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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 ..."
Abstract

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.
Computing Rational Forms of Integer Matrices
 J. SYMBOLIC COMPUT
, 2000
"... A new algorithm is presented for finding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially on the fastest previously known a ..."
Abstract

Cited by 10 (3 self)
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A new algorithm is presented for finding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially 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.