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# NAG Toolbox: nag_1d_minimax_polynomial (e02al)

## Purpose

nag_1d_minimax_polynomial (e02al) calculates a minimax polynomial fit to a set of data points.

## Syntax

[a, ref, ifail] = e02al(x, y, m, 'n', n)
[a, ref, ifail] = nag_1d_minimax_polynomial(x, y, m, 'n', n)

## Description

Given a set of data points $\left({x}_{i},{y}_{i}\right)$, for $\mathit{i}=1,2,\dots ,n$, nag_1d_minimax_polynomial (e02al) uses the exchange algorithm to compute an $m$th-degree polynomial
 $Px = a0 + a1x + a2 x2 + ⋯ + am xm$
such that $\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left|\mathrm{P}\left({x}_{i}\right)-{y}_{i}\right|$ is a minimum.
The function also returns a number whose absolute value is the final reference deviation (see Arguments). The function is an adaptation of Boothroyd (1967).

## References

Boothroyd J B (1967) Algorithm 318 Comm. ACM 10 801
Stieffel E (1959) Numerical methods of Tchebycheff approximation On Numerical Approximation (ed R E Langer) 217–232 University of Wisconsin Press

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The values of the $x$ coordinates, ${x}_{i}$, for $\mathit{i}=1,2,\dots ,n$.
Constraint: ${x}_{1}<{x}_{2}<\dots <{x}_{n}$.
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The values of the $y$ coordinates, ${y}_{i}$, for $\mathit{i}=1,2,\dots ,n$.
3:     $\mathrm{m}$int64int32nag_int scalar
$m$, where $m$ is the degree of the polynomial to be found.
Constraint: $0\le {\mathbf{m}}<\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(100,{\mathbf{n}}-1\right)$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
$n$, the number of data points.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{a}\left({\mathbf{m}}+1\right)$ – double array
The coefficients ${a}_{i}$ of the minimax polynomial, for $\mathit{i}=0,1,\dots ,m$.
2:     $\mathrm{ref}$ – double scalar
The final reference deviation, i.e., the maximum deviation of the computed polynomial evaluated at ${x}_{\mathit{i}}$ from the reference values ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. ref may return a negative value which indicates that the algorithm started to cycle due to round-off errors.
3:     $\mathrm{ifail}$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:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{m}}<100$.
Constraint: ${\mathbf{m}}<{\mathbf{n}}-1$.
Constraint: ${\mathbf{m}}\ge 0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{x}}\left(\mathit{i}+1\right)>{\mathbf{x}}\left(\mathit{i}\right)$.
${\mathbf{ifail}}=-99$
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

This is dependent on the given data points and on the degree of the polynomial. The data points should represent a fairly smooth function which does not contain regions with markedly different behaviours. For large numbers of data points (${\mathbf{n}}>100$, say), rounding error will affect the computation regardless of the quality of the data; in this case, relatively small degree polynomials (${\mathbf{m}}\ll \sqrt{{\mathbf{n}}}$) may be used when this is consistent with the required approximation. A limit of $99$ is placed on the degree of polynomial since it is known from experiment that a complete loss of accuracy often results from using such high degree polynomials in this form of the algorithm.

## Further Comments

The time taken increases with $m$.

## Example

This example calculates a minimax fit with a polynomial of degree $5$ to the exponential function evaluated at $21$ points over the interval $\left[0,1\right]$. It then prints values of the function and the fitted polynomial.
```function e02al_example

fprintf('e02al example results\n\n');

% Minimax polynomial of degree 5 that fits exp(x) on grid in [0,1].
x = [0:0.05:1];
y = exp(x);
m = int64(5);
[a, ref, ifail] = e02al(x, y, m);

fprintf('   Polynomial coefficients\n');
fprintf('        %7.4f\n',a(1:m+1));
fprintf('\n   Reference deviation = %8.2e\n\n', ref);
fprintf('   x     Fit      exp(x)   Residual\n');

xx = [0:0.1:1];
p  = a(m+1:-1:1);
s  = polyval(p,xx);
yy = exp(xx);

for i=1:11
fprintf('%6.2f%9.4f%9.4f%11.2e\n',xx(i), s(i), yy(i), s(i) - yy(i));
end

```
```e02al example results

Polynomial coefficients
1.0000
1.0001
0.4991
0.1704
0.0348
0.0139

Reference deviation = 1.09e-06

x     Fit      exp(x)   Residual
0.00   1.0000   1.0000  -1.09e-06
0.10   1.1052   1.1052   9.74e-07
0.20   1.2214   1.2214  -7.44e-07
0.30   1.3499   1.3499  -9.18e-07
0.40   1.4918   1.4918   2.99e-07
0.50   1.6487   1.6487   1.09e-06
0.60   1.8221   1.8221   4.59e-07
0.70   2.0138   2.0138  -8.16e-07
0.80   2.2255   2.2255  -8.42e-07
0.90   2.4596   2.4596   8.75e-07
1.00   2.7183   2.7183  -1.09e-06
```

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