nag_ode_bvp_ps_lin_quad_weights (d02uyc) (PDF version)
d02 Chapter Contents
d02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_ode_bvp_ps_lin_quad_weights (d02uyc)

+ Contents

    1  Purpose
    7  Accuracy

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 wi, for i=0,1,,n, and function values fi=fti (where ti=-cosi×π/n, for i=0,1,,n, are the Chebyshev Gauss–Lobatto points), then -1 1 fx dx i=0 n wi fi .
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:     nIntegerInput
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, wi, for i=0,1,,n.
3:     failNagError *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  Further Comments

A real array of length 2n is internally allocated.

9  Example

This example approximates the integral -1 3 3 x2 dx using 65 Clenshaw–Curtis weights and a 65-point Chebyshev Gauss–Lobatto grid on -1,3.

9.1  Program Text

Program Text (d02uyce.c)

9.2  Program Data

Program Data (d02uyce.d)

9.3  Program Results

Program Results (d02uyce.r)


nag_ode_bvp_ps_lin_quad_weights (d02uyc) (PDF version)
d02 Chapter Contents
d02 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012