nag_quad_1d_inf_wtrig (d01as) is based on the QUADPACK routine QAWFE (see Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form

g (x) w (x)

over a semi-infinite interval, where

w (x)

is either

\sin (ω x)

\cos (ω x)

Over successive intervals

C_{k} = [a + (k - 1) c, a + k c], k = 1, 2, \dots, lst

integration is performed by the same algorithm as is used by nag_quad_1d_fin_wtrig (d01an). The intervals

C_{k}

are of constant length

c = \{2 [|ω|] + 1\} π / |ω|, ω \neq 0,

where

[|ω|]

represents the largest integer less than or equal to

|ω|

. Since

c

equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function

g

is positive and monotonically decreasing over

[a, \infty)

. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the

ε

-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described by Piessens et al. (1983).

ω = 0

and

key = 1

, the function uses the same algorithm as nag_quad_1d_inf (d01am) (with

epsrel = 0.0

In contrast to the other functions in Chapter D01, nag_quad_1d_inf_wtrig (d01as) works only with an absolute error tolerance (epsabs). Over the interval

C_{k}

it attempts to satisfy the absolute accuracy requirement

{EPSA}_{k} = U_{k} \times epsabs,

where

U_{k} = (1 - p) p^{k - 1}

, for

k = 1, 2, \dots

and

p = 0.9

However, when difficulties occur during the integration over the

k

th sub-interval

C_{k}

such that the error flag

ierlst (k)

is nonzero, the accuracy requirement over subsequent intervals is relaxed. See Piessens et al. (1983) for more details.

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

Wynn P (1956) On a device for computing the

e_{m} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

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: $omega$ – double scalar

The argument

ω

in the weight function of the transform.

4: $key$ – int64int32nag_int scalar

Indicates which integral is to be computed.

$key = 1$: $w (x) = \cos (ω x)$ .
$key = 2$: $w (x) = \sin (ω x)$ .

Constraint:

key = 1

2

5: $epsabs$ – double scalar

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

Optional Input Parameters

1: $limlst$ – int64int32nag_int scalar: Suggested value: $limlst = 50$ is adequate for most problems.
Default: $50$
An upper bound on the number of intervals $C_{k}$ needed for the integration.

Constraint: $limlst \geq 3$ .
2: $lw$ – int64int32nag_int scalar: Default: a value in the range $800$ to $2000$ is adequate for most problems.
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 each interval $C_{k}$ 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: $lw \geq 4$ .
3: $liw$ – int64int32nag_int scalar: Default: $lw / 2$
The dimension of the array iw. the number of sub-intervals into which each interval $C_{k}$ may be divided cannot exceed $liw / 2$ .

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: $lst$ – int64int32nag_int scalar: The number of intervals $C_{k}$ actually used for the integration.
4: $erlst (limlst)$ – double array: $erlst (k)$ contains the error estimate corresponding to the integral contribution over the interval $C_{k}$ , for $k = 1, 2, \dots, lst$ .
5: $rslst (limlst)$ – double array: $rslst (k)$ contains the integral contribution over the interval $C_{k}$ , for $k = 1, 2, \dots, lst$ .
6: $ierlst (limlst)$ – int64int32nag_int array: $ierlst (k)$ contains the error flag corresponding to $rslst (k)$ , for $k = 1, 2, \dots, lst$ . See Error Indicators and Warnings.
7: $iw (liw)$ – int64int32nag_int array: $iw (1)$ contains the maximum number of sub-intervals actually used for integrating over any of the intervals $C_{k}$ . The rest of the array is used as workspace.
8: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_quad_1d_inf_wtrig (d01as) 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 local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity, etc.) you will probably gain from splitting up the interval at this point and calling nag_quad_1d_inf_wtrig (d01as) on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs 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 $ifail = 1$ .

W $ifail = 4$: 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 $ifail = 1$ .

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.) you will probably gain from splitting up the interval at this point and calling nag_quad_1d_inf_wtrig (d01as) on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs or increasing the amount of workspace.

Please note that divergence can occur with any nonzero value of ifail.

W $ifail = 5$: The integral is probably divergent, or slowly convergent. Please note that divergence can occur with any nonzero value of ifail.

$ifail = 6$

On entry,	$key \neq 1$ or $2$ ,
or	$limlst < 3$ .

W $ifail = 7$: Bad integration behaviour occurs within one or more of the intervals $C_{k}$ . Location and type of the difficulty involved can be determined from the vector ierlst.

W $ifail = 8$: Maximum number of intervals $C_{k}$ ( $= limlst$ ) allowed has been achieved. Increase the value of limlst to allow more cycles.

W $ifail = 9$: The extrapolation table constructed for convergence acceleration of the series formed by the integral contribution over the intervals $C_{k}$ , does not converge to the required accuracy.

$ifail = 10$

On entry,	$lw < 4$ ,
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.

In the cases

ifail = 7

8

9

, additional information about the cause of the error can be obtained from the array ierlst, as follows:

$ierlst (k) = 1$: The maximum number of $subdivisions = \min (lw / 4, liw / 2)$ has been achieved on the $k$ th interval.
$ierlst (k) = 2$: Occurrence of round-off error is detected and prevents the tolerance imposed on the $k$ th interval from being achieved.
$ierlst (k) = 3$: Extremely bad integrand behaviour occurs at some points of the $k$ th interval.
$ierlst (k) = 4$: The integration procedure over the $k$ th interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.
$ierlst (k) = 5$: The integral over the $k$ th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of $ierlst (k)$ .

Accuracy

nag_quad_1d_inf_wtrig (d01as) cannot guarantee, but in practice usually achieves, the following accuracy:

|I - result| \leq |epsabs|,

where epsabs is the user-specified absolute error tolerance. Moreover, it returns the quantity abserr, which, in normal circumstances, satisfies

|I - result| \leq abserr \leq |epsabs| .

Further Comments

None.

Example

This example computes

\int_{0}^{\infty} \frac{1}{\sqrt{x}} \cos (π x / 2) d x .

Open in the MATLAB editor: d01as_example

function d01as_example


fprintf('d01as example results\n\n');

a = 0;
omega = pi/2;
key = int64(1);
epsabs = 0.001;
[result, abserr, lst, erlst, rslst, ierlst, iw, ifail] = ...
  d01as( ...
	 @g, a, omega, key, epsabs);

formr = '%-6s - %-30s = %8.5f\n';
forms = '%-6s - %-30s = %8s\n';
formd = '%-6s - %-30s = %8d\n';
fprintf(formr, 'a', 'lower limit of integration', a);
fprintf(forms, 'b', 'upper limit of integration', 'infinity');
fprintf(formr, 'epsabs', 'absolute accuracy requested', epsabs);
fprintf('\n');
fprintf(formr, 'result', 'approximation to the integral', result);
fprintf(formr, 'abserr', 'estimate of the absolute error', abserr);
fprintf(formd, 'lst', 'number of intervals used', lst);
fprintf(formd, 'iw(1)', 'maximum number of subintervals', iw(1));



function [result] = g(x)
  if (x > 0)
    result = 1/sqrt(x);
  else
    result = 0;
  end

d01as example results

a      - lower limit of integration     =  0.00000
b      - upper limit of integration     = infinity
epsabs - absolute accuracy requested    =  0.00100

result - approximation to the integral  =  1.00000
abserr - estimate of the absolute error =  0.00059
lst    - number of intervals used       =        6
iw(1)  - maximum number of subintervals =        8

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

Chapter Contents

Chapter Introduction

NAG Toolbox