Fast parallel matrix and GCD computations

Borodin, Allan; Gathen, Joachim von zur; Hopcroft, John E.

doi:10.1016/s0019-9958(82)90766-5

Cited by 165 publications

(122 citation statements)

References 14 publications

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“…The claim about the sizes easily follows from the last remark because we may apply Berkowitz's algorithm to matrices of the form BB t (with B a submatrix of M) whose entries have size bounded by t + log n. The reader can see these as well as other parallel algorithms for computer algebra in [5]. 9…”

Section: Parallel Linear Algebramentioning

confidence: 99%

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NC algorithms for real algebraic numbers

Cucker

Lanneau

Mishra

et al. 1992

AAECC

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Abstract. Characterizing real algebraic numbers by a sign-sequence (according to Thorn's lemma), using a variant of Ben Or Kozan and Reif algorithm for reducing the solving of systems of polynomial inequalities to the problem of counting real zeroes satisfying one polynomial inequality, and using a multivariate Sturm theory generalizing Hermite quadratic forms method for counting real zeroes, we prove that the computations on real algebraic numbers are in NC.

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Section: Parallel Linear Algebramentioning

confidence: 99%

“…Details about it appear in [14] or [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] where the Sturm-Habicht sequence is introduced and studied.…”

Section: Sturm-sylvester Sequencementioning

confidence: 99%

NC algorithms for real algebraic numbers

Cucker

Lanneau

Mishra

et al. 1992

AAECC

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“…Let NonSingular-Equations(n) be the problem of solving a non-singular n × n system of linear equations [BGH82].…”

Section: Breaking Nisan's Prg With Polynomially Many Passesmentioning

confidence: 99%

“…Our distinguisher uses some elementary algebraic properties. It relies on a deep result due to Mulmuley [Mul87] (which derandomizes Borodin, Gathen and Hopcroft [BGH82]) for solving a non-singular system of linear equations with polynomial size, constant fan-in circuits of O(log 2 n) depth, which in particular implies a O(log 2 n) space algorithm. Our distinguisher works in logarithmic space -it uses the algorithm of [Mul87] in linear systems of size O(2 c √ log n ), where n is the length of the whole input.…”

Section: Introductionmentioning

confidence: 99%

How strong is Nisanʼs pseudo-random generator?

David

Papakonstantinou

Sidiropoulos

2011

Information Processing Letters

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We study the resilience of the classical pseudo-random generator (PRG) of Nisan [Nis92] against space-bounded machines that make multiple passes over the input. Our motivation comes from the derandomization of BPNC 1 . Observe that if for every log-space machine that reads its input n O(1) times there is a PRG fooling this machine, then in particular we fool NC 1 circuits. Nisan's PRG is known to fool log-space machines that read the input once. We ask what are the limits of this PRG regarding log-space machines that make multiple passes over the input. We show that for every constant k Nisan's PRG fools log-space machines that make log k n passes over the input, using a seed of length log k n, for some k > k. We complement this result by showing that in general Nisan's-PRG cannot fool log-space machines that make n O(1) passes even for a seed of length 2 Θ( √ log n) . The observations made in this note outline a more general approach in understanding the difficulty of derandomizing BPNC 1 .

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“…(x). We very briefly describe the procedures in [3,25] to compute a maximal linearly independent set of columns. Over an abstract field, for a set of columns B"(B , .…”

Section: J ) For the Symbolic Jordan Form Conformable To J Can mentioning

confidence: 99%

Fast Parallel Algorithms for Matrix Reduction to Normal Forms

Villard¹

1997

Applicable Algebra in Engineering, Communication and Computing

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Abstract. We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M L L (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F"P\ BP is in Frobenius normal form can be done in NC ) . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix to compute unimodular matrices º(x) and »(x) such that S(x)"º(x)A(x)»(x) can be done in NC) . We get that over concrete fields such as the rationals, these problems are in NC .Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in NC ) . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K [x].

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Fast parallel matrix and GCD computations

Cited by 165 publications

References 14 publications

NC algorithms for real algebraic numbers

NC algorithms for real algebraic numbers

How strong is Nisanʼs pseudo-random generator?

Fast Parallel Algorithms for Matrix Reduction to Normal Forms

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