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## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine d02uyf ( n, w,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Out) :: w(n+1)
#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.

## 3Description

d02uyf obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points.
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: $\mathbf{n}$Integer Input
On entry: $n$, where the number of grid points is $n+1$.
Constraint: ${\mathbf{n}}>0$ and n is even.
2: $\mathbf{w}\left({\mathbf{n}}+1\right)$Real (Kind=nag_wp) array Output
On exit: the Clenshaw–Curtis quadrature weights, ${w}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$.
3: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: n is even.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

The accuracy should be close to machine precision.

## 8Parallelism and Performance

d02uyf is not threaded in any implementation.

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 $65$-point Chebyshev Gauss–Lobatto grid on $\left[-1,3\right]$.

### 10.1Program Text

Program Text (d02uyfe.f90)

### 10.2Program Data

Program Data (d02uyfe.d)

### 10.3Program Results

Program Results (d02uyfe.r)