Interpolation and Approximation

Polynomials in MATLAB

Plot the polynomial y = 50 + 15x – 4x² + 2x³ – 12x⁴ – 2x⁶ for x values between -2.8 and 2.8

Compute polynomial values directly

xx = -2.8 : 0.01 : 2.8;
yy1 = 50 + 15*xx - 4*xx.^2 + 2*xx.^3 - 12*xx.^4 - 2*xx.^6;

Compute polynomial values using polyval

Represent polynomial as a vector of coefficients

coefs =

If polynomial has degree N, then length(coefs) is

Use polyval to evaluate the polynomial at desired x values

yy2 =

Interpolation, approximation, and plotting

Given N data points, (x_k,y_k), find a smooth function f (e.g., to be able to find values between data points).

Interpolation:

for all data points (x_k,y_k),

Approximation:

for all data points (x_k,y_k),

Having found the smooth function, we now want to plot data values as distinct points and the interpolating/approximating function as a smooth curve.

x_data = x coordinates of data values
y_data = y coordinates of data values
xx = vector of closely spaced x values over range to plot
yy = vector of y values obtained by evaluating the function at xx

Using one plot command:

Using multiple plot commands:

Interpolation

Using polynomials

Given N data points, there is a unique polynomial of degree (at most)

Use polyfit( x_data , y_data , degree )

Example: Given data points (1, 1), (3, 3), and (5, 2), find the interpolating polynomial through these data points.

Estimate the value at x = 4:

What if you have a lot of data points and need to do interpolation?

Problem: high-degree polynomials
tend to oscillate wildly

Option 1:

Option 2:

Using splines

yy = spline( x_data , y_data , xx )

Approximation

Goal: minimize the error between
data points & approximating function

Using polynomials

What if approximating function is not a polynomial?

Idea: convert approximating function into something that looks like a polynomial

Example: