Efficiant polynomial interpolation algorithms

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Efficiant polynomial interpolation algorithms
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Overview

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

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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

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The Vandermonde Matrix
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Generalized Vandermonde
where
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Determinant of a Vandermonde
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Determinant of a Vandermonde
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Determinant of a Vandermonde
The Vandermonde matrix is non-singular ? the ki
are distinct
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The previous result can not be applyed for
generalized Vandermonde matrices

• Example
• wich is 0 also when

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Non-singularity of generalized Vandermonde
matrices

• Proposition 1
• If the ki are distinct positiv real numbers
• gt the matrix is non-zero

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The inverse of a Vandermonde matrix
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The inverse of a Vandermonde matrix
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Solving a Vandermonde system of equations
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Solving a Vandermonde system of equations
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Solving a Vandermonde system of equations
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The algorithm to solve the system
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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
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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

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Univariate Interpolation

• Lagrange Interpolation
• Newton Interpolation
• Abstract Interpolation

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

• f(a)f(x)(mod (x-a))

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The Chinese remainder algorithm over Z
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Chinese remainder with polinoms

• When given and

Then we change it to the following
situation Given Compute
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Newton Interpolation algorithm

1. Let f(x)0, q(x)1
2. Loop for n times doing following
3. f(x)f(x)q(ki)-1q(x)(wi-f(ki))
4. q(x)(x-ki)q(x)

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Newtons interpolation formula

• Let
• Newtons interpolation formula claims that there
  exist constants such that
• In fact, and is the solution of

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Newtons interpolation formula
Then And more generally Solving the gives
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Multivariate Interpolation

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

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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

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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(k1)0,...,xn0)

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Multivariate dense Interpolation
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The complexity of the dense interpolation

• Let I(d) be the complexity of interpolating d1
  values to produce a univariate plynomial of
  degree d and Nk the complexity for the first k
  variables

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Probabilistic Sparse Interpolation

• Formal Presentation
• Example
• Analysis

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Probabilistic Sparse Interpolation

• Assume we want to dermine P(X1,..., Xn) which is
  an element of LX 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

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Probabilistic Sparse Interpolation
Def is a precise evaluation point if
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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
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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

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Probabilistic Sparse Interpolation

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

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Probabilistic Sparse Interpolation
Assume we want to dermine P(X1,..., Xn) which is
an element of LX 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
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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
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Probabilistic Sparse Interpolation
So each of the can be computed using O(n2) and
we can avoid computing the other interpolations
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Probabilistic Sparse Interpolation

• The probability that the Vandermonde system of
  equation is non-singular is bounded by

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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 (k1)th stage

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Probabilistic Sparse Interpolation

• Example
• Lets assume we are given a Black Box representing
  the following polinom

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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

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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)
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Deterministic Sparse Interpolation without degree
bounds

• Let
• and consider
• so
• consider also

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Deterministic Sparse Interpolation without degree
bounds

• Then we get the following Toeplitz system of
  linear equations

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

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• The End
• Questions?
• Bibliography Richard Zippel

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Appendixpseudo-codes for the interpolation
algorithms
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Code for Vandermonde Matrices
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Code for Lagrange Interpolation
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Code for Newton Interpolation