Symbolic-numeric sparse interpolation of multivariate polynomials

Citation

Giesbrecht M, Labahn G & Lee W (2009) Symbolic-numeric sparse interpolation of multivariate polynomials. Journal of Symbolic Computation, 44 (8), pp. 943-959. https://doi.org/10.1016/j.jsc.2008.11.003

Abstract
We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony's method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony's method. We analyse the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications.

Keywords
Symbolic-numeric computing; multivariate interpolation

Journal
Journal of Symbolic Computation: Volume 44, Issue 8