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On p-adic computation of the rational form of a matrix

✍ Scribed by Marie-Hélène Mathieu; David Ford

Publisher
Elsevier Science
Year
1990
Tongue
English
Weight
546 KB
Volume
10
Category
Article
ISSN
0747-7171

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✦ Synopsis

We consider the problem of bringing a given matrix into "cyclic form," from which the rational form can be computed easily. Matrices are taken to have p-adic integer entries, and computations are done with rational integer approximations to p-adic integers. We give bounds on the precision necessary to ensure that the resulting cyclic form is indeed similar to the original matrix. We also give a criterion for deciding whether the cyclic form is correct.

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A method for computing the inverse of an (n × n) integer matrix A using p-adic approximation is given. The method is similar to Dixon's algorithm, but ours has a quadratic convergence rate. The complexity of this algorithm (without using FFT or fast matrix multiplication) is O(n 4 (log n) 2 ), the s