# NAG Library Routine Document

## 1Purpose

e02aef evaluates a polynomial from its Chebyshev series representation.

## 2Specification

Fortran Interface
 Subroutine e02aef ( a, xcap, p,
 Integer, Intent (In) :: nplus1 Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(nplus1), xcap Real (Kind=nag_wp), Intent (Out) :: p
#include nagmk26.h
 void e02aef_ ( const Integer *nplus1, const double a[], const double *xcap, double *p, Integer *ifail)

## 3Description

e02aef evaluates the polynomial
 $12a1T0x-+a2T1x-+a3T2x-+⋯+an+1Tnx-$
for any value of $\stackrel{-}{x}$ satisfying $-1\le \stackrel{-}{x}\le 1$. Here ${T}_{j}\left(\stackrel{-}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\stackrel{-}{x}$. The value of $n$ is prescribed by you.
In practice, the variable $\stackrel{-}{x}$ will usually have been obtained from an original variable $x$, where ${x}_{\mathrm{min}}\le x\le {x}_{\mathrm{max}}$ and
 $x-=x-xmin-xmax-x xmax-xmin$
Note that this form of the transformation should be used computationally rather than the mathematical equivalent
 $x-= 2x-xmin-xmax xmax-xmin$
since the former guarantees that the computed value of $\stackrel{-}{x}$ differs from its true value by at most $4\epsilon$, where $\epsilon$ is the machine precision, whereas the latter has no such guarantee.
The method employed is based on the three-term recurrence relation due to Clenshaw (1955), with modifications to give greater numerical stability due to Reinsch and Gentleman (see Gentleman (1969)).
For further details of the algorithm and its use see Cox (1974) and Cox and Hayes (1973).
Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120
Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press
Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory
Gentleman W M (1969) An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients Comput. J. 12 160–165

## 5Arguments

1:     $\mathbf{nplus1}$ – IntegerInput
On entry: the number $n+1$ of terms in the series (i.e., one greater than the degree of the polynomial).
Constraint: ${\mathbf{nplus1}}\ge 1$.
2:     $\mathbf{a}\left({\mathbf{nplus1}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{a}}\left(\mathit{i}\right)$ must be set to the value of the $\mathit{i}$th coefficient in the series, for $\mathit{i}=1,2,\dots ,n+1$.
3:     $\mathbf{xcap}$ – Real (Kind=nag_wp)Input
On entry: $\stackrel{-}{x}$, the argument at which the polynomial is to be evaluated. It should lie in the range $-1$ to $+1$, but a value just outside this range is permitted (see Section 6) to allow for possible rounding errors committed in the transformation from $x$ to $\stackrel{-}{x}$ discussed in Section 3. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of $x$ lies in the range ${x}_{\mathrm{min}}$ to ${x}_{\mathrm{max}}$.
4:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the value of the polynomial.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, 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).

## 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:
${\mathbf{ifail}}=1$
$\mathrm{ABS}\left({\mathbf{xcap}}\right)>1.0+4\epsilon$, where $\epsilon$ is the machine precision. In this case the value of p is set arbitrarily to zero.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{nplus1}}<1$.
${\mathbf{ifail}}=-99$
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 rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients ${a}_{i}+\delta {a}_{i}$. The ratio of the sum of the absolute values of the $\delta {a}_{i}$ to the sum of the absolute values of the ${a}_{i}$ is less than a small multiple of .

## 8Parallelism and Performance

e02aef is not threaded in any implementation.

The time taken is approximately proportional to $n+1$.
It is expected that a common use of e02aef will be the evaluation of the polynomial approximations produced by e02adf and e02aff.

## 10Example

Evaluate at $11$ equally-spaced points in the interval $-1\le \stackrel{-}{x}\le 1$ the polynomial of degree $4$ with Chebyshev coefficients, $2.0$, $0.5$, $0.25$, $0.125$, $0.0625$.
The example program is written in a general form that will enable a polynomial of degree $n$ in its Chebyshev series form to be evaluated at $m$ equally-spaced points in the interval $-1\le \stackrel{-}{x}\le 1$. The program is self-starting in that any number of datasets can be supplied.

### 10.1Program Text

Program Text (e02aefe.f90)

### 10.2Program Data

Program Data (e02aefe.d)

### 10.3Program Results

Program Results (e02aefe.r)

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