NAG Library Routine Document

D02UYF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

D02UYF 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

SUBROUTINE D02UYF (

N, W, IFAIL)

INTEGER	N, IFAIL
REAL (KIND=nag_wp)	W(N+1)

3 Description

D02UYF 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 Parameters

1: N – INTEGERInput: On entry: $n$ , where the number of grid points is $n + 1$ .
Constraint: $N > 0$ and N is even.
2: W( $N + 1$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the Clenshaw–Curtis quadrature weights, $w_{i}$ , for $i = 0, 1, \dots, n$ .
3: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

N \leq 0

or N is odd.

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

The accuracy should be close to machine precision.

8 Further Comments

A real array of length

2 n

is internally allocated.

9 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 Routine DocumentD02UYF