. The strategy is a modification of that in nag_quad_1d_fin_osc (d01ak). We start by bisecting the original interval and applying modified Clenshaw–Curtis integration of orders

12

and

24

to both halves. Clenshaw–Curtis integration is then used on all sub-intervals which have

a

b

as one of their end points (see Piessens et al. (1974)). On the other sub-intervals Gauss–Kronrod (

7

–

15

point) integration is carried out.

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

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

Piessens R, Mertens I and Branders M (1974) Integration of functions having end-point singularities Angew. Inf. 16 65–68

Parameters

Compulsory Input Parameters

1: $g$ – function handle or string containing name of m-file

g must return the value of the function

g

at a given point x.

[result] = g(x)

Input Parameters

1: $x$ – double scalar: The point at which the function $g$ must be evaluated.

Output Parameters

1: $result$ – double scalar: The value of $g (x)$ evaluated at x.

2: $a$ – double scalar

a

, the lower limit of integration.

3: $b$ – double scalar

b

, the upper limit of integration.

Constraint:

b > a

4: $alfa$ – double scalar

The argument

α

in the weight function.

Constraint:

alfa > - 1.0

5: $beta$ – double scalar

The argument

β

in the weight function.

Constraint:

beta > - 1.0

6: $key$ – int64int32nag_int scalar

Indicates which weight function is to be used.

$key = 1$: $w (x) = {(x - a)}^{α} {(b - x)}^{β}$ .
$key = 2$: $w (x) = {(x - a)}^{α} {(b - x)}^{β} \ln (x - a)$ .
$key = 3$: $w (x) = {(x - a)}^{α} {(b - x)}^{β} \ln (b - x)$ .
$key = 4$: $w (x) = {(x - a)}^{α} {(b - x)}^{β} \ln (x - a) \ln (b - x)$ .

Constraint:

key = 1

2

3

4

7: $epsabs$ – double scalar

The absolute accuracy required. If epsabs is negative, the absolute value is used. See Accuracy.

8: $epsrel$ – double scalar

The relative accuracy required. If epsrel is negative, the absolute value is used. See Accuracy.

Optional Input Parameters

1: $lw$ – int64int32nag_int scalar: Suggested value: $lw = 800$ to $2000$ is adequate for most problems.
Default: $800$
The dimension of the array w. 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 function. The number of sub-intervals cannot exceed $lw / 4$ . The more difficult the integrand, the larger lw should be.

Constraint: $lw \geq 8$ .
2: $liw$ – int64int32nag_int scalar: Default: $lw / 4$
The dimension of the array iw. the number of sub-intervals into which the interval of integration may be divided cannot exceed liw.

Constraint: $liw \geq 2$ .

Output Parameters

1: $result$ – double scalar: The approximation to the integral $I$ .
2: $abserr$ – double scalar: An estimate of the modulus of the absolute error, which should be an upper bound for $|I - result|$ .
3: $w (lw)$ – double array: Details of the computation see Further Comments for more information.
4: $iw (liw)$ – int64int32nag_int array: $iw (1)$ contains the actual number of sub-intervals used. The rest of the array is used as workspace.
5: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_quad_1d_fin_wsing (d01ap) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: 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 discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called 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.

W $ifail = 2$: Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.

W $ifail = 3$: 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,	$b \leq a$ ,
or	$alfa \leq - 1.0$ ,
or	$beta \leq - 1.0$ ,
or	$key \neq 1$ , $2$ , $3$ or $4$ .

$ifail = 5$

On entry,	$lw < 8$ ,
or	$liw < 2$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

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

Accuracy

nag_quad_1d_fin_wsing (d01ap) 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 .

Further Comments

The time taken by nag_quad_1d_fin_wsing (d01ap) 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 nag_quad_1d_fin_wsing (d01ap) 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) w (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
$r_{i} = w (3 n + i)$ .

Example

This example computes

\int_{0}^{1} \ln x \cos (10 π x) d x and \int_{0}^{1} \frac{\sin (10 x)}{\sqrt{x (1 - x)}} d x .

Open in the MATLAB editor: d01ap_example

function d01ap_example


fprintf('d01ap example results\n\n');

a    = 0;
b    = 1;
alfa = 0;
beta = 0;
key  = int64(2);
epsabs = 0;
epsrel = 0.0001;

g = @(x) cos(10*pi*x);

[result, abserr, w, iw, ifail] = ...
  d01ap( ...
	 g, a, b, alfa, beta, key, epsabs, epsrel);

fprintf('Result = %13.6f,  Standard error = %10.2e\n', result, abserr);

d01ap example results

Result =     -0.048989,  Standard error =   1.14e-07

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox