Approximating data with polynomials using Matlab

Approximating data points involves selecting a function $y=f\left(x\right)$ such that $f\left(x_k\right)$ comes close to $y_k$ Common functions to use are polynomials.

$a_1{x}^n+a_2{x}^{n-1}+a_3{x}^{n-2}+\cdots +a_{n-1}{x}^2+a_{n}x+a_{n+1}$

If there are n+1 data points then this nth-degree polynomial will interpolate the data that is the function will pass through each point. For approximation more commonly the polynomial will be of low degree, such as a linear function, and there are many data points. In this case the approximating polynomial is meant to come close to the data, which might be statistically scattered around a linear trend.

Matlab conveniently computes the coefficients of a polynomial that comes close to a set of data points in the least squares sense. If x and y are vectors of n+1 $\left(x_k,y_k\right)$ data values entered into Matlab, then the function polyfit

returns the coefficients of a polynomial of degree m that approximates the n+1 data points in vectors x and y. The variable a is a vector of length m+1 that holds the coefficients

$a_1,a_2,\dots a_n,a_{m+1}$

With these coefficients a linear polynomial for instance could be written in Matlab as

However, Matlab has a more convenient way to evaluate a polynomial with coefficients computed by polyfit. This is with a companion function called polyval.

If x_values is a vector of x values, then the polyval function uses the coefficients in the a vector to compute the value of the polynomial at each of the x values and returns these results to the vector y_values.