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Concept
Lagrange polynomial
Summary
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.
Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node,
Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.
Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory.
For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation.
Given a set of nodes , which must all be distinct, for indices , the Lagrange basis for polynomials of degree for those nodes is the set of polynomials each of degree which take values if and . Using the Kronecker delta this can be written Each basis polynomial can be explicitly described by the product:
Notice that the numerator has roots at the nodes while the denominator scales the resulting polynomial so that
The Lagrange interpolating polynomial for those nodes through the corresponding values is the linear combination:
Each basis polynomial has degree , so the sum has degree , and it interpolates the data because
The interpolating polynomial is unique. Proof: assume the polynomial of degree interpolates the data. Then the difference is zero at distinct nodes But the only polynomial of degree with more than roots is the constant zero function, so or
Each Lagrange basis polynomial can be rewritten as the product of three parts, a function common to every basis polynomial, a node-specific constant (called the barycentric weight), and a part representing the displacement from to :
By factoring out from the sum, we can write the Lagrange polynomial in the so-called first barycentric form:
If the weights have been pre-computed, this requires only operations compared to for evaluating each Lagrange basis polynomial individually.
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