# NAG Library Function Document

## 1Purpose

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 $\left[-1,1\right]$.

## 2Specification

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

## 3Description

Given the (Clenshaw–Curtis) weights ${w}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$, and function values ${f}_{\mathit{i}}=f\left({t}_{\mathit{i}}\right)$ (where ${t}_{\mathit{i}}=-\mathrm{cos}\left(\mathit{i}×\pi /n\right)$, for $\mathit{i}=0,1,\dots ,n$, are the Chebyshev Gauss–Lobatto points), then $\underset{-1}{\overset{1}{\int }}f\left(x\right)dx\approx \sum _{\mathit{i}=0}^{n}{w}_{i}{f}_{i}$.
For a function discretized on a Chebyshev Gauss–Lobatto grid on $\left[a,b\right]$ the resultant summation must be multiplied by the factor $\left(b-a\right)/2$.

## 4References

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, where the number of grid points is $n+1$.
Constraint: ${\mathbf{n}}>0$ and n is even.
2:    $\mathbf{w}\left[{\mathbf{n}}+1\right]$doubleOutput
On exit: the Clenshaw–Curtis quadrature weights, ${w}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$.
3:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The accuracy should be close to machine precision.

## 8Parallelism and Performance

A real array of length $2n$ is internally allocated.

## 10Example

This example approximates the integral $\underset{-1}{\overset{3}{\int }}3{x}^{2}dx$ using $65$ Clenshaw–Curtis weights and a $\mathrm{65}$-point Chebyshev Gauss–Lobatto grid on $\left[-1,3\right]$.

### 10.1Program Text

Program Text (d02uyce.c)

### 10.2Program Data

Program Data (d02uyce.d)

### 10.3Program Results

Program Results (d02uyce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017