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16
Near Optimal Algorithms for Computing Smith Normal Forms of Integer Matrices
, 1996
"... We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d. For the case of matrices over ZZ d , we present an algorithm that computes the Smith form S of an A 2 ZZ n\Thetam d in only O(n `\Gamma1 m) operations from ZZ d . Here, ` is t ..."
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Cited by 53 (5 self)
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We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d. For the case of matrices over ZZ d , we present an algorithm that computes the Smith form S of an A 2 ZZ n\Thetam d in only O(n `\Gamma1 m) operations from ZZ d . Here, ` is the exponent for matrix multiplication over rings: two n \Theta n matrices over a ring R can be multiplied in O(n ` ) operations from R. We apply our algorithm for matrices over ZZ d to get an algorithm for computing the Smith form S of an A 2 ZZ n\Thetam in O~(n `\Gamma1 m \Delta M(n log jjAjj)) bit operations (where jjAjj = max jA i;j j and M(t) bounds the cost of multiplying two dtebit integers). These complexity results improve significantly on the complexity of previously best known Smith form algorithms (both deterministic and probabilistic) which guarantee correctness. 1 Introduction The Smith normal form is a canonical diagonal form for equivalence of matrices over a princ...
The Complexity of Matrix Rank and Feasible Systems of Linear Equations
"... We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other p ..."
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Cited by 40 (8 self)
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We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other problems), are complete under logspace reductions. As an important
Computing Hermite and Smith Normal Forms of Triangular Integer Matrices
 Linear Algebra Appl
, 1996
"... This paper considers the problem of transforming a triangular integer input matrix to canonical Hermite and Smith normal form. We provide algorithms and prove deterministic running times for both transformation problems that are linear (hence optimal) in the matrix dimension. The algorithms are easi ..."
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Cited by 22 (4 self)
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This paper considers the problem of transforming a triangular integer input matrix to canonical Hermite and Smith normal form. We provide algorithms and prove deterministic running times for both transformation problems that are linear (hence optimal) in the matrix dimension. The algorithms are easily implemented, assume standard integer multiplication, and admit excellent performance in practice. The results presented here lead to faster practical algorithms for computing the Hermite and Smith normal form of an arbitrary (non triangular) integer input matrix. 1 Introduction It follows from Hermite [Her51] that any m \Theta n rank n integer matrix A can be transformed using a sequence of integer row operations to an upper triangular matrix H that has jth diagonal entry h j positive for 1 j n and offdiagonal entries ¯ h ij satisfying 0 ¯ h ij ! h j for 1 i ! j n. The matrix H  called the Hermite normal form of A  always exists and is unique. In this paper we consider the...
Minimal Triangulations and Normal Surfaces
, 2003
"... This thesis examines three distinct problems relating to the combinatorial structures of minimal 3manifold triangulations and to the study of normal surfaces within these triangulations. These problems include the formation and analysis of a census of 3manifold triangulations, a study of splitting ..."
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Cited by 12 (3 self)
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This thesis examines three distinct problems relating to the combinatorial structures of minimal 3manifold triangulations and to the study of normal surfaces within these triangulations. These problems include the formation and analysis of a census of 3manifold triangulations, a study of splitting surfaces within 3manifold triangulations and an investigation into the complexity of the normal surface solution space. An algorithm for generating a census of all closed prime minimal 3manifold triangulations is presented, extending the algorithms of earlier authors in several ways. Automorphisms of face pairings are utilised to improve the e±ciency of the generation of triangulations. 0e±ciency tests and searches for particular subcomplexes within a triangulation are introduced to aid the subsequent processing of these triangulations. Results involving face pairing graphs are proven for the purpose of eliminating large classes of triangulations at di®erent stages of the algorithm. Using this algorithm, a census is formed of all closed prime minimal triangulations containing at most six tetrahedra. The census of nonorientable triangulations in particular is the ¯rst such census to be published. A detailed analysis is performed of the underlying combinatorial structures of
A Solution to the Extended GCD Problem with Applications
 IN PROC. INT'L. SYMP. ON SYMBOLIC AND ALGEBRAIC COMPUTATION: ISSAC '97
, 1997
"... This paper considers a variation of the extended gcd problem: the "modulo N extended gcd problem". Given an integer row vector [a i ] n i=1 , the modulo N extended gcd problem asks for an integer vector [c i ] n i=1 such that gcd( n X i=1 c i a i ; N) = gcd(a1 ; a2 ; : : : ; an ; N): ..."
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Cited by 7 (4 self)
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This paper considers a variation of the extended gcd problem: the "modulo N extended gcd problem". Given an integer row vector [a i ] n i=1 , the modulo N extended gcd problem asks for an integer vector [c i ] n i=1 such that gcd( n X i=1 c i a i ; N) = gcd(a1 ; a2 ; : : : ; an ; N): A deterministic algorithm is presented which returns an exceptionally small solution for a given instance of the problem: both max n i=1 jc i j and the number of nonzero c i 's will be bounded by O(log N ). The gcd algorithm presented here has numerous applications and has already led to faster algorithms for computing row reduced echelon forms of integer matrices and solving systems of linear Diophantine equations. In this paper we show how to apply our gcd algorithm to the problem of computing small pre and postmultipliers for the Smith normal of an integer matrix.
On the Computation of Elementary Divisors of Integer Matrices
, 2000
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Fast parallel algorithms for matrix reduction to normal forms
 IN ENGINEERING, COMMUNICATION AND CONTROL
, 1997
"... We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M (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 ..."
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Cited by 5 (2 self)
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We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M (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 A(x) inM (K[x]); 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 NC². 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 polynomialtime sequential algorithm to compute transformations for the Smith form over K[x].
A Fast+Practical+Deterministic Algorithm for Triangularizing Integer Matrices
, 1996
"... This paper presents a new algorithm for computing the row reduced echelon form triangularization H of an n \Theta m integer input matrix A. The cost of the algorithm is O(nmr 2 log 2 rjjAjj + r 4 log 3 rjjAjj) bit operations where r is the rank of A and jjAjj = max ij jA ij j. This complexi ..."
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Cited by 4 (2 self)
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This paper presents a new algorithm for computing the row reduced echelon form triangularization H of an n \Theta m integer input matrix A. The cost of the algorithm is O(nmr 2 log 2 rjjAjj + r 4 log 3 rjjAjj) bit operations where r is the rank of A and jjAjj = max ij jA ij j. This complexity result assumes standard (quadratic) integer arithmetic but still matches, in the paramaters n, m and r, the best bit complexity we can reasonably hope for under the assumption of standard matrix arithmetic. A unimodular transforming matrix U which satisfies UA = H is also computed within the same running time. As a direct application of our triangularization algorithm we give a fast algorithm for solving a system A~x = ~ b of linear Diophantine equations. The algorithms presented here are both fast and practical. They are easily implemented, handle the case of input matrices having arbitrary shape and rank profile, and allow integer arithmetic to be performed in a residue number system. ...
Elementary Algebra Revisited: Randomized Algorithms
 In Randomization Methods in Algorithm Design, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 43
, 1999
"... Abstract. We look at some simple algorithms for elementary problems in algebra that yield dramatic efficiency improvements over standard methods through randomization. The randomized algorithms are, in a sense, “obvious”. Their formal statement was delayed partly by the need for rigorous analysis, b ..."
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Cited by 2 (1 self)
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Abstract. We look at some simple algorithms for elementary problems in algebra that yield dramatic efficiency improvements over standard methods through randomization. The randomized algorithms are, in a sense, “obvious”. Their formal statement was delayed partly by the need for rigorous analysis, but more so by the need to rethink traditional approaches to elementary algorithms. We illustrate this philosophy with some basic problems in computational number theory (GCD of many integers), linear algebra (lowrank Gaussian elimination) and group theory (random subproducts for subgroup construction), along with a brief survey of other areas. 1.
Parallel Algorithms for Computing the Smith Normal Form of Large Matrices
"... Abstract. Smith normal form computation has many applications in group theory, module theory and number theory. As the entries of the matrix and of its corresponding transformation matrices can explode during the computation, it is a very difficult problem to compute the Smith normal form of large d ..."
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Abstract. Smith normal form computation has many applications in group theory, module theory and number theory. As the entries of the matrix and of its corresponding transformation matrices can explode during the computation, it is a very difficult problem to compute the Smith normal form of large dense matrices. The computation has two main problems: the high execution time and the memory requirements, which might exceed the memory of one processor. To avoid these problems, we develop two parallel Smith normal form algorithms using MPI. These are the first algorithms computing the Smith normal form with corresponding transformation matrices, both over the rings Z and F[x]. We show that our parallel algorithms both have a good efficiency, i.e. by doubling the processes, the execution time is nearly halved, and succeed in computing the Smith normal form of dense example matrices over the rings Z and F2[x] with more than thousand rows and columns. 1