Black Box Linear Algebra with the LinBox Library

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Title: Black Box Linear Algebra with the LinBox Library
Author: Turner, William J.
Advisors: Erich Kaltofen, Committee Chair
Carl D. Meyer, Committee Member
Ralph C. Smith, Committee Member
B. David Saunders, Committee Member
Hoon Hong, Committee Member
Abstract: Black box algorithms for exact linear algebra view a matrix as a linear operator on a vector space, gathering information about the matrix only though matrix-vector products and not by directly accessing the matrix elements. Wiedemann's approach to black box linear algebra uses the fact that the minimal polynomial of a matrix generates the Krylov sequences of the matrix and their projections. By preconditioning the matrix, this approach can be used to solve a linear system, find the determinant of the matrix, or to find the matrix's rank. This dissertation discusses preconditioners based on Benes networks to localize the linear independence of a black box matrix and introduces a technique to use determinantal divisors to find preconditioners that ensure the cyclicity of nonzero eigenvalues. This technique, in turn, introduces a new determinant-preserving preconditioner for a dense integer matrix determinant algorithm based on the Wiedemann approach to black box linear algebra and relaxes a condition on the preconditioner for the Kaltofen-Saunders black box rank algorithm. The dissertation also investigates the minimal generating matrix polynomial of Coppersmith's block Wiedemann algorithm, how to compute it using Beckermann and Labahn's Fast Power Hermite-Pade Solver, and a block algorithm for computing the rank of a black box matrix. Finally, it discusses the design of the LinBox library for symbolic linear algebra.
Date: 2002-07-02
Degree: PhD
Discipline: Computational Mathematics

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