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

Fortran Interface

Subroutine d02uyf (

n, w, ifail)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Out)	::	w(n+1)

C Header Interface

#include <nag.h>

void	d02uyf_ (const Integer n, double w[], Integer ifail)

The routine may be called by the names d02uyf or nagf_ode_bvp_ps_lin_quad_weights.

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 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: $w (n + 1)$ – Real (Kind=nag_wp) array Output: On exit: the Clenshaw–Curtis quadrature weights, $w_{i}$ , for $i = 0, 1, \dots, n$ .
3: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: n is even.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy should be close to machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

d02uyf 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 PY MB

NAG FL Interfaced02uyf (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 FL Interface
d02uyf (bvp_ps_lin_quad_weights)