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44
Linbox: A Generic Library For Exact Linear Algebra
, 2002
"... Base Class pointers Concrete Field virtual functions Field Archetype Linbox field archetype Figure 1: Black box design. The LinBox black box matrix archetype is simpler than the field archetype because the design constraints are less stringent. As with the field type, we need a common object ..."
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Cited by 39 (12 self)
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Base Class pointers Concrete Field virtual functions Field Archetype Linbox field archetype Figure 1: Black box design. The LinBox black box matrix archetype is simpler than the field archetype because the design constraints are less stringent. As with the field type, we need a common object interface to describe how algorithms are to access black box matrices, but it only requires functions to access the matrix's dimensions and to apply the matrix or its transpose to a vector. Thus our black box matrix archetype is simply an abstract class, and all actual black box matrices are subclasses of the archetype class. We note that the overhead involved with this inheritance mechanism is negligible in comparison with the execution time of the methods, unlike for our field element types.
Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
, 2002
"... We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices an ..."
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Cited by 36 (2 self)
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We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
Efficient computation of the characteristic polynomial
 Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation
, 2005
"... We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of ..."
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Cited by 18 (13 self)
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We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of KellerGehrig. Then we show that a generalization of KellerGehrig’s third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices.
Dense linear algebra over wordsize prime fields: the FFLAS and FFPACK packages
, 2009
"... In the past two decades, some major efforts have been made to reduce exact (e.g. integer, rational, polynomial) linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be c ..."
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Cited by 15 (5 self)
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In the past two decades, some major efforts have been made to reduce exact (e.g. integer, rational, polynomial) linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be careful with the underlying arithmetic. It is well known that modular techniques such as the Chinese remainder algorithm or the padic lifting allow very good practical performance, especially when word size arithmetic are used. Therefore, finite field arithmetic becomes an important core for efficient exact linear algebra libraries. In this paper, we study high performance implementations of basic linear algebra routines over word size prime fields: specially the matrix multiplication; our goal being to provide an exact alternate to the numerical BLAS library. We show that this is made possible by a careful combination of numerical computations and asymptotically faster algorithms. Our kernel has
COREDUCTION HOMOLOGY ALGORITHM FOR INCLUSIONS AND PERSISTENT HOMOLOGY
"... Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the p ..."
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Cited by 15 (4 self)
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Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to twosided filtrations. In addition to describing the theoretical background, we present results of numerical experiments, as well as several applications to concrete problems in materials science. 1.
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
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Cited by 14 (5 self)
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Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.
Dense linear algebra over finite fields: the FFLAS and FFPACK packages
, 2009
"... In the last past two decades, several efforts have been made to reduce exact linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be careful with the underlying arithmet ..."
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Cited by 7 (1 self)
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In the last past two decades, several efforts have been made to reduce exact linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be careful with the underlying arithmetic. It is well know that modular technique such as Chinese remainder algorithm or padic lifting allow in practice to achieve better performances especially when word size arithmetic are used. Therefore, finite field arithmetics becomes an important core for efficient exact linear algebra libraries. In this paper we study different implementations of finite field in order to achieve efficiency for basic linear algebra routines such as dot product or matrix multiplication; our goal being to provide an exact alternate to numerical BLAS library. Following matrix multiplication reductions, our kernel has many symbolic linear algebra applications: symbolic triangularization, system solving, exact determinant computation and matrix inversion are then studied and we demonstrate the efficiency of these reductions in practice.
Computing the Rank of Large Sparse Matrices over Finite Fields
"... We want to achieve efficient exact computations, such as the rank, of sparse matrices over finite fields. We therefore compare the practical behaviors, on a wide range of sparse matrices of the deterministic Gaussian elimination technique, using reordering heuristics, with the probabilistic, blackbo ..."
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Cited by 7 (2 self)
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We want to achieve efficient exact computations, such as the rank, of sparse matrices over finite fields. We therefore compare the practical behaviors, on a wide range of sparse matrices of the deterministic Gaussian elimination technique, using reordering heuristics, with the probabilistic, blackbox, Wiedemann algorithm. Indeed, we prove here that the latter is the fastest iterative variant of the Krylov methods to compute the minimal polynomial or the rank of a sparse matrix.
Bounds on the coefficients of the characteristic and minimal polynomials
 Journal of Inequalities in Pure and Applied Mathematics
, 2007
"... Abstract. This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise inputdependant bounds on these coefficients. Such bou ..."
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Cited by 5 (4 self)
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Abstract. This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise inputdependant bounds on these coefficients. Such bounds are e.g. useful to perform deterministic chinese remaindering of the characteristic or minimal polynomial of an integer matrix. 1.