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Polynomial curve fitting

`p = polyfit(x,y,n)`

```
[p,S] =
polyfit(x,y,n)
```

```
[p,S,mu]
= polyfit(x,y,n)
```

`[`

also
returns `p`

,`S`

,`mu`

]
= polyfit(`x`

,`y`

,`n`

)`mu`

, which is a two-element vector with
centering and scaling values. `mu(1)`

is `mean(x)`

,
and `mu(2)`

is `std(x)`

. Using these
values, `polyfit`

centers `x`

at
zero and scales it to have unit standard deviation

$$\widehat{x}=\frac{x-\overline{x}}{{\sigma}_{x}}\text{\hspace{0.17em}}.$$

This centering and scaling transformation improves the numerical properties of both the polynomial and the fitting algorithm.

In problems with many points, increasing the degree of the polynomial fit using

`polyfit`

does not always result in a better fit. High-order polynomials can be oscillatory between the data points, leading to a*poorer*fit to the data. In those cases, you might use a low-order polynomial fit (which tends to be smoother between points) or a different technique, depending on the problem.Polynomials are unbounded, oscillatory functions by nature. Therefore, they are not well-suited to extrapolating bounded data or monotonic (increasing or decreasing) data.

`polyfit`

uses `x`

to form
Vandermonde matrix `V`

with `n+1`

columns
and `m = length(x)`

rows, resulting in the linear
system

$$\left(\begin{array}{cccc}{x}_{1}^{n}& {x}_{1}^{n-1}& \cdots & 1\\ {x}_{2}^{n}& {x}_{2}^{n-1}& \cdots & 1\\ \vdots & \vdots & \ddots & \vdots \\ {x}_{m}^{n}& {x}_{m}^{n-1}& \cdots & 1\end{array}\right)\left(\begin{array}{c}{p}_{1}\\ {p}_{2}\\ \vdots \\ {p}_{n}\end{array}\right)=\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\\ \vdots \\ {y}_{m}\end{array}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}},$$

which `polyfit`

solves with `p = V\y`

.
Since the columns in the Vandermonde matrix are powers of the vector `x`

,
the condition number of `V`

is often large for high-order
fits, resulting in a singular coefficient matrix. In those cases centering
and scaling can improve the numerical properties of the system to
produce a more reliable fit.

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