The solution of symbolic computation problems can be accelerated in at least two ways: one is to integrate limited precision floating point arithmetic on the scalars, and the other is to use parallel processes. We propose to investigate those approaches on problems in real algebraic optimization and exact linear algebra.
Semidefinite numerical optimization produces sum-of-squares representations for polynomial inequalities, which express global optimality. But the representations are numeric and approximate, and exact rational certificates for the inequalities are derived via exact symbolic means, thus leading to truly hybrid symbolic-numeric algorithms. When combining floating point arithmetic and randomization, analysis of the expected condition numbers of the random intermediate problems can guarantee success. We propose to introduce fraction-free algorithms and analyze the arising determinantal condition numbers.
LinBox is a C++ library for exact linear algebra. We propose to participate in the parallelization of the LinBox library by investigating problems with memory contention and by creating interactive symbolic supercomputing environments. Several related problems in computational algebraic complexity that will receive attention are: quadratic-time certificates for linear algebra problems, determinantal representation of polynomials by symmetric linear matrix forms, and interpolation of supersparse (lacunary) rational functions.
The PI's research expands the infrastructure in symbolic computation and the understanding of the underlying complexity. Aside from certifying optima, exact sums-of-squares certificates have proved theorems in mathematics, physics and control theory. Some of the important applications of LinBox are sparse linear algebra over finite fields and integer Smith forms for computational topology data. The PI is making the developed software freely available.
Algorithmic thinking and computation have become a cornerstone of modern science (pure and applied) and modern life. Symbolic computation programs, such as Mathematica, Maple, and the SAGE platform, have many applications, such as modeling data sets by symbolic expression for Toyota and Shell Oil, generating programs for signal processing, and program and protocol verification.
