NAG Library Function Document

nag_ode_bvp_ps_lin_quad_weights (d02uyc)

+− 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

1 Purpose

nag_ode_bvp_ps_lin_quad_weights (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>

#include <nagd02.h>

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

3 Description

nag_ode_bvp_ps_lin_quad_weights (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 – IntegerInput: On entry: $n$ , where the number of grid points is $n + 1$ .
Constraint: $n > 0$ and n is even.
2: w[ $n + 1$ ] – doubleOutput: 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

7 Accuracy

The accuracy should be close to machine precision.

8 Parallelism and Performance

Not applicable.

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]

NAG Library Function Documentnag_ode_bvp_ps_lin_quad_weights (d02uyc)