Polynomial Interpolation by Definition

Summary

The interpolating polynomial can be written in the monomial basis and found by solving a Vandermonde system. Existence and uniqueness for distinct nodes guarantee a single solution of degree at most $n$ .

Prerequisites

Existence and Uniqueness of Interpolating Polynomial, Systems of Linear Equations

Method Definition

Seek $P(x)=c_0+c_{1}x+\cdots +c_n{x}^n$ such that $P(x_i)=y_i$ for distinct $x_0,\dots ,x_n$ . In matrix form,

Worked Example

Nodes $(0,2),(1,3)$ : solve $c_0=2$ , $c_0+c_1=3$ → $P(x)=2+x$ .

Common Failure Modes

• Repeated nodes make the Vandermonde matrix singular.
• High $n$ yields ill-conditioned Vandermonde systems; prefer Newton form in practice.

Connections

• Related: Lagrange Polynomial, Newton Polynomial

References

Vandermonde interpolation is the monomial-basis form of the unique interpolant.^[1]