AF: Small: Collaborative Research: High Performance Exact Linear Algebra Kernels

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This project will improve the state of the art of the implementation and optimization of algorithms for exact linear algebra computation. With exact computation, solving systems of linear equations is advanced from limited accuracy to exact solutions. This greatly increases the scope of accessible applications and allows matrix invariants such as rank, determinant, characteristic and minimal polynomial, and Smith and Frobenius normal forms to be computed.We will combine a newly developed theoretical basis for block blackbox methods in linear algebra with high performance implementation, in hardware and software, of the computational kernels from which these implementations are constructed. The resulting implementations will be made publicly available in the framework of the LinBox software library. A system for automatically tuning the underlying computer algebra kernels will be developed and distributed as part of the LinBox library. The autotuning framework will benefit other computer algebra systems as well. The resulting advances for computation in finite domains, such as modular numbers and finite algebraic field extensions, will benefit many areas including cryptography and coding theory.The project has many practical impacts as follows:Experimental mathematics will be enhanced. In experimental mathematics, symbolic computation provides for testing of conjectures. And, perhaps more importantly, data from symbolic computations can guide the formulation of conjectures that are then candidates for formal proof. By permitting larger exact linear algebra computations, this project will increase the usefulness of such computation in mathematics.The broadest, and perhaps most significant, outcome of this project is an ability to solve many problems which currently have no solution method at all. This project will make it possible to efficiently solve linear systems where numerical methods fail due to ill-condition of the problem instance, yet the exact result, could it be obtained, is valid and meaningful despite the approximate nature of the data.