nag_quad_1d_fin_wtrig (d01an) is based on the QUADPACK routine QFOUR (see Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form

g (x) w (x)

, where

w (x)

is either

\sin (ω x)

\cos (ω x)

. If a sub-interval has length

L = |b - a| 2^{- l}

then the integration over this sub-interval is performed by means of a modified Clenshaw–Curtis procedure (see Piessens and Branders (1975)) if

L ω > 4

and

l \leq 20 .

In this case a Chebyshev series approximation of degree

24

is used to approximate

g (x)

, while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree

12

. If the above conditions do not hold then Gauss

7

-point and Kronrod

15

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

ε

-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation 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 and Branders M (1975) Algorithm 002: computation of oscillating integrals J. Comput. Appl. Math. 1 153–164

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: $b$ – double scalar

b

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

a < b

4: $omega$ – double scalar

The argument

ω

in the weight function of the transform.

5: $key$ – 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

6: $epsabs$ – double scalar

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

7: $epsrel$ – 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: $lw \geq 4$ .
2: $liw$ – int64int32nag_int scalar: Default: $lw / 2$
The dimension of the array iw. the number of sub-intervals into which the interval of integration 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: $w (lw)$ – 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_wtrig (d01an) 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 the integrator 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 behaviour of $g (x)$ 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 $ifail = 1$ .

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 $2$ .

$ifail = 7$

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.

Accuracy

nag_quad_1d_fin_wtrig (d01an) 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 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_wtrig (d01an) 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_wtrig (d01an) 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}} g (x) w (x) d x ≃ r_{i}

and

result = \sum_{i = 1}^{n} r_{i}

unless nag_quad_1d_fin_wtrig (d01an) terminates while testing for divergence of the integral (see Section 3.4.3 of Piessens et al. (1983)). In this case, result (and abserr) are taken to be the values returned from the extrapolation process. 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 \sin (10 π x) d x .

Open in the MATLAB editor: d01an_example

function d01an_example


fprintf('d01an example results\n\n');

a = 0;
b = 1;
omega = 10*pi;
key = int64(2);
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = ...
  d01an( ...
	 @g, a, b, omega, key, epsabs, epsrel);

fprintf('Result = %13.5f,  Standard error = %10.2e\n', result, abserr);



function [result] = g(x)
  if (x > 0)
    result = log(x);
  else
    result = 0;
  end

d01an example results

Result =      -0.12814,  Standard error =   3.58e-06

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

Chapter Contents

Chapter Introduction

NAG Toolbox