NAG Library Routine Document

D01DAF

+− 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

D01DAF attempts to evaluate a double integral to a specified absolute accuracy by repeated applications of the method described by Patterson (1968) and Patterson (1969).

2 Specification

SUBROUTINE D01DAF (

YA, YB, PHI1, PHI2, F, ABSACC, ANS, NPTS, IFAIL)

INTEGER	NPTS, IFAIL
REAL (KIND=nag_wp)	YA, YB, PHI1, PHI2, F, ABSACC, ANS
EXTERNAL	PHI1, PHI2, F

3 Description

D01DAF attempts to evaluate a definite integral of the form

I = \int_{a}^{b} \int_{ϕ_{1} (y)}^{ϕ_{2} (y)} f (x, y) d x d y

where

a

and

b

are constants and

ϕ_{1} (y)

and

ϕ_{2} (y)

are functions of the variable

y

The integral is evaluated by expressing it as

I = \int_{a}^{b} F (y) d y,   where F (y) = \int_{ϕ_{1} (y)}^{ϕ_{2} (y)} f (x, y) d x .

Both the outer integral

I

and the inner integrals

F (y)

are evaluated by the method, described by Patterson (1968) and Patterson (1969), of the optimum addition of points to Gauss quadrature formulae.

This method uses a family of interlacing common point formulae. Beginning with the

3

-point Gauss rule, formulae using

7

15

31

63

127

and finally

255

points are derived. Each new formula contains all the points of the earlier formulae so that no function evaluations are wasted. Each integral is evaluated by applying these formulae successively until two results are obtained which differ by less than the specified absolute accuracy.

4 References

Patterson T N L (1968) On some Gauss and Lobatto based integration formulae Math. Comput. 22 877–881

Patterson T N L (1969) The optimum addition of points to quadrature formulae, errata Math. Comput. 23 892

5 Parameters

1: YA – REAL (KIND=nag_wp)Input

On entry:

a

, the lower limit of the integral.

2: YB – REAL (KIND=nag_wp)Input

On entry:

b

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

a < b

3: PHI1 – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

PHI1 must return the lower limit of the inner integral for a given value of

y

The specification of PHI1 is:

FUNCTION PHI1 (

REAL (KIND=nag_wp) PHI1

REAL (KIND=nag_wp)

1: Y – REAL (KIND=nag_wp)Input: On entry: the value of $y$ for which the lower limit must be evaluated.

PHI1 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01DAF is called. Parameters denoted as Input must not be changed by this procedure.

4: PHI2 – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

PHI2 must return the upper limit of the inner integral for a given value of

y

The specification of PHI2 is:

FUNCTION PHI2 (

REAL (KIND=nag_wp) PHI2

REAL (KIND=nag_wp)

1: Y – REAL (KIND=nag_wp)Input: On entry: the value of $y$ for which the upper limit must be evaluated.

PHI2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01DAF is called. Parameters denoted as Input must not be changed by this procedure.

5: 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 (

X, Y)

REAL (KIND=nag_wp) F

REAL (KIND=nag_wp)

X, Y

1: X – REAL (KIND=nag_wp)Input
2: Y – REAL (KIND=nag_wp)Input: On entry: the coordinates of the point $(x, y)$ 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 D01DAF is called. Parameters denoted as Input must not be changed by this procedure.

6: ABSACC – REAL (KIND=nag_wp)Input

On entry: the absolute accuracy requested.

7: ANS – REAL (KIND=nag_wp)Output

On exit: the estimated value of the integral.

8: NPTS – INTEGEROutput

On exit: the total number of function evaluations.

9: 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: D01DAF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: This indicates that $255$ points have been used in the outer integral and convergence has not been obtained. All the inner integrals have, however, converged. In this case ANS may still contain an approximate estimate of the integral.

$IFAIL = 10 \times n$: This indicates that the outer integral has converged but $n$ inner integrals have failed to converge with the use of $255$ points. In this case ANS may still contain an approximate estimate of the integral, but its reliability will decrease as $n$ increases.

$IFAIL = 10 \times n + 1$: This indicates that both the outer integral and $n$ of the inner integrals have not converged. ANS may still contain an approximate estimate of the integral, but its reliability will decrease as $n$ increases.

7 Accuracy

The absolute accuracy is specified by the variable ABSACC. If, on exit,

IFAIL = 0

then the result is most likely correct to this accuracy. Even if IFAIL is nonzero on exit, it is still possible that the calculated result could differ from the true value by less than the given accuracy.

8 Further Comments

The time taken by D01DAF depends upon the complexity of the integrand and the accuracy requested.

With Patterson's method accidental convergence may occasionally occur, when two estimates of an integral agree to within the requested accuracy, but both estimates differ considerably from the true result. This could occur in either the outer integral or in one or more of the inner integrals.

If it occurs in the outer integral then apparent convergence is likely to be obtained with considerably fewer integrand evaluations than may be expected. If it occurs in an inner integral, the incorrect value could make the function

F (y)

appear to be badly behaved, in which case a very large number of pivots may be needed for the overall evaluation of the integral. Thus both unexpectedly small and unexpectedly large numbers of integrand evaluations should be considered as indicating possible trouble. If accidental convergence is suspected, the integral may be recomputed, requesting better accuracy; if the new request is more stringent than the degree of accidental agreement (which is of course unknown), improved results should be obtained. This is only possible when the accidental agreement is not better than machine accuracy. It should be noted that the routine requests the same accuracy for the inner integrals as for the outer integral. In practice it has been found that in the vast majority of cases this has proved to be adequate for the overall result of the double integral to be accurate to within the specified value.

The routine is not well-suited to non-smooth integrands, i.e., integrands having some kind of analytic discontinuity (such as a discontinuous or infinite partial derivative of some low-order) in, on the boundary of, or near, the region of integration. Warning: such singularities may be induced by incautiously presenting an apparently smooth interval over the positive quadrant of the unit circle,

R

I = \int_{R} (x + y) d x d y .

This may be presented to D01DAF as

I = \int_{0}^{1} d y \int_{0}^{\sqrt{1 - y^{2}}} (x + y) d x = \int_{0}^{1} (\frac{1}{2} (1 - y^{2}) + y \sqrt{1 - y^{2}}) d y

but here the outer integral has an induced square-root singularity stemming from the way the region has been presented to D01DAF. This situation should be avoided by re-casting the problem. For the example given, the use of polar coordinates would avoid the difficulty: