NAG C Library Function Document
nag_1d_cheb_eval (e02aec)
1
Purpose
nag_1d_cheb_eval (e02aec) evaluates a polynomial from its Chebyshev series representation.
2
Specification
#include <nag.h> 
#include <nage02.h> 
void 
nag_1d_cheb_eval (Integer nplus1,
const double a[],
double xcap,
double *p,
NagError *fail) 

3
Description
nag_1d_cheb_eval (e02aec) evaluates the polynomial
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
Note that this form of the transformation should be used computationally rather than the mathematical equivalent
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 upon the threeterm 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),
Cox and Hayes (1973).
4
References
Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120
Cox M G (1974) A datafitting package for the nonspecialist 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 nonspecialist 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
5
Arguments
 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:
$\mathbf{a}\left[{\mathbf{nplus1}}\right]$ – const doubleInput

On entry: ${\mathbf{a}}\left[\mathit{i}1\right]$ must be set to the value of the $\mathit{i}$th coefficient in the series, for $\mathit{i}=1,2,\dots ,n+1$.
 3:
$\mathbf{xcap}$ – doubleInput

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 9) 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:
$\mathbf{p}$ – double *Output

On exit: the value of the polynomial.
 5:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_INT_ARG_LT

On entry,
nplus1 must not be less than 1:
${\mathbf{nplus1}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INVALID_XCAP

On entry,
$\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.
7
Accuracy
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 $\left(n+1\right)\times $ machine precision.
8
Parallelism and Performance
nag_1d_cheb_eval (e02aec) is not threaded in any implementation.
The time taken by nag_1d_cheb_eval (e02aec) is approximately proportional to $n+1$.
It is expected that a common use of
nag_1d_cheb_eval (e02aec) will be the evaluation of the polynomial approximations produced by
nag_1d_cheb_fit (e02adc) and
nag_1d_cheb_interp_fit (e02afc).
10
Example
Evaluate at 11 equallyspaced 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 $m$ equallyspaced points in the interval $1\le \stackrel{}{x}\le 1$. The program is selfstarting in that any number of datasets can be supplied.
10.1
Program Text
Program Text (e02aece.c)
10.2
Program Data
Program Data (e02aece.d)
10.3
Program Results
Program Results (e02aece.r)