NAG Toolbox: nag_quad_1d_fin_sing (d01al)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{f}$ – function handle or string containing name of m-file

f must return the value of the integrand $f$ at a given point.

[result] = f(x)

Input Parameters

1:     $\mathrm{x}$ – double scalar

The point at which the integrand $f$ must be evaluated.

Output Parameters

1:     $\mathrm{result}$ – double scalar

The value of the integrand at x

2:     $\mathrm{a}$ – double scalar

$a$, the lower limit of integration.

3:     $\mathrm{b}$ – double scalar

$b$, the upper limit of integration. It is not necessary that $a<b$.

4:     $\mathrm{points}\left(:\right)$ – double array

The dimension of the array points must be at least $\max \left(1,\mathbf{npts}\right)$

The user-specified break-points.

Constraint: the break-points must all lie within the interval of integration (but may be supplied in any order).

5:     $\mathrm{epsabs}$ – double scalar

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

6:     $\mathrm{epsrel}$ – double scalar

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

Optional Input Parameters

1:     $\mathrm{npts}$ – int64int32nag_int scalar

Default: the dimension of the array points.

The number of user-supplied break-points within the integration interval.

Constraint: $\mathbf{npts}\ge 0$ and $\mathbf{npts}<\min \left(\left(\mathbf{lw}-2\times \mathbf{npts}-4\right)/4,\left(\mathbf{liw}-\mathbf{npts}-2\right)/2\right)$.

2:     $\mathrm{lw}$ – 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 the interval of integration may be divided by the function. The number of sub-intervals cannot exceed $\left(\mathbf{lw}-2\times \mathbf{npts}-4\right)/4$. The more difficult the integrand, the larger lw should be.

Constraint: $\mathbf{lw}\ge 2\times \mathbf{npts}+8$.

3:     $\mathrm{liw}$ – int64int32nag_int scalar

Default: $\mathbf{lw}/2$ 

The dimension of the array iw. the number of sub-intervals into which the interval of integration may be divided cannot exceed $\left(\mathbf{liw}-\mathbf{npts}-2\right)/2$.

Constraint: $\mathbf{liw}\ge \mathbf{npts}+4$.

Output Parameters

1:     $\mathrm{result}$ – double scalar

The approximation to the integral $I$.

2:     $\mathrm{abserr}$ – double scalar

An estimate of the modulus of the absolute error, which should be an upper bound for $\left|I-\mathbf{result}\right|$.

3:     $\mathrm{w}\left(\mathbf{lw}\right)$ – double array

Details of the computation see Further Comments for more information.

4:     $\mathrm{iw}\left(\mathbf{liw}\right)$ – int64int32nag_int array

$\mathbf{iw}\left(1\right)$ contains the actual number of sub-intervals used. The rest of the array is used as workspace.

5:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

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

W  $\mathbf{ifail}=1$

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.) it should be supplied to the function as an element of the vector points. 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  $\mathbf{ifail}=2$

Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.

W  $\mathbf{ifail}=3$

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 $\mathbf{ifail}=\mathbf{1}$.

W  $\mathbf{ifail}=4$

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 $\mathbf{ifail}=\mathbf{1}$.

W  $\mathbf{ifail}=5$

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

   $\mathbf{ifail}=6$

The input is invalid: break-points are specified outside the integration range, $\mathbf{npts}>\min \left(\left(\mathbf{lw}-2\times \mathbf{npts}-4\right)/4,\left(\mathbf{liw}-\mathbf{npts}-2\right)/2\right)$ or $\mathbf{npts}<0$. result and abserr are set to zero.

   $\mathbf{ifail}=7$

On entry,	$\mathbf{lw}<2\times \mathbf{npts}+8$,
or	$\mathbf{liw}<\mathbf{npts}+4$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d01al_example

function d01al_example


fprintf('d01al example results\n\n');

a = 0;
b = 1;
points = [1/7];
epsabs = 0;
epsrel = 0.001;
[result, abserr, w, iw, ifail] = ...
  d01al( ...
	 @f, a, b, points, epsabs, epsrel);

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



function [result] = f(x)
  a = abs(x-1/7);
  if (a ~= 0)
    result = a^(-0.5);
  else
    result = 0;
  end

d01al example results

Result =       2.60757,  Standard error =   6.39e-14