A Practical Implementation of Modular Algorithms for Frobenius Normal Forms of Rational Matrices

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  • Practical Implementation of Modular Algorithms for Frobenius Normal Forms of Rational Matrices
  • アルゴリズム理論

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Abstract

Modular algorithms for computing the Frobenius normal forms of integer and rational matrices are presented and their implementation is reported. These methods compute Frobenius normal forms over Zpi where pi's are distinct primes and then construct the normal forms over Z or Q by the Chinese remainder theorem. Our implementation includes: (1) detection of unlucky primes (2) a new formula for the efficient computation of a transformation matrix and (3) extension of our preceding algorithm over Z to one over Q. Through experiments using a number of test matrices we confirm that our modular algorithm is more efficient in practical terms than the straightforward implementation of conventional methods.

Modular algorithms for computing the Frobenius normal forms of integer and rational matrices are presented and their implementation is reported. These methods compute Frobenius normal forms over Zpi, where pi's are distinct primes, and then construct the normal forms over Z or Q by the Chinese remainder theorem. Our implementation includes: (1) detection of unlucky primes, (2) a new formula for the efficient computation of a transformation matrix, and (3) extension of our preceding algorithm over Z to one over Q. Through experiments using a number of test matrices, we confirm that our modular algorithm is more efficient in practical terms than the straightforward implementation of conventional methods.

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