Computing rational forms of integer matrices

Authors:
Mark Giesbrecht;Arne Storjohann
Affiliations:
School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1;School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Venue:
Journal of Symbolic Computation
Year:
2002

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Abstract

A new algorithm is presented for finding the Frobenius rational form F ∈ Zn×n of any A ∈ Zn×n which requires an expected number of O(n4(logn + log ||A||) + n3(log n + log ||A||)2) word operations using standard integer and matrix arithmetic (where ||A||= maxij |Aij|). This substantially improves on the fastest previously known algorithms. The algorithm is probabilistic of the Las Vegas type: it assumes a source of random bits but always produces the correct answer. Las Vegas algorithms are also presented for computing a transformation matrix to the Frobenius form, and for computing the rational Jordan form of an integer matrix.