Lagrange Interpolation Polynomials

Interpolacion

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Interpolations

Numerical Analysis

Teacher: Alejandro Haro

Yessica Medina Gutiérrez 3040902

Actuary

21-02-2018

Lagrange Interpolation Polynomials

Suppose that are n+1 points other than the real axis and that f (x) is a function of real value defined over some interval that contains these points. We want to build a polynomial p(x) of degree ≤n that interpolates f (x) and this satisfies

Next we will show that there is at least a degenerate polynomia of degree ≤n that interpolates f (x)

In this case we look for a polynomial that is canceled in all the points except for .The function

=(x-(x-

it's such a polynomial.Also since x are assumed to be different, so that

And this is the solution of the interpolation’s problema in this special case

But then for an arbitrary function f (x)

Theorem 4.1.-Given a function of real values f (x) and n + 1 different points there exists exactly a polynomial of degree ≤n that interposes in f (x) in those points the summation is called Lagrange form for the points .for example if we have two different points..

This is the case of linear interpolation written in one of its multiple equivalent forms

Newton Interpolation Polynomials

The Newton form of the equation of a straight line passing through two points and is

And the general form of the equation of the polynomila passing trough n points

To ilustrate the method for finding the coefficients consider the problem of finding the values for example in the parábola of that pass in the points , and

Sustituting

gives

The calculations can be performed in a systematic manner by using the “divided differences” of the function values.