Efficiant polynomial interpolation algorithms

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. 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