D02 Chapter Contents
D02 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentD02UBF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

D02UBF evaluates a function, or one of its lower order derivatives, from its Chebyshev series representation at Chebyshev Gauss–Lobatto points on $\left[a,b\right]$. The coefficients of the Chebyshev series representation required are usually derived from those returned by D02UAF or D02UEF.

## 2  Specification

 SUBROUTINE D02UBF ( N, A, B, Q, C, F, IFAIL)
 INTEGER N, Q, IFAIL REAL (KIND=nag_wp) A, B, C(N+1), F(N+1)

## 3  Description

D02UBF evaluates the Chebyshev series
 $S x- = 12 c1 T0 x- + c2 T1 x- + c3T2 x- +⋯+ cn+1 Tn x- ,$
or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on $\left[a,b\right]$. Here ${T}_{j}\left(\stackrel{-}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\stackrel{-}{x}$ defined on $\left[-1,1\right]$. In terms of your original variable, $x$ say, the input values at which the function values are to be provided are
 $xr = - 12 b - a cos πr-1 /n + 1 2 b + a , r=1,2,…,n+1 , ​$
where $b$ and $a$ are respectively the upper and lower ends of the range of $x$ over which the function is required.
The calculation is implemented by a forward one-dimensional discrete Fast Fourier Transform (DFT).
Canuto C (1988) Spectral Methods in Fluid Dynamics 502 Springer
Canuto C, Hussaini M Y, Quarteroni A and Zang T A (2006) Spectral Methods: Fundamentals in Single Domains Springer
Trefethen L N (2000) Spectral Methods in MATLAB SIAM

## 5  Parameters

1:     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:     A – REAL (KIND=nag_wp)Input
On entry: $a$, the lower bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{A}}<{\mathbf{B}}$.
3:     B – REAL (KIND=nag_wp)Input
On entry: $b$, the upper bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{B}}>{\mathbf{A}}$.
4:     Q – INTEGERInput
On entry: the order, $q$, of the derivative to evaluate.
Constraint: $0\le {\mathbf{Q}}\le 4$.
5:     C(${\mathbf{N}}+1$) – REAL (KIND=nag_wp) arrayInput
On entry: the Chebyshev coefficients, ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$.
6:     F(${\mathbf{N}}+1$) – REAL (KIND=nag_wp) arrayOutput
On exit: the derivatives ${S}^{\left(q\right)}{x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$, of the Chebyshev series, $S$.
7:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{N}}\le 0$ or N is odd.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{A}}\ge {\mathbf{B}}$.
${\mathbf{IFAIL}}=3$
${\mathbf{Q}}\ne 0$, $1$, $2$, $3$ or $4$.
${\mathbf{IFAIL}}=-999$
Internal memory allocation failed.

## 7  Accuracy

Evaluations of DFT to obtain function or derivative values should be an order $n$ multiple of machine precision assuming full accuracy to machine precision in the given Chebyshev series representation.

The number of operations is of the order $n\mathrm{log}n$ and the memory requirements are $\mathit{O}\left(n\right)$; thus the computation remains efficient and practical for very fine discretizations (very large values of $n$).