Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

A Essanhaji,M Errachid,Saeid Abbasbandy

doi:10.1155/2022/8227086

Abstract

The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called Z , z -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.

Highlights

Given a finite interpolation set Zn = fz1, ⋯zng ⊂ Kp of distincts nodes, the Lagrange interpolation problem consists of finding, for a given data vector R = ðrz : z ∈ ZnÞ ∈ KZn, a polynomial P ∈ Πp such that
We define a notion of ðZ, zÞ-partition, and we present a random algorithm for computing the polynomials Qi, for i ∈ 1⁄21⁄21 ; n, which will be used for giving the algorithm RLMVPIA
This work contributes to solve the problem of Lagrange multivariate polynomial interpolation with any finite set of interpolation nodes, using a recursive algorithm RLMVPIA with a random approach based on the ðZ, zÞ -partition concept

Summary

Introduction

Let K be a commutative field and p, n ∈ N∗. By Πp = K1⁄2x1, ⋯, xp, we denote the algebra of all polynomials in p variables, and we denote by Πpd the subspace of all polynomials of total degree less than or equal to d, where d is a nonnull integer. We construct a random algorithm for finding several sub-spaces solutions of the problem P ðZnÞ It is well-known that in the univariate case (p = 1) the Lagrange interpolation problem with respect to n distinct points is always uniquely solvable, if one takes P to be the space of all polynomials of degree less than or equal to n − 1. This new approach takes into consideration the distribution of the nodes of the considered interpolation set Zn by introducing a new concept described in the following. Applying the algorithm ZPNA several times, we get the following ðZ, zÞ-partitions and the associated polynomials.

RLMVPIA Random Approach

Example 1

Example 3

Example 4

Conclusion