \sin (30 x) \cos x d x .

9.1 Program Text

Program Text (d01akfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (d01akfe.r)

D01AKF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentD01AKF