1. Efficiant polynomial interpolation algorithms

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

3. 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

4. The Vandermonde Matrix

5. Generalized Vandermonde
where

6. Determinant of a Vandermonde

7. Determinant of a Vandermonde

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

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

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

11. The inverse of a Vandermonde matrix

12. The inverse of a Vandermonde matrix

13. Solving a Vandermonde system of equations

14. Solving a Vandermonde system of equations

15. Solving a Vandermonde system of equations

16. The algorithm to solve the system

17. 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

18. 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

19. Univariate Interpolation
Lagrange Interpolation
Newton Interpolation
Abstract Interpolation

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

21. Lagrange Interpolation
This is a Vandermonde system where

22. Lagrange Interpolation

23. Lagrange Interpolation

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

25. The Chinese remainder algorithm over Z

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

27. 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)

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

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

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

31. 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

32. 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)

33. Multivariate dense Interpolation

34. 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

35. Probabilistic Sparse Interpolation
Formal Presentation
Example
Analysis

36. 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

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

38. 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

39. 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

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

41. 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

42. 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

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

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

45. 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

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

47. 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:

48. 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)

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

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

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

52. 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

53. The End
Questions?
Bibliography: Richard Zippel

54. Appendix pseudo-codes for the interpolation algorithms

55. Code for Vandermonde Matrices

56. Code for Lagrange Interpolation

57. Code for Newton Interpolation