The primary characteristic of symbolic computation with large rational numbers is that the computations may be slow but exact -- sometimes more exact than needed. The primary characteristic of numerical computation with floating point numbers is that the computations are fast but approximate -- sometimes more approximate than acceptable. In this research, accuracy and speed of computation are achieved by using hybrids of symbolic and numeric methods. This work involves improvements for linear system solving, matrix inertia computation, and minimal polynomial computation. The methods and implementations that this project creates can be taken up by software systems used widely in science, engineering, and education such as Maple, Mathematica, and Matlab. This research contributes to the fundamental understanding of the interplay between the exact and the approximate. The investigators create and demonstrate computational methods to solve several classes of linear systems of equations. Techniques heretofore undeveloped in this arena include matrix bordering techniques and iterative refinement. Additionally, the research improves symbolic-numeric capability to compute inertia of matrices, a measure that is important in control theory and arises in the study of Lie groups. The research includes finessing numerical loss of accuracy and handling of singular cases. The experimental approach to mathematics is enhanced by enabling large problems to be solved, particularly in number theory, combinatorics, algebraic geometry, thus providing data for conjecture formation and for experimental verification of conjectures. For science and engineering, this project creates a capability to solve a class of problems for which no solution method currently exists at all, specifically it is to solve linear systems where (1) numerical methods fail due to ill-condition of the problem instance, yet (2) the exact result is valid and meaningful despite the approximate nature of the input data.
