Lagrange polynomials for interpolation

Linear interpolation consists of approximating a function [latex size=”1″]f(x)[/latex] as:

[latex size=”1″]f(x)=sum_{i=1}^{N}a_i phi_i(x);;;;;;;;;;;(1)[/latex]

where the [latex size=”1″]a_i[/latex]’s are the interpolation coefficients and the [latex size=”1″]phi_i[/latex]’s are prefixed interpolation functions.

Lagrange interpolation, which is one of the simplest and mostly employed interpolation methods, consists of finding the interpolation coefficients as the solution of the linear system

[latex size=”1″]f(x_j)=sum_{i=1}^{N}a_i phi_i(x_j), ;;;;;; j=1,ldots,N;;;;;;;;;;;(2)[/latex]

where the [latex size=”1″]x_j[/latex]’s are interpolation points.

A common case is when the interpolation functions are polynonials, say

[latex size=”1″]f(x_j)=sum_{i=1}^{N}alpha_i x^{i-1}_j, ;;;;;; j=1,ldots,N.;;;;;;;;;;;(3)[/latex]

The determinant of such a system is a Vandermonde determinant which is always non-vanishing and therefore the system always admits a unique solution, provided that the interpolation points are all different.
Accordingly, polynomial Lagrange interpolation is always unique.
Polynomial interpolation functions in eq. (1) can be found by assuming that [latex size=”1″]phi_i(x)[/latex] is a polynomial of degree [latex size=”1″]N-1[/latex], that [latex size=”1″]phi_i(x_j)=0[/latex], [latex size=”1″]j=1,ldots,N[/latex] and [latex size=”1″]jneq i[/latex], and that [latex size=”1″]phi_i(x_i)=1[/latex]

Accordingly,

[latex size=”1″]phi_i(x)=prod_{jneq i}frac{x-x_j}{x_i-x_j};;;;;;;;;;;(4)[/latex]

and

[latex size=”1″]f(x)=sum_{i=1}^{N}f(x_i)prod_{jneq i}frac{x-x_j}{x_i-x_j};;;;;;;;;;;(5)[/latex]

Reference: N.S. Bakhvalov, Numerical Methods, Mir Publishers Moscow, 1981.

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