Polynomial Interpolation Matlab

Polynomial interpolation is a method used in mathematics to estimate a polynomial function that passes through a set of given data points. This method is widely used in various fields of science and engineering, such as signal processing, image processing, and data analysis.

Matlab is a powerful software tool for scientific computing and data analysis. It provides a variety of functions for polynomial interpolation, making it easy to perform interpolation tasks quickly and accurately. In this article, we will explore polynomial interpolation using Matlab.

Before we start, let's review some basic concepts related to polynomial interpolation. A polynomial function of degree n is defined as:

                    f(x) = a0 + a1x + a2x^2 + ... + anx^n

where

a0, a1, a2, ..., an 

are the coefficients of the polynomial. The degree of the polynomial is the highest power of x in the polynomial. In polynomial interpolation, we want to estimate the coefficients of the polynomial that passes through a set of given data points.

Let's consider a set of n+1 data points:

 (x0, y0), (x1, y1), (x2, y2), ..., (xn, yn). 

Our goal is to find a polynomial function f(x) of degree n that passes through these data points. We can use the Lagrange interpolation formula to estimate the coefficients of the polynomial:

                            f(x) = L0(x)y0 + L1(x)y1 + L2(x)y2 + ... + Ln(x)yn

                              where L0(x), L1(x), L2(x), ..., Ln(x) are the Lagrange basis polynomials, defined as:

                              Lk(x) = ∏(i=0,i≠k)n(x-xi)/(xk-xi)

Using this formula, we can estimate the coefficients of the polynomial that passes through the given data points.

Now, let's see how we can implement polynomial interpolation in Matlab. The polyfit function is used to estimate the coefficients of a polynomial that passes through a set of data points. The syntax of the polyfit function is:

                        p = polyfit(x, y, n)

where x and y are the vectors containing the data points, and n is the degree of the polynomial. The output of the polyfit function is the vector p containing the coefficients of the polynomial.

For example, let's consider the following set of data points:

                        x = [0 1 2 3 4]
                        y = [1 2 4 8 16]

We want to estimate a polynomial of degree 4 that passes through these data points. We can use the polyfit function as follows:

                    p = polyfit(x, y, 4)

The output of this function is:

                    p = [0.0139 -0.2361 1.9444 -7.9583 16.0000]

These are the coefficients of the polynomial that passes through the given data points. We can use the polyval function to evaluate the polynomial at any point. The syntax of the polyval function is:

                    y = polyval(p, x)

where p is the vector containing the coefficients of the polynomial, and x is the vector containing the points where we want to evaluate the polynomial.

For example, let's evaluate the polynomial at x = 2.5:

                    y = polyval(p, 2.5)

The output of this function is:

                    y = 3.8906

This is the value of the polynomial at x = 2.5.