d02uyc 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]

2 Specification

#include <nag.h>

void	d02uyc (Integer n, double w[], NagError *fail)

The function may be called by the names: d02uyc or nag_ode_bvp_ps_lin_quad_weights.

3 Description

d02uyc 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

4 References

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , where the number of grid points is $n + 1$ .

Constraint: $n > 0$ and n is even.
2: $w [n + 1]$ – double Output: On exit: the Clenshaw–Curtis quadrature weights, $w_{i}$ , for $i = 0, 1, \dots, n$ .
3: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: n is even.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The accuracy should be close to machine precision.

8 Parallelism and Performance

d02uyc is not threaded in any implementation.

9 Further Comments

A real array of length

2 n

is internally allocated.

10 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]

d02uy: FL CL CPP AD

NAG CL Interfaced02uyc (bvp_​ps_​lin_​quad_​weights)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d02uyc (bvp_ps_lin_quad_weights)