# NAG Library Routine Document

## 1Purpose

d02ucf returns the Chebyshev Gauss–Lobatto grid points on $\left[a,b\right]$.

## 2Specification

Fortran Interface
 Subroutine d02ucf ( n, a, b, x,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a, b Real (Kind=nag_wp), Intent (Out) :: x(n+1)
#include <nagmk26.h>
 void d02ucf_ (const Integer *n, const double *a, const double *b, double x[], Integer *ifail)

## 3Description

d02ucf returns the Chebyshev Gauss–Lobatto grid points on $\left[a,b\right]$. The Chebyshev Gauss–Lobatto points on $\left[-1,1\right]$ are computed as ${t}_{\mathit{i}}=-\mathrm{cos}\left(\frac{\left(\mathit{i}-1\right)\pi }{n}\right)$, for $\mathit{i}=1,2,\dots ,n+1$. The Chebyshev Gauss–Lobatto points on an arbitrary domain $\left[a,b\right]$ are:
 $xi = b-a 2 ti + a+b 2 , i=1,2,…,n+1 .$

## 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 $n+1$. This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.
Constraint: ${\mathbf{n}}>0$ and n is even.
2:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $a$, the lower bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $b$, the upper bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.
4:     $\mathbf{x}\left({\mathbf{n}}+1\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the Chebyshev Gauss–Lobatto grid points, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$, on $\left[a,b\right]$.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  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.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$ and ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The Chebyshev Gauss–Lobatto grid points computed should be accurate to within a small multiple of machine precision.

## 8Parallelism and Performance

d02ucf is not threaded in any implementation.

The number of operations is of the order $n\mathrm{log}\left(n\right)$ and there are no internal memory requirements; thus the computation remains efficient and practical for very fine discretizations (very large values of $n$).