Berlekamp/Massey Algorithms for Linearly Generated Matrix Sequences

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Title: Berlekamp/Massey Algorithms for Linearly Generated Matrix Sequences
Author: Yuhasz, George Louis
Advisors: Dr. Agnes Szanto, Committee Member
Dr. Michael Singer, Committee Member
Dr. Ilse Ipsen, Committee Member
Dr. Erich Kaltofen, Committee Chair
Abstract: The Berlekamp/Massey algorithm computes the unique minimal generator of a linearly generated scalar sequence. The matrix generalization of the Berlekamp/Massey algorithm, the Matrix Berlekamp/Massey algorithm, computes a minimal matrix genera- tor of a linearly generated matrix sequence. The Matrix Berlekamp/Massey algorithm has applications in multivariable control theory and exact sparse linear algebra. The fraction free version of the Matrix Berlekamp/Massey algorithm can be adapted into a linear solver for block Hankel matrices. A thorough investigation of the Matrix Berlekamp/Massey algo- rithm and the fraction free Matrix Berlekamp/Massey algorithm is presented. A description of the algorithms and their invariants are given. The underlying linear algebra of the algo- rithms is explored. The connection between the update procedures of the algorithms and the nullspaces of the related matrix systems is detailed. A full definition and description of linearly generated matrix sequences and their various generators is given first as background. A new classification of all linearly generated matrix sequences is proven to exist. A complete description of the Matrix Berlekamp/ Massey algorithm and its invariants is then given. We describe a new early termination criterion for the algorithm and give a full proof of correctness for the algorithm. Our version and proof of the algorithm removes all rank and dimension constraints present in previous versions in the literature. Next a new variation of the Matrix Berlekamp/Massey algorithm is described. The fraction free Matrix Berlekamp/Massey algorithm performs its operations over integral domains. The divisions performed by the algorithm are exact. A full proof of the algorithm and its exact division is given. Finally, we describe two implementations of the Matrix Berlekamp/Massey algorithm, a Maple implementation and a C++ implementation, and compare the two implementations. The C++ implementation is done in the generic LinBox library for exact linear algebra and modeled after the Standard Template Library.
Date: 2009-04-28
Degree: PhD
Discipline: Mathematics

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