User:Egm6341.s11.team4.hylon/hw4

=Problem 4.6 Fix Runge phenomenon of Lagrangian Interpolation using nodes of Legendre Polynomial, Chebyshev Polynomial and equidistant nodes =

Given
Refer to lecture notes [[media:nm1.s11.mtg21.djvu|21-1]] for more details.

Objective
1. Use roots of Legendre Polynomial to the accuracy of $$10^{-6}$$. Plot $$f _n^L(x)$$ and $$f(x)$$.

2. Use roots of Chebyshev Polynomial, to the accuracy of $$10^{-6}$$ by Newton-cotes methods. Plot $$f _n^L(x)$$ and $$f(x)$$.

3. Comparison of Lagrangian Interpolation using three different types of nodes.

Using nodes of Legendre Polynomial




Using nodes of Chebyshev Polynomial




Since there the difference in above figure is not much clear, we plot above picture using $$n=6$$ again.



Comment
From above results, we see that:

Among three different types of interpolation nodes, using roots of Chebyshev Polynomial is much accurate and can best fix Runge phenomenon, although using roots of Legendre Polynomial can also settle it.

In this problem, Runge phenomenon happens only when we use odd number of nodes to do Lagrangian interpolation.

Problem is solved by Hailong Chen

Reference
1.	Lagrange_polynomial 2.	Legendre_polynomials 3.     Chebyshev_polynomials 4.     Gauss-Legendre_quadrature