NAG Library Routine Document
E02AEF
1 Purpose
E02AEF evaluates a polynomial from its Chebyshev series representation.
2 Specification
INTEGER 
NPLUS1, IFAIL 
REAL (KIND=nag_wp) 
A(NPLUS1), XCAP, P 

3 Description
E02AEF 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 on 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) and
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 Parameters
 1: 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: A(NPLUS1) – 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: 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: P – REAL (KIND=nag_wp)Output
On exit: the value of the polynomial.
 5: 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:
 ${\mathbf{IFAIL}}=1$
$\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,  ${\mathbf{NPLUS1}}<1$. 
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 $\left(n+1\right)\times \mathit{machineprecision}$.
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.
9 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.
9.1 Program Text
Program Text (e02aefe.f90)
9.2 Program Data
Program Data (e02aefe.d)
9.3 Program Results
Program Results (e02aefe.r)