Output-sensitive modular algorithms for polynomial matrix normal forms

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case, we define lucky homomorphisms, determine the...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 42; no. 7; pp. 733 - 750
Main Authors: Cheng, Howard, Labahn, George
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.07.2007
Subjects:
Matrices
Modular algorithm
Popov form
Row-reduced form
Weak Popov form
Popov form
Row-reduced form
Matrices
Weak Popov form
Modular algorithm
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary:We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case, we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorithms have the advantage that they are output-sensitive; that is, the number of homomorphic images required depends on the size of the output. Furthermore, there is no need to verify the result by trial division or multiplication. Our algorithms can be used to compute normalized one-sided greatest common divisors and least common multiples of polynomial matrices, along with irreducible matrix-fraction descriptions of matrix rational functions. When our algorithm is used to compute polynomial greatest common divisors, we obtain a new output-sensitive modular algorithm.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2007.03.001