NAG Toolbox: nag_quad_1d_gauss_wgen (d01bc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates the type of quadrature rule.

$\mathbf{itype}=0$

Gauss–Legendre, with normal weights.

$\mathbf{itype}=1$

Gauss–Jacobi, with normal weights.

$\mathbf{itype}=-1$

Gauss–Jacobi, with adjusted weights.

$\mathbf{itype}=2$

Exponential Gauss, with normal weights.

$\mathbf{itype}=-2$

Exponential Gauss, with adjusted weights.

$\mathbf{itype}=3$

Gauss–Laguerre, with normal weights.

$\mathbf{itype}=-3$

Gauss–Laguerre, with adjusted weights.

$\mathbf{itype}=4$

Gauss–Hermite, with normal weights.

$\mathbf{itype}=-4$

Gauss–Hermite, with adjusted weights.

$\mathbf{itype}=5$

Rational Gauss, with normal weights.

$\mathbf{itype}=-5$

Rational Gauss, with adjusted weights.

Constraint: $\mathbf{itype}=0$, $1$, $-1$, $2$, $-2$, $3$, $-3$, $4$, $-4$, $5$ or $-5$.

2:     $\mathrm{a}$ – double scalar

3:     $\mathrm{b}$ – double scalar

4:     $\mathrm{c}$ – double scalar

5:     $\mathrm{d}$ – double scalar

The parameters $a$, $b$, $c$ and $d$ which occur in the quadrature formulae. c is not used if $\mathbf{itype}=0$; d is not used unless $\mathbf{itype}=1$, $-1$, $5$ or $-5$. For some rules c and d must not be too large (see Error Indicators and Warnings).

6:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of weights and abscissae to be returned. If $\mathbf{itype}=-2$ or $-4$ and $\mathbf{c}\ne 0.0$, an odd value of n may raise problems (see $\mathbf{ifail}=\mathbf{6}$).

Constraint: $\mathbf{n}>0$.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{weight}\left(\mathbf{n}\right)$ – double array

The n weights.

2:     $\mathrm{abscis}\left(\mathbf{n}\right)$ – double array

The n abscissae.

3:     $\mathrm{ifail}$ – 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 algorithm for computing eigenvalues of a tridiagonal matrix has failed to obtain convergence. If the soft fail option is used, the values of the weights and abscissae on return are indeterminate.

   $\mathbf{ifail}=2$

On entry,	$\mathbf{n}<1$,
or	$\mathbf{itype}<-5$,
or	$\mathbf{itype}>5$.

If the soft fail option is used, weights and abscissae are returned as zero.

   $\mathbf{ifail}=3$

a, b, c or d is not in the allowed range:

• if $\mathbf{itype}=0$, $\mathbf{a}\ge \mathbf{b}$;
• if $\mathbf{itype}=\pm 1$, $\mathbf{a}\ge \mathbf{b}$ or $\mathbf{c}\le -1.0$ or $\mathbf{d}\le -1.0$ or $\mathbf{c}+\mathbf{d}+2.0>\mathit{gmax}$;
• if $\mathbf{itype}=\pm 2$, $\mathbf{a}\ge \mathbf{b}$ or $\mathbf{c}\le -1.0$;
• if $\mathbf{itype}=\pm 3$, $\mathbf{b}=0.0$ or $\mathbf{c}\le -1.0$ or $\mathbf{c}+1.0>\mathit{gmax}$;
• if $\mathbf{itype}=\pm 4$, $\mathbf{b}\le 0.0$ or $\mathbf{c}\le -1.0$ or $\left(\mathbf{c}+1.0/2.0\right)>\mathit{gmax}$;
• if $\mathbf{itype}=\pm 5$, $\mathbf{a}+\mathbf{b}=0.0$ or $\mathbf{c}\le -1.0$ or $\mathbf{d}\le \mathbf{c}+1.0$.

Here $\mathit{gmax}$ is the (machine-dependent) largest integer value such that $\Gamma \left(\mathit{gmax}\right)$ can be computed without overflow.

If the soft fail option is used, weights and abscissae are returned as zero.

W  $\mathbf{ifail}=4$

One or more of the weights are larger than $\mathit{rmax}$, the largest floating-point number on this machine. $\mathit{rmax}$ is given by the function nag_machine_real_largest (x02al). If the soft fail option is used, the overflowing weights are returned as $\mathit{rmax}$. Possible solutions are to use a smaller value of n; or, if using adjusted weights, to change to normal weights.

W  $\mathbf{ifail}=5$

One or more of the weights are too small to be distinguished from zero on this machine. If the soft fail option is used, the underflowing weights are returned as zero, which may be a usable approximation. Possible solutions are to use a smaller value of n; or, if using normal weights, to change to adjusted weights.

W  $\mathbf{ifail}=6$

Exponential Gauss or Gauss–Hermite adjusted weights with n odd and $\mathbf{c}\ne 0.0$. Theoretically, in these cases:

• for $\mathbf{c}>0.0$, the central adjusted weight is infinite, and the exact function $f\left(x\right)$ is zero at the central abscissa.
• for $\mathbf{c}<0.0$, the central adjusted weight is zero, and the exact function $f\left(x\right)$ is infinite at the central abscissa.

In either case, the contribution of the central abscissa to the summation is indeterminate.

In practice, the central weight may not have overflowed or underflowed, if there is sufficient rounding error in the value of the central abscissa.

If the soft fail option is used, the weights and abscissa returned may be usable; you must be particularly careful not to ‘round’ the central abscissa to its true value without simultaneously ‘rounding’ the central weight to zero or $\infty$ as appropriate, or the summation will suffer. It would be preferable to use normal weights, if possible.

Note:  remember that, when switching from normal weights to adjusted weights or vice versa, redefinition of $f\left(x\right)$ is involved.

   $\mathbf{ifail}=-99$

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: d01bc_example

function d01bc_example


fprintf('d01bc example results\n\n');

% Initialize variables and arrays.
itype = -3;
a =0;
b = 1;
c = 0;
d = 0;
n = 7;

[weight, abscis, ifail] = d01bc(int64(itype), a, b, c, d, int64(n));
if ifail ~= 0
    % Convergence problems, or input variables out of bounds.
    % Print message and exit.
    error('Warning: d01bc returned with ifail = %1d ',ifail);
end

% Output abscissae and weights.
fprintf('   Laguerre formula, %d points \n', n);
fprintf('    Abscissae         Weights \n')
for i = 1:n
    fprintf('   %10.5e     %10.5e \n',abscis(i),weight(i))
end
% Plot results.
fig1 = figure;
display_plot(abscis, weight);


function plot(abscis, weight)
% Use a bar graph, and set the bar width.
bar1 = bar(abscis, weight);
set(bar1, 'BarWidth', 0.1);
% Set the title.
% Label the axes.
xlabel('x');
ylabel('Weights at Abscissae');

function display_plot(abscis, weight)
% Use a bar graph, and set the bar width.
bar1 = bar(abscis, weight);
set(bar1, 'BarWidth', 0.1);
% Set the title.
title({'Abscissae and Weights for',...
          'the 7-point Gauss-Laguerre Formula (a=0, b=1)'});
% Label the axes.
xlabel('x');
ylabel('Weights at Abscissae');

d01bc example results

   Laguerre formula, 7 points 
    Abscissae         Weights 
   1.93044e-01     4.96478e-01 
   1.02666e+00     1.17764e+00 
   2.56788e+00     1.91825e+00 
   4.90035e+00     2.77185e+00 
   8.18215e+00     3.84125e+00 
   1.27342e+01     5.38068e+00 
   1.93957e+01     8.40543e+00