Answer to Question #93978 in Quantitative Methods for ABDULLAHI AA

Question

The interpolating polynominal of degree \\(\\|eqn\\) With the nodes \\(x_{0},x_{1},\\ c dots,x_{n}u) can be written as?

Answer 1

There are two different kinds of the interpolating polynomials:

1)Lagrange polynomial:

$L(x)=\sum_{i=0}^nf(x_i)\prod_{0\le m \le n, m\ne i}\frac{x-x_m}{x_i-x_m}$

2)Newton polynomial:

Forward divided:

$N(x)=[y_0]+[y_0,y_1](x-x_0)+ \newline +[y_0,y_1,y_2](x-x_0)(x-x_1)+...+ \newline +[y_0,...,y_n](x-x_0)(x-x_1)...(x-x_{n-1})$

Backward divided:

$N(x)=[y_n]+[y_n,y_{n-1}](x-x_n)+ \newline +[y_n,y_{n-1},y_{n-2}](x-x_n)(x-x_{n-1})+...+ \newline +[y_n,y_{n-1},...y_0](x-x_n)(x-x_{n-1})...(x-x_1)$

where

$y_i=f(x_i),i=0,1,...,n; [y_i]=y_i,[y_i,y_{i+1}]= \newline =\frac{y_{i+1}-y_i}{x_{i+1}-x_i}, [y_i,y_{i+1},y_{i+2}]= \newline \frac{[y_{i+2},y_{i+1}]-[y_{i+1},y_i]}{x_{i+2}-x_i},...,[y_0,...,y_n]= \newline =\frac{[y_n,...,y_1]-[y_{n-1},...,y_0]}{x_n-x_0}.$