On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms

Algebraic randomization techniques can be applied to hybrid symbolic-numeric algorithms. Here we consider the problem of interpolating a sparse rational function from noisy values. We develop a new hybrid algorithm based on Zippel’s original sparse polynomial interpolation technique. We show experimentally that our algorithm can handle sparse polynomials with large degrees. We also give a (partial) mathematical justification why the Zippel’s algebraic randomization technique can be used with our approximate data: the randomly generated non-zero values are expected to be bounded away from zero. We show that the random Fourier-like matrices arising in our algorithm, have the desired rank property in the exact case, and appear usable numerically. Furthermore, we show that Sylvester matrices of polynomials with nonidentically distributed random coefficients have large condition numbers. That phenomenon has precluded several algebraic randomization techniques from use in the approxima...

• Type: Conference
• Year: 2007
• Where: ISSAC
• Authors: Erich Kaltofen, Zhengfeng Yang, Lihong Zhi
• Keywords: Algebraic Randomization Techniques, ISSAC 2007, Sparse Polynomials, Zippel’s Algebraic Randomization