Gilles Villard

LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA

Black box techniques [12] are enabling exact linear algebra computations of a scale well beyond anything previously possible. The development of new and interesting algorithms has proceeded apace for the past two decades. It is time for the dissemination of these algorithms in an easily used software library so that the mathematical community may readily take advantage of their power. LinBox is that library. In this paper, we describe the design of this generic library, sketch its current range of capabilities, and give several examples of its use. The examples include a solution of Trefethen’s “Hundred Digit Challenge” problem #7 [14] and the computation of all the homology groups of simplicial complexes using the Smith normal form [8]. Exact black box methods are currently successful on sparse matrices with hundreds of thousands of rows and columns and having several million nonzero entries. The main reason large problems can be solved by black box methods is that they require much less memory in general than traditional eliminationbased methods do. This fact is widely used in the numerical computation area. We refer for instance to the templates for linear system solution and eigenvalue problems [2,1]. This has also led the computer algebra community to a considerable interest in black box methods. Since Wiedemann’s seminal paper [16], many developments have been proposed especially to adapt Krylov or Lanczos methods to fast exact algorithms. We refer to [5] and references therein for a review of problems and solutions. LinBox supplies efficient black box solutions for a variety of problems including linear equations and matrix normal forms with the guiding design principle of re-usability. The most essential and driving design criterion for LinBox is that it is generic with respect to the domain of computation. This is because there are many and various representations of finite fields each of which is advantageous to use for some algorithm under some circumstance. The integral and rational number capabilities depend heavily on modular

On computing the determinant and Smith form of an integer matrix

A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A/spl isin/Z/sup n/spl times/n/ the algorithm requires O(n/sup 3.5/(log n)/sup 4.5/) bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O(n/sup 2+/spl theta//2//spl middot/log/sup 2/nloglogn) bit operations, where n/spl times/n matrices can be multiplied with O(n/sup /spl theta//) operations. The determinant is found by computing the Smith form of the integer matrix an extremely useful canonical form in itself. Our algorithm is probabilistic of the Monte Carlo type. That is, it assumes a source of random bits and on any invocation of the algorithm there is a small probability of error.

Efficient matrix preconditioners for black box linear algebra

Abstract The main idea of the “black box” approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials. In most cases preconditioning is necessary to obtain the desired result. Here good preconditioners will be used to ensure geometrical/algebraic properties on matrices, rather than numerical ones, so we do not address a condition number. We offer a review of problems for which (algebraic) preconditioning is used, provide a bestiary of preconditioning problems, and discuss several preconditioner types to solve these problems. We present new conditioners, including conditioners to preserve low displacement rank for Toeplitz-like matrices. We also provide new analyses of preconditioner performance and results on the relations among preconditioning problems and with linear algebra problems. Thus, improvements are offered for the efficiency and applicability of preconditioners. The focus is on linear algebra problems over finite fields, but most results are valid for entries from arbitrary fields.

On Efficient Sparse Integer Matrix Smith Normal Form Computations

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (elimination and/or black box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.

Shifted normal forms of polynomial matrices

In t,his paper we st,ucly the problen~ of transforrniug, via invertible colu1tln opcrat.ious~ it matrix polyioruial into a varicty Of .shiftcd forms. Esarnplcs of forms c:overed in out frmwa-ork include a colunm rctluccd form: il triangular fornlz R I%!rInite IlOrInd fOrll1 or it Popov IlorIllal fOrIll alollg wit,11 their shifted courltcrpart,s. I3y obt.aiuiug tlcgrvc bounds for uuiniodiilar niiill,iplicrs of shifted Popor fornis we are able t,o c11lbct1 tlic probleni of conqmtiug il normal forni into 0Iic of deterniiJIing a sliift.ctl forni Of R InininIal pOl~IlOIIliill hiISiS fur all XWXiiltWl IIliLtris polyioruial. Shifted niiuind polynomial lXPX?S cm be conlpu1.d via sigma bases [2! 31 iIlld ill POpOv forui Vii1 Mahler s~sbenls [il. Tl ic d 1. t,t, cr Iut:t.lIotl gives a fractiorl-frw algorithm for computing niatris riormal forms.

Parallel lattice basis reduction

The famous L3 algorithm for lattice basis reduction k parallelizecl. Using a dktributed memory architecture compntationaJ model, the algorithm we propose efficient y uses 0( n2 ) processors, where n is the dimension of the bask to reduce. Its implementation, realized on a massively parallel machine, allows us to conduct many experimentations. The first results are presented in this paper. We show that high speed-ups are obtained even for large amounts of processors, and give new ernpiricti knowledge of the L3 sequential complexity. ●This work was supported in part by the PRC Math &natiques et Injormatique and by the GToupement C3 of the french Centre National de la Recherche Scientifique. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direot commercial advantage, the ACM copyright notice and tlhe title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. LSSAC ‘92-7192/CA, USA ~ 1992 ACM 0-89791 -490 -21921000710269 . ..

Further analysis of Coppersmith's block Wiedemann algorithm for the solution of sparse linear systems (extended abstract)

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An LLL-reduction algorithm with quasi-linear time complexity: extended abstract

We analyse the probability of success of the block algorithm proposed by Coppersmith for solving large sparse systems Aw = O of linear equations over a field K. Itis based on a modification of a scheme proposed by Wiedemann. An open question was to prove that the block algorithm may produce a solution for small finite fields e.g. for K =GF(2). Our investigations allow us to answer this question nearly completely. We prove that the input parameters of the algorithm may be tuned such that, for any input system, a solution is computed with high probability for any field. Conversely, for particular input systems, we show that the conditions on the input parameters may be relaxed to ensure the success. We also improve the previous probability measurements in the case of large cardkmlity fields.

Computing Popov and Hermite forms of polynomial matrices

We devise an algorithm, L1, with the following specifications: It takes as input an arbitrary basis B=(bi)i ∈ Zd x d of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(d5+ε β + dω+1+ε β1+ε) where β = log max |bi| (for any ε>0 and ω is a valid exponent for matrix multiplication). This is the first LLL-reducing algorithm with a time complexity that is quasi-linear in β and polynomial in d. The backbone structure of L1 is able to mimic the Knuth-Schönhage fast gcd algorithm thanks to a combination of cutting-edge ingredients. First the bit-size of our lattice bases can be decreased via truncations whose validity are backed by recent numerical stability results on the QR matrix factorization. Also we establish a new framework for analyzing unimodular transformation matrices which reduce shifts of reduced bases, this includes bit-size control and new perturbation tools. We illustrate the power of this framework by generating a family of reduction algorithms.

Computing the rank and a small nullspace basis of a polynomial matrix

For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(log~+’ (ml)) pamlel arithmetic steps and O(L(nd) ) processors if the same processor bound holds with time O (logX (rid)) for determining the lexicographically first maximal linearly independent subset of the set of the columns of an nd x nd matrix over K. These results are obtamed by applying in the matrix case, the techniques used in the scalar case of the gcd of polynomials.