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Nearly Optimal Algorithms For Canonical Matrix Forms
, 1993
"... A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nea ..."
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Cited by 62 (13 self)
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A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processorefficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.
Arithmetic and factorization of polynomials over F_2
 PROC. ISSAC 96
, 1996
"... We describe algorithms for polynomial multiplication and polynomial factorization over the binary field F2 and their implementation. They allow polynomials of degree up to 100,000 to be factored in about one day of CPU time. ..."
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Cited by 7 (1 self)
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We describe algorithms for polynomial multiplication and polynomial factorization over the binary field F2 and their implementation. They allow polynomials of degree up to 100,000 to be factored in about one day of CPU time.
Modular Algorithms for Polynomial Basis Conversion and Greatest Factorial Factorization
, 2000
"... We give new algorithms for converting between representations of polynomials with respect to certain kinds of bases, comprising the usual monomial basis and the falling factorial basis, for fast multiplication and Taylor shift in the falling factorial basis, and for computing the greatest factori ..."
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Cited by 7 (0 self)
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We give new algorithms for converting between representations of polynomials with respect to certain kinds of bases, comprising the usual monomial basis and the falling factorial basis, for fast multiplication and Taylor shift in the falling factorial basis, and for computing the greatest factorial factorization. We analyze both the classical and the new algorithms in terms of arithmetic coefficient operations. For the special case of polynomials with integer coefficients, we present modular variants of these methods and give cost estimates in terms of word operations.