Faster inversion and other black box matrix computations using efficient block projections

Eberly, Wayne; Giesbrecht, Mark; Giorgi, Pascal; Storjohann, Arne; Villard, Gilles

doi:10.1145/1277548.1277569

Cited by 28 publications

(33 citation statements)

References 27 publications

(71 reference statements)

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“…In particular, a recent algorithm for dense matrices [22] might be adapted for black-box matrices. In this regard, extending the block projections of [13] to the case of similarity transformations would play a crucial role.…”

Section: Resultsmentioning

confidence: 99%

On finding multiplicities of characteristic polynomial factors of black-box matrices

Dumas

Pernet²,

Saunders

2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.

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Section: Resultsmentioning

confidence: 99%

On finding multiplicities of characteristic polynomial factors of black-box matrices

Dumas

Pernet²,

Saunders

2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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“…[12]) based on column and row reductions to calculate the Smith Normal form for integer matrices. For faster algorithms we refer to [30,45]. For an excellent survey on the available strategies see also [14].…”

Section: The Solution Of Problem Dr For the General Casementioning

confidence: 99%

Minimal Representations and Algebraic Relations for Single Nested Products

Schneider

2020

Program Comput Soft

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Recently, it has been shown constructively how a finite set of hypergeometric products, multibasic hypergeometric products or their mixed versions can be modeled properly in the setting of formal difference rings. Here special emphasis is put on robust constructions: whenever further products have to be considered, one can reuse -up to some mild modifications-the already existing difference ring. In this article we relax this robustness criteria and seek for another form of optimality. We will elaborate a general framework to represent a finite set of products in a formal difference ring where the number of transcendental product generators is minimal. As a bonus we are able to describe explicitly all relations among the given input products.This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).3 In this example the evaluation of an element from É(ι)(x) is carried out by replacing x with concrete values n ∈ AE. Later we will generalize this simplest case to formal difference rings equipped with an evaluation function acting on the ring elements.4 More generally, if R is a commutative ring with 1, we define the ideal I generated by a 1 , . . . , ar ∈ R with I = a 1 , . . . , ar R = {f 1 a 1 + · · · + fr ar | f 1 , . . . , fr ∈ R}.

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“…A similar approach also applies to the blackbox computation model, where computing the minimal polynomial is the building block to which most problems reduce to [4,13,12].…”

Section: Reductions To Building Blocksmentioning

confidence: 99%

Exact Linear Algebra Algorithmic

Pernet

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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Exact linear algebra is a core component of many symbolic and algebraic computations, as it often delivers competitive theoretical complexities and also better harnesses the efficiency of modern computing infrastructures. In this tutorial we will present an overview on the recent advances in exact linear algebra algorithmic and implementation techniques, and highlight the few key ideas that have proven successful in their design. As an illustration, we will study in more details the computation of some matrix normal forms over a finite field or the ring of polynomials, specific to computer algebra. In particular, we will give a special care to the design and implementation of parallel exact linear algebra routines, trying to emphasize the similarities and distinctness with parallel numerical linear algebra. We aim to provide the working computer algebraist with a set of best practices for the use or the design of exact linear algebra software, together with an overview on a few still unresolved algorithmic problems in the field.

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Faster inversion and other black box matrix computations using efficient block projections

Cited by 28 publications

References 27 publications

On finding multiplicities of characteristic polynomial factors of black-box matrices

On finding multiplicities of characteristic polynomial factors of black-box matrices

Minimal Representations and Algebraic Relations for Single Nested Products

Exact Linear Algebra Algorithmic

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