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# NAG Toolbox: nag_fit_1dmmax (e02ac)

## Purpose

nag_fit_1dmmax (e02ac) calculates a minimax polynomial fit to a set of data points.

## Syntax

[a, ref] = e02ac(x, y, m1, 'n', n)
[a, ref] = nag_fit_1dmmax(x, y, m1, 'n', n)

## Description

Given a set of data points (xi,yi)$\left({x}_{i},{y}_{i}\right)$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$, nag_fit_1dmmax (e02ac) uses the exchange algorithm to compute an m$m$th-order polynomial
 P(x) = a1 + a2x + a3x2 + ⋯ + am + 1xm $P(x)=a1+a2x+a3x2+⋯+am+1xm$
such that maxi 2|P(xi)yi|$\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}2|\mathrm{P}\left({x}_{i}\right)-{y}_{i}|$ is a minimum.
The function also returns a number whose absolute value is the final reference deviation (see Section [Error Indicators and Warnings]). 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:     x(n) – double array
The values of the x$x$ coordinates, xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
Constraint: x1 < x2 < < xn${x}_{1}<{x}_{2}<\cdots <{x}_{n}$.
2:     y(n) – double array
The values of the y$y$ coordinates, yi${y}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
3:     m1 – int64int32nag_int scalar
m + 1$m+1$, where m$m$ is the order of the polynomial to be found.
Constraint: m1 < min (n,100)${\mathbf{m1}}<\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},100\right)$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
The number n$n$ of data points.

None.

### Output Parameters

1:     a(m1) – double array
The coefficients ai${a}_{\mathit{i}}$ of the final polynomial, for i = 1,2,,m + 1$\mathit{i}=1,2,\dots ,m+1$.
2:     ref – double scalar
The final reference deviation (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

With exact arithmetic the algorithm should terminate after a finite number of steps. This need not necessarily be the case using computer arithmetic. Should the function start cycling then an exit is made with ref given a negative value. This is by no means an indicator that a catastrophic error has occurred and does not preclude useful results being obtained.
The absolute value of ref is the final reference deviation. See Stieffel (1959) for an explanation of this term.
If an error is detected in an input parameter nag_fit_1dmmax (e02ac) will act as if a soft noisy exit has been requested (see Section [Soft Fail Option] in the (essin)).

## Accuracy

This is wholly dependent on the given data points.

The time taken increases with m$m$.

## Example

```function nag_fit_1dmmax_example
x = [0;
0.05;
0.1;
0.15;
0.2;
0.25;
0.3;
0.35;
0.4;
0.45;
0.5;
0.55;
0.6;
0.65;
0.7;
0.75;
0.8;
0.85;
0.9;
0.95;
1];
y = [1;
1.051271096376024;
1.105170918075648;
1.161834242728283;
1.22140275816017;
1.284025416687741;
1.349858807576003;
1.419067548593257;
1.49182469764127;
1.568312185490169;
1.648721270700128;
1.733253017867395;
1.822118800390509;
1.915540829013896;
2.013752707470477;
2.117000016612675;
2.225540928492468;
2.339646851925991;
2.45960311115695;
2.585709659315846;
2.718281828459045];
m1 = int64(6);
[a, ref] = nag_fit_1dmmax(x, y, m1)
```
```

a =

1.0000
1.0001
0.4991
0.1704
0.0348
0.0139

ref =

1.0915e-06

```
```function e02ac_example
x = [0;
0.05;
0.1;
0.15;
0.2;
0.25;
0.3;
0.35;
0.4;
0.45;
0.5;
0.55;
0.6;
0.65;
0.7;
0.75;
0.8;
0.85;
0.9;
0.95;
1];
y = [1;
1.051271096376024;
1.105170918075648;
1.161834242728283;
1.22140275816017;
1.284025416687741;
1.349858807576003;
1.419067548593257;
1.49182469764127;
1.568312185490169;
1.648721270700128;
1.733253017867395;
1.822118800390509;
1.915540829013896;
2.013752707470477;
2.117000016612675;
2.225540928492468;
2.339646851925991;
2.45960311115695;
2.585709659315846;
2.718281828459045];
m1 = int64(6);
[a, ref] = e02ac(x, y, m1)
```
```

a =

1.0000
1.0001
0.4991
0.1704
0.0348
0.0139

ref =

1.0915e-06

```

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Chapter Contents
Chapter Introduction
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