**Chapter 3 Interpolation and Polynomial Approximation**

**Lagrange Interpolation**

If are distinct numbers and is a function whose values are given at these numbers, then there exists a unique polynomial of degree at most with the property that

for each

This Lagrange interpolation polynomial is give by

where

, for each If , then for each , exists a number with

where is particularly shown in the software to evaluate the quality of distribution of interpolation nodes.

In the software, several styles of distribution of interpolation nodes can be chosen. They are uniform distribution, Legendre-Gauss points, Legendre-Gauss-Radau points, Legendre-Gauss-Lobatto points, Chebyshev-Gauss points, Chebyshev-Gauss-Radau, and Chebyshev-Gauss-Lobatto points. Except the uniform distribution, the others are defined as below considering .

Let be the Legendre polynomial of degree , then

Legendre-Gauss points are the roots of ;

Legendre-Gauss-Radau points are the roots of ;

Legendre-Gauss-Lobatto points are -1,1 and the roots of .

Also, Chebyshev-Gauss points are ;

Chebyshev-Gauss-Radau points are ;

Chebyshev-Gauss-Lobatto points are , for each These distributions of interpolation nodes related to Legendre and Chebyshev polynomials have special effects on equally distributing over . For general other than , an additional linear mapping is required.

In the software, the default function is called Runge function, for which its Lagrange polynomial based on uniform distribution fails to converge in . However, this does not happen and full convergence on is recovered for the other distributions related to Legendre and Chebyshev polynomials.

References:

【1】
R. L. Burden and J. D. Faires, *Numerical Analysis*, PWS,

【2】
K. E. Atkinson, *An
Introduction to Numerical Analysis*, Wiley,

【3】
B. Fornberg, *A
Practical Guide to Pseudospectral Methods*,