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North Carolina State University

1. Turner, William J. Black Box Linear Algebra with the LinBox Library.

Degree: PhD, Computational Mathematics, 2002, North Carolina State University

URL: http://www.lib.ncsu.edu/resolver/1840.16/3025

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.
*Advisors/Committee Members: Erich Kaltofen, Committee Chair (advisor), Carl D. Meyer, Committee Member (advisor), Ralph C. Smith, Committee Member (advisor), B. David Saunders, Committee Member (advisor), Hoon Hong, Committee Member (advisor).*

Subjects/Keywords: black box linear algebra; Wiedemann method; block Wiedemann method; linear algebra; randomized algorithm; LinBox library

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APA (6^{th} Edition):

Turner, W. J. (2002). Black Box Linear Algebra with the LinBox Library. (Doctoral Dissertation). North Carolina State University. Retrieved from http://www.lib.ncsu.edu/resolver/1840.16/3025

Chicago Manual of Style (16^{th} Edition):

Turner, William J. “Black Box Linear Algebra with the LinBox Library.” 2002. Doctoral Dissertation, North Carolina State University. Accessed July 11, 2020. http://www.lib.ncsu.edu/resolver/1840.16/3025.

MLA Handbook (7^{th} Edition):

Turner, William J. “Black Box Linear Algebra with the LinBox Library.” 2002. Web. 11 Jul 2020.

Vancouver:

Turner WJ. Black Box Linear Algebra with the LinBox Library. [Internet] [Doctoral dissertation]. North Carolina State University; 2002. [cited 2020 Jul 11]. Available from: http://www.lib.ncsu.edu/resolver/1840.16/3025.

Council of Science Editors:

Turner WJ. Black Box Linear Algebra with the LinBox Library. [Doctoral Dissertation]. North Carolina State University; 2002. Available from: http://www.lib.ncsu.edu/resolver/1840.16/3025