Berlekamp/Massey Algorithms for Linearly Generated Matrix Sequences
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Date
2009-04-28
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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.
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Keywords
exact linear algebra, linear recurrances, generic programming
Citation
Degree
PhD
Discipline
Mathematics
