D01RGF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

D01RGF

+− 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

D01RGF is a general purpose integrator 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 .

The routine is suitable as a general purpose integrator, and can be used when the integrand has singularities and infinities. In particular, the routine can continue if the subroutine F explicitly returns a quiet or signalling NaN or a signed infinity.

2 Specification

SUBROUTINE D01RGF (

A, B, F, EPSABS, EPSREL, DINEST, ERREST, NEVALS, IUSER, RUSER, IFAIL)

INTEGER	NEVALS, IUSER(*), IFAIL
REAL (KIND=nag_wp)	A, B, EPSABS, EPSREL, DINEST, ERREST, RUSER(*)
EXTERNAL	F

3 Description

D01RGF uses the algorithm described in Gonnet (2010). It is an adaptive algorithm, similar to the QUADPACK routine QAGS (see Piessens et al. (1983), see also D01ATF, D01AUF and D01RAF) but includes significant differences regarding how the integrand is represented, how the integration error is estimated and how singularities and divergent integrals are treated. The local error estimation is described in Gonnet (2010).

D01RGF requires a subroutine to evaluate the integrand at an array of different points and is therefore particularly efficient when the evaluation can be performed in vector mode on a vector-processing machine.

4 References

Gonnet P (2010) Increasing the reliability of adaptive quadrature using explicit interpolants ACM Trans. Math. software 37 26

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

5 Parameters

1: A – REAL (KIND=nag_wp)Input

On entry:

a

, the lower limit of integration.

2: B – REAL (KIND=nag_wp)Input

On entry:

b

, the upper limit of integration. It is not necessary that

a < b

.

Note: if

A = B

, the routine will immediately return

DINEST = 0.0

,

ERREST = 0.0

and

NEVALS = 0

.

3: F – SUBROUTINE, supplied by the user.External Procedure

F must return the value of the integrand

f

at a set of points.

The specification of F is:

SUBROUTINE F (

X, NX, FV, IFLAG, IUSER, RUSER)

INTEGER	NX, IFLAG, IUSER(*)
REAL (KIND=nag_wp)	X(NX), FV(NX), RUSER(*)

1: X(NX) – REAL (KIND=nag_wp) arrayInput: On entry: the abscissae, $x_{i}$ , for $i = 1, 2, \dots, NX$ , at which function values are required.
2: NX – INTEGERInput: On entry: the number of abscissae at which a function value is required.
3: FV(NX) – REAL (KIND=nag_wp) arrayOutput: On exit: FV must contain the values of the integrand $f$ . $FV (i) = f (x_{i})$ for all $i = 1, 2, \dots, NX$ .
4: IFLAG – INTEGERInput/Output: On entry: $IFLAG = 0$ .

On exit: set $IFLAG < 0$ to force an immediate exit with $IFAIL = - 1$ .
5: IUSER( $*$ ) – INTEGER arrayUser Workspace
6: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: F is called with the parameters IUSER and RUSER as supplied to D01RGF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01RGF is called. Parameters denoted as Input must not be changed by this procedure.

4: EPSABS – REAL (KIND=nag_wp)Input

On entry: the absolute accuracy required.

If EPSABS is negative,

|EPSABS|

is used. See Section 7.

If

EPSABS = 0.0

, only the relative error will be used.

5: EPSREL – REAL (KIND=nag_wp)Input

On entry: the relative accuracy required.

If EPSREL is negative,

|EPSREL|

is used. See Section 7.

If

EPSREL = 0.0

, only the absolute error will be used otherwise the actual value of EPSREL used by D01RGF is

\max (machine precision, |EPSREL|)

.

Constraint: at least one of EPSABS and EPSREL must be nonzero.

6: DINEST – REAL (KIND=nag_wp)Output

On exit: the estimate of the definite integral F.

7: ERREST – REAL (KIND=nag_wp)Output

On exit: the error estimate of the definite integral F.

8: NEVALS – INTEGEROutput

On exit: the number of function evaluations.

9: IUSER( $*$ ) – INTEGER arrayUser Workspace

10: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by D01RGF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.

11: 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

or

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: D01RGF 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 requested accuracy was not achieved. Consider using larger values of EPSABS and EPSREL.

$IFAIL = 2$: The integral is probably divergent or slowly convergent.

$IFAIL = 14$: Both $EPSABS = 0.0$ and $EPSREL = 0.0$ .

$IFAIL = - 1$: Exit requested from F with $IFLAG = ⟨value⟩$ .

$IFAIL = - 999$: Dynamic memory allocation failed.

7 Accuracy

D01RGF cannot guarantee, but in practice usually achieves, the following accuracy:

|I - DINEST| \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 ERREST which, in normal circumstances, satisfies

|I - DINEST| \leq ERREST \leq tol .

8 Further Comments

The time taken by D01RGF depends on the integrand and the accuracy required.

D01RGF is suitable for evaluating integrals that have singularities within the requested interval.

In particular, D01RGF accepts non-finite values on return from the user-supplied subroutine F, and will adapt the integration rule accordingly to eliminate such points. Non-finite values include NaNs and infinities.

9 Example

This example computes

\int_{- 1}^{1} \frac{\sin (x)}{x} \ln (10 (1 - x)) .

9.1 Program Text

Program Text (d01rgfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (d01rgfe.r)

D01RGF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012