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# NAG Toolbox: nag_fit_1dcheb_eval (e02ae)

## Purpose

nag_fit_1dcheb_eval (e02ae) evaluates a polynomial from its Chebyshev series representation.

## Syntax

[p, ifail] = e02ae(a, xcap, 'nplus1', nplus1)
[p, ifail] = nag_fit_1dcheb_eval(a, xcap, 'nplus1', nplus1)

## Description

nag_fit_1dcheb_eval (e02ae) 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).

## 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 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left({\mathbf{nplus1}}\right)$ – double array
${\mathbf{a}}\left(\mathit{i}\right)$ must be set to the value of the $\mathit{i}$th coefficient in the series, for $\mathit{i}=1,2,\dots ,n+1$.
2:     $\mathrm{xcap}$ – double scalar
$\stackrel{-}{x}$, 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 Error Indicators and Warnings) to allow for possible rounding errors committed in the transformation from $x$ to $\stackrel{-}{x}$ discussed in Description. 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 ${x}_{\mathrm{max}}$.

### Optional Input Parameters

1:     $\mathrm{nplus1}$int64int32nag_int scalar
Default: the dimension of the array a.
The number $n+1$ of terms in the series (i.e., one greater than the degree of the polynomial).
Constraint: ${\mathbf{nplus1}}\ge 1$.

### Output Parameters

1:     $\mathrm{p}$ – double scalar
The value of the polynomial.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
$\mathrm{ABS}\left({\mathbf{xcap}}\right)>1.0+4\epsilon$, where $\epsilon$ is the machine precision. In this case the value of p is set arbitrarily to zero.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{nplus1}}<1$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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 .

The time taken is approximately proportional to $n+1$.
It is expected that a common use of nag_fit_1dcheb_eval (e02ae) will be the evaluation of the polynomial approximations produced by nag_fit_1dcheb_arb (e02ad) and nag_fit_1dcheb_glp (e02af).

## Example

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.
```function e02ae_example

fprintf('e02ae example results\n\n');

a = [2     0.5     0.25     0.125     0.0625];

xcap = [-1:0.2:1]';
for i=1:11
[p(i), ifail] = e02ae(a, xcap(i));
end
disp('      x        p(x)');
disp([xcap p']);

```
```e02ae example results

x        p(x)
-1.0000    0.6875
-0.8000    0.6613
-0.6000    0.6943
-0.4000    0.7433
-0.2000    0.7843
0    0.8125
0.2000    0.8423
0.4000    0.9073
0.6000    1.0603
0.8000    1.3733
1.0000    1.9375

```

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