Lagrange polynomial Guide, Meaning , Facts, Information and Description

In the mathematical subfield of numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. It was first discovered by Edward Waring in 1779 and later rediscovered by Leonhard Euler in 1783.

As there is only one interpolation polynomial for a given set of data points it is a bit misleading to call the polynomial Langrange interpolation polynomial. The more precise name is interpolation polynomial in the Langrange form.

Table of contents

1 Definition 2 Proof 3 Main idea 4 Usage 5 See also

Definition

[[image:lagrangepolys.png|frame|This image shows, for 4 random points ((-9, 5), (-4, 2), (-1, -2), (7, 9)), the (cubic) interpolation polynomial L(x), which is the sum of the scaled basis polynomials y₀l_0(x), y₁l₁(x), y₂l₂(x) and y₃l₃(x). The interpolation polynomial passes through all 4 control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.]]

Given a set of k+1 data points

where no two x_j are the same, the interpolation polynomial in the Lagrange form is a linear combination of Lagrange basis polynomials

with the Lagrange basis polynomials defined as

Proof

The function we are looking for has to be polynomial function L(x) of degree k with

According to the Stone-Weierstrass theorem such a function exists and is unique. The Langrange polynomial is the solution to the interpolation problem.

As can be easily seen

1. is a polynomial and has degree k

Thus the function L(x) is a polynomial with degree k and

.

Therefore L(x) is our unique interpolation polynomial.

Main idea

Solving an interpolation problems leads to a problem in linear algebra where we have to solve a matrix. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Langrange basis we get the much simpler identity matrix = δ_i,j; which we can solve instantly.

Usage

The Langrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore it is preferred in proofs and theoretical arguments. But as can be seen from the construction each time a node x_k changes, all Langrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the Newton polynomial.

In numerical integration the Lagrange basis polynomials are used to derive the Newton-Cotes formulas.

See also

• Polynomial interpolation
• Newton form of the interpolation polynomial
• Bernstein form of the interpolation polynomial
• Newton-Cotes formulas

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