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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_fit_1dcheb_eval2 (e02ak)

## Purpose

nag_fit_1dcheb_eval2 (e02ak) evaluates a polynomial from its Chebyshev series representation, allowing an arbitrary index increment for accessing the array of coefficients.

## Syntax

[result, ifail] = e02ak(n, xmin, xmax, a, ia1, x)
[result, ifail] = nag_fit_1dcheb_eval2(n, xmin, xmax, a, ia1, x)

## Description

If supplied with the coefficients ${a}_{i}$, for $\mathit{i}=0,1,\dots ,n$, of a polynomial $p\left(\stackrel{-}{x}\right)$ of degree $n$, where
 $px-=12a0+a1T1x-+⋯+anTnx-,$
nag_fit_1dcheb_eval2 (e02ak) returns the value of $p\left(\stackrel{-}{x}\right)$ at a user-specified value of the variable $x$. Here ${T}_{j}\left(\stackrel{-}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\stackrel{-}{x}$. It is assumed that the independent variable $\stackrel{-}{x}$ in the interval $\left[-1,+1\right]$ was obtained from your original variable $x$ in the interval $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$ by the linear transformation
 $x-=2x-xmax+xmin xmax-xmin .$
The coefficients ${a}_{i}$ may be supplied in the array a, with any increment between the indices of array elements which contain successive coefficients. This enables the function to be used in surface fitting and other applications, in which the array might have two or more dimensions.
The method employed is based on the three-term recurrence relation due to Clenshaw (see Clenshaw (1955)), with modifications due to Reinsch and Gentleman (see Gentleman (1969)). For further details of the algorithm and its use see Cox (1973) 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 (1973) A data-fitting package for the non-specialist user NPL Report NAC 40 National Physical Laboratory
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:     $\mathrm{n}$int64int32nag_int scalar
$n$, the degree of the given polynomial $p\left(\stackrel{-}{x}\right)$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{xmin}$ – double scalar
3:     $\mathrm{xmax}$ – double scalar
The lower and upper end points respectively of the interval $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$. The Chebyshev series representation is in terms of the normalized variable $\stackrel{-}{x}$, where
 $x-=2x-xmax+xmin xmax-xmin .$
Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$.
4:     $\mathrm{a}\left(\mathit{la}\right)$ – double array
la, the dimension of the array, must satisfy the constraint $\mathit{la}\ge \left(\mathit{np1}-1\right)×{\mathbf{ia1}}+1$.
The Chebyshev coefficients of the polynomial $p\left(\stackrel{-}{x}\right)$. Specifically, element $\mathit{i}×{\mathbf{ia1}}+1$ must contain the coefficient ${a}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$. Only these $n+1$ elements will be accessed.
5:     $\mathrm{ia1}$int64int32nag_int scalar
The index increment of a. Most frequently, the Chebyshev coefficients are stored in adjacent elements of a, and ia1 must be set to $1$. However, if, for example, they are stored in ${\mathbf{a}}\left(1\right),{\mathbf{a}}\left(4\right),{\mathbf{a}}\left(7\right),\dots \text{}$, then the value of ia1 must be $3$.
Constraint: ${\mathbf{ia1}}\ge 1$.
6:     $\mathrm{x}$ – double scalar
The argument $x$ at which the polynomial is to be evaluated.
Constraint: ${\mathbf{xmin}}\le {\mathbf{x}}\le {\mathbf{xmax}}$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The value of the polynomial $p\left(\stackrel{-}{x}\right)$.
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:
${\mathbf{ifail}}=1$
 On entry, $\mathit{np1}<1$, or ${\mathbf{ia1}}<1$, or $\mathit{la}\le \left(\mathit{np1}-1\right)×{\mathbf{ia1}}$, or ${\mathbf{xmin}}\ge {\mathbf{xmax}}$.
${\mathbf{ifail}}=2$
x does not satisfy the restriction ${\mathbf{xmin}}\le {\mathbf{x}}\le {\mathbf{xmax}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The rounding errors 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 $n+1$.

## Example

Suppose a polynomial has been computed in Chebyshev series form to fit data over the interval $\left[-0.5,2.5\right]$. The following program evaluates the polynomial at $4$ equally spaced points over the interval. (For the purposes of this example, xmin, xmax and the Chebyshev coefficients are supplied . Normally a program would first read in or generate data and compute the fitted polynomial.)
```function e02ak_example

fprintf('e02ak example results\n\n');

xmin = -0.5; xmax = 2.5;
a = [2.53213  1.13032  0.2715  0.04434   0.00547   0.00054  4e-05];

n = int64(6);
ia1 = int64(1);

% Evaluate polynomial from coefficients a at x
m = 21;
dx =  (xmax-xmin)/(m-1);
x = [xmin:dx:xmax];

for i = 1:m
[fit(i), ifail] = e02ak( ...
n, xmin, xmax, a, ia1, x(i));
end

sol = [x; fit];
fprintf('     x           p(x)\n');
fprintf('%9.4f    %9.4f\n',sol(1:2,1:5:21));

fig1 = figure;
plot(x,fit);
title('Evaluation of Chebyshev Polynomial');
xlabel('x');
ylabel('p(x)');
legend('fit','exp(x)');

```
```e02ak example results

x           p(x)
-0.5000       0.3679
0.2500       0.6065
1.0000       1.0000
1.7500       1.6487
2.5000       2.7183
``` 