# Vandermonde System

Learned in CS370.

So Vandermonde matrix is one of the ways to do Polynomial Interpolation using a monomial basis.

A vandermonde system is a linear system characterized by a Vandermonde matrix used for polynomial interpolation.

Using a vandermonde system, we compute the interpolating polynomial by reducing it down to solving a linear system of equations.

Monomial Basis

The familiar form $p(x)=c_{1}+c_{2}x+c_{3}x_{2}+c_{4}x_{3}+...+c_{n}x_{nโ1}$ is called the monomial form, and can also be written $p(x)=โ_{i=1}c_{i}x_{iโ1}$ The sequence $1,x,x_{2},...$ is called the

monomial basis.Monomial form is a sum of coefficients $c_{i}$ times basis functions x^i$.

This is in contrast with the Lagrange Basis.

In general, if we had a set of data $(x_{1},y_{1}),...,(x_{n},y_{n})$, and want a polynomial of the form (2.1.1) then we can set up the linear system V ยท c = y where

Matrices of the form V are called Vandermonde matrices. All of the data required for creating one is contained in its second column, i.e., Vi,2 = x_i.

These facts have both practical and theoretical implications.

- The theoretical implication is that we can prove the basic theorem by showing that $V$ is non-singular
- The practical implication is that we have reduced computing the interpolating polynomial to solving a linear system of equations.