Efficiant polynomial interpolation algorithms

Overview Introduction to Vandermonde Matrices and its utilities Univariate Interpolation Multivariate Interpolation

Properties of Vandermonde Matrices Easy to ensure that they are non-singular Systems of linear equations whose coefficients form Vandermonde matrices are easy to solve exactly

The Vandermonde Matrix

Generalized Vandermonde where

Determinant of a Vandermonde

Determinant of a Vandermonde

Determinant of a Vandermonde The Vandermonde matrix is non-singular  the ki are distinct

Example wich is 0 also when The previous result can not be applyed for generalized Vandermonde matrices Example wich is 0 also when

Non-singularity of generalized Vandermonde matrices Proposition 1: If the ki are distinct positiv real numbers => the matrix is non-zero

The inverse of a Vandermonde matrix

The inverse of a Vandermonde matrix

Solving a Vandermonde system of equations

Solving a Vandermonde system of equations

Solving a Vandermonde system of equations

The algorithm to solve the system

The algorithm to solve the system The computation of the xi is arranged as follows: Calculate each vector and add it to the accumulating X

Analysis of the algorithm By calculating the vectors one after the other we only need to compute one Pi(Z) at the time Each Pi(Z) only needs O(n) time and since we have n polinoms to compute, the complexity is O(n2) and the space needed is O(n) Because the inverse of the transposed matrix is the transpose of the inverse of the matrix, the algorithm only need a little adjustment to solve a transposed Vandermonde system of equations On the Appendix there is an example of this alorithm taken from Zippel

Univariate Interpolation Lagrange Interpolation Newton Interpolation Abstract Interpolation

Lagrange Interpolation Giving are a set of distinct evaluation points with its correspondating functional values The goal is to find the polinome

Lagrange Interpolation This is a Vandermonde system where

Lagrange Interpolation

Lagrange Interpolation

Newton Interpolation f(a)=f(x)(mod (x-a))

The Chinese remainder algorithm over Z

Chinese remainder with polinoms When given and Then we change it to the following situation: Given Compute

Newton Interpolation algorithm Let f(x)=0, q(x)=1 Loop for n times doing following: f(x)=f(x)+q(ki)-1q(x)(wi-f(ki)) q(x)=(x-ki)q(x)

Newton´s interpolation formula Let Newton´s interpolation formula claims that there exist constants such that In fact, and is the solution of

Newton´s interpolation formula Then And more generally Solving the gives

Multivariate Interpolation Dense Interpolation Probabilistic Sparse Interpolation Deterministic Sparse Interpolation without degree bounds

Multivariate dense Interpolation We are given a black box with a degree bound „d“ for the polinom P(xi,..,xn) So we can assume that P has the form

Multivariate dense Interpolation So we get the values of which are the coeficients found by interpolating P on X1 By doing this procedure we compute recursively P(X1,...,Xk,x(k+1)0,...,xn0)

Multivariate dense Interpolation

The complexity of the dense interpolation Let I(d) be the complexity of interpolating d+1 values to produce a univariate plynomial of degree „d“ and Nk the complexity for the first k variables

Probabilistic Sparse Interpolation Formal Presentation Example Analysis

Probabilistic Sparse Interpolation Assume we want to dermine P(X1,..., Xn) which is an element of L[X] where L is a field of cardinal q and the degree of each Xi is bounded by „d“ and there are no more than T non-zero monomials

Probabilistic Sparse Interpolation Def: is a precise evaluation point if:

Probabilistic Sparse Interpolation The probability by wich is an imprecise evaluation point: For each k we can write It is an imprecise evaluation point if one of the cik = 0 And the probability that this happends is no more than

Probabilistic Sparse Interpolation Given is a k-1 tuple The probability that is 0 if we are we are working on a field of characteristic 0 or at least When working on a field of q elements the probability is bounded by

Probabilistic Sparse Interpolation So the following probability is then one that underlines the Probabilistic Sparse Interpolation

Probabilistic Sparse Interpolation Assume we want to dermine P(X1,..., Xn) which is an element of L[X] where L is a field of cardinal q and the degree of each Xi is bounded by „d“ and there are no more than T non-zero monomials As in the dense interpolation we Interpolate

Probabilistic Sparse Interpolation At the kth stage the first computation gives us: We then assume that The probability of that being the right skeleton is We then pick a (k-1) tuple And we set up the following transposed Vandermonde system of linear ecuations

Probabilistic Sparse Interpolation So each of the can be computed using O(n2) and we can avoid computing the other interpolations

Probabilistic Sparse Interpolation The probability that the Vandermonde system of equation is non-singular is bounded by

Probabilistic Sparse Interpolation So we get for each k Then we solve trough the dense interpolation We then expand it and we get And we are ready to compute the (k+1)th stage

Probabilistic Sparse Interpolation Example Lets assume we are given a Black Box representing the following polinom

Deterministic Sparse Interpolation without degree bounds Given are a bound on the number of non-zero terms „T“ and the number of variables „n“ We want to compute By choosing a distinct prime for each Xi then the quantities will all be distinct. Let Then we get:

Deterministic Sparse Interpolation without degree bounds The rank of the system of equations is exactly the number of non-zero monomials in P This could be easily done by taking the first T equations and computing their rank which requires O(T3)

Deterministic Sparse Interpolation without degree bounds Let and consider so consider also

Deterministic Sparse Interpolation without degree bounds Then we get the following Toeplitz system of linear equations

Deterministic Sparse Interpolation without degree bounds So the system is non-singular if the mi are distinct

Deterministic Sparse Interpolation without degree bounds So the system can be solved by Gaussian elimination O(t3) So then we get Q(Z) In order to find the mi we just need to find the zeroes of Q which are in fact positive integers making this procedure much easier Knowing the mi, the ei can easily be determined by factoring each of the mi, which is in fact very easy because the possible divisors are the first n primes allready known By knowing the mi it is allso easy to compute the ci just by solving the Vandermonde system, formed by the first t equations

The End Questions? Bibliography: Richard Zippel

Appendix pseudo-codes for the interpolation algorithms

Code for Vandermonde Matrices

Code for Lagrange Interpolation

Code for Newton Interpolation