nag_ode_bvp_ps_lin_quad_weights (d02uy) obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points. This allows for fast approximations of integrals for functions specified on Chebyshev Gauss–Lobatto points on

[- 1, 1]

Syntax

[w, ifail] = d02uy(n)

[w, ifail] = nag_ode_bvp_ps_lin_quad_weights(n)

Description

nag_ode_bvp_ps_lin_quad_weights (d02uy) obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points.

Given the (Clenshaw–Curtis) weights

w_{i}

, for

i = 0, 1, \dots, n

, and function values

f_{i} = f (t_{i})

(where

t_{i} = - \cos (i \times π / n)

, for

i = 0, 1, \dots, n

, are the Chebyshev Gauss–Lobatto points), then

\int_{- 1}^{1} f (x) d x \approx \sum_{i = 0}^{n} w_{i} f_{i}

For a function discretized on a Chebyshev Gauss–Lobatto grid on

[a, b]

the resultant summation must be multiplied by the factor

(b - a) / 2

References

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , where the number of grid points is $n + 1$ .

Constraint: $n > 0$ and n is even.

Optional Input Parameters

None.

Output Parameters

1: $w (n + 1)$ – double array: The Clenshaw–Curtis quadrature weights, $w_{i}$ , for $i = 0, 1, \dots, n$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n > 0$ .

Constraint: n is even.

$ifail = - 99$: 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

The accuracy should be close to machine precision.

Further Comments

A real array of length

2 n

is internally allocated.

Example

This example approximates the integral

\int_{- 1}^{3} 3 x^{2} d x

using

65

Clenshaw–Curtis weights and a

65

-point Chebyshev Gauss–Lobatto grid on

[- 1, 3]

Open in the MATLAB editor: d02uy_example

function d02uy_example


fprintf('d02uy example results\n\n');

n = int64(64);
a = -1;
b =  3;

% Set up solution grid
[x, ifail] = d02uc(n, a, b);

% Get integrand values on grid
f = 3*x.^2;
scale = 0.5*(b-a);

% Get quadrature weights
[w, ifail] = d02uy(n);

% Evaluate apprimation to definite integral
integ = dot(w, f)*scale;

% Print solution
fprintf('Integral of f(x) from %4.1f to %4.1f = %7.4f\n', a, b, integ);
fprintf('\nError in approximation = %12.2e\n', integ-28);

d02uy example results

Integral of f(x) from -1.0 to  3.0 = 28.0000

Error in approximation =     3.55e-15

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

Chapter Contents

Chapter Introduction

NAG Toolbox