What is Hybrid Symbolic-Numeric Computation?

Hybrid symbolic-numeric computation constitutes the Fifth of my "Seven Dwarfs" of Symbolic Computation [1], which I have listed in my SNSC talk in Hagenberg in 2008. Hybridization requires that the solution of a computational mathematical problem should utilize both a numeric and a symbolic algorithmic component. The growth of the symbolic parts in scientific computing is driven by industry, as the proliferation of MapleSIM, a hybrid symbolic-numeric engineering design platform, testifies. In my talk, I will introduce two hybrid algorithms: one is our ArtinProver agorithm for proving the globality of an optimum of a rational function via an exact sum-of-squares certificate. The second is a randomized algorithm for recovering a sparse signal via a series of Hankel matrix condition number estimates. Specifically, I will discuss the important questions: what is an exact certificate, and what is the meaning of "success with high probability" in randomized algorithms with imprecise floating point data.