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Analysis of Coppersmith's Block Wiedemann Algorithm for the Parallel Solution of Sparse Linear Systems

Erich Kaltofen
1995 Mathematics of Computation  
By using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems.  ...  We prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors.  ...  Acknowledgments Thanks to Austin Lobo for discussions on the theory and implementation of the block Wiedemann method and generalized Levinson-Durbin method, and to Martin Morf for his advice on the asymptotically  ... 
doi:10.2307/2153451 fatcat:vftjxfl2vjhfjnybkonhgdmlym

Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems [chapter]

Erich Kaltofen
1993 Lecture Notes in Computer Science  
By using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems.  ...  We prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors.  ...  Acknowledgments Thanks to Austin Lobo for discussions on the theory and implementation of the block Wiedemann method and generalized Levinson-Durbin method, and to Martin Morf for his advice on the asymptotically  ... 
doi:10.1007/3-540-56686-4_44 fatcat:3i4shd2vxjdphfdjndcsh4jvne

Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems

Erich Kaltofen
1995 Mathematics of Computation  
By using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems.  ...  We prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors.  ...  Acknowledgments Thanks to Austin Lobo for discussions on the theory and implementation of the block Wiedemann method and generalized Levinson-Durbin method, and to Martin Morf for his advice on the asymptotically  ... 
doi:10.1090/s0025-5718-1995-1270621-1 fatcat:lvcvkwpsrbhpbhkhhemdxri4i4

Fast computation of linear generators for matrix sequences and application to the block Wiedemann algorithm

Emmanuel Thomé
2001 Proceedings of the 2001 international symposium on Symbolic and algebraic computation - ISSAC '01  
In this paper we describe how the half-gcd algorithm can be adapted in order to speed up the sequential stage of Coppersmith's block Wiedemann algorithm for solving large sparse linear systems over any  ...  We discuss the implications of this improvement for the overall cost of the block Wiedemann algorithm and how its parameters should be chosen for best efficiency.  ...  INTRODUCTION Coppersmith's block Wiedemann algorithm [9] applies to the solving of large, sparse linear systems over finite fields.  ... 
doi:10.1145/384101.384145 dblp:conf/issac/Thome01 fatcat:nagxyxl6tncwzcynoldrpi25bq

Solving Quadratic Equations with XL on Parallel Architectures [chapter]

Chen-Mou Cheng, Tung Chou, Ruben Niederhagen, Bo-Yin Yang
2012 Lecture Notes in Computer Science  
This paper describes such an implementation of XL using the block Wiedemann algorithm.  ...  Under reasonable assumptions, the best way to solve generic MQ systems is the XL algorithm implemented with a sparse matrix solver such as Wiedemann's algorithm.  ...  Block Wiedemann Algorithm The computationally most expensive task in XL is to nd a solution for the sparse linear system M of equations over a nite eld.  ... 
doi:10.1007/978-3-642-33027-8_21 fatcat:j5regqqc3zeallw6t5mcqaoepq

Page 3587 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
Kellogg (1-MD-T; College Park, MD) 95f:65094 65F50 15A06 65Y05 Kaltofen, Erich (1-RSP-C; Troy, NY) Analysis of Coppersmith’s block Wiedemann algorithm for the parallel solution of sparse linear systems  ...  The algorithm is designed to solve large, sparse, possibly singu- lar linear systems over finite fields.  ... 

Nearly sparse linear algebra and application to discrete logarithms computations [chapter]

Antoine Joux, Cécile Pierrot
2016 Contemporary Developments in Finite Fields and Applications  
This is achieved by modifying the Block Wiedemann algorithm.  ...  Under some precisely stated conditions on the choices of initial vectors in the algorithm, we show that our variation not only produces a random solution of a linear system but gives a full basis of the  ...  Coppersmith's Block Wiedemann algorithm The Block Wiedemann algorithm is a parallelization of the previous Wiedemann algorithm introduced by Don Coppersmith.  ... 
doi:10.1142/9789814719261_0008 fatcat:hqh53xj32vcedbsg3v6zyu4oeu

Process Scheduling in DSC and the Large Sparse Linear Systems Challenge

A. Dı́az, M. Hitz, E. Kaltofen, A. Lobo, T. Valente
1995 Journal of symbolic computation  
In the second a parallel version of a sparse linear system solver is used to compute the solution of sparse linear systems over finite fields.  ...  We are able to find the solution of a 100,000 by 100,000 linear system with about 10.3 million non-zero entries over the Galois field with 2 elements using 3 computers in about 54 hours CPU time.  ...  Solving Large Sparse Linear Systems We now report on the use of DSC in experiments with the Block Wiedemann Algorithm.  ... 
doi:10.1006/jsco.1995.1015 fatcat:ozsef3adbfeh3pfyrmbmt4cw5e

Page 6236 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
{For the entire collection see MR 94h:68006.} 94k:11134 11Y16 68Q25 Kaltofen, Erich (1-RSP-C; Troy, NY) Analysis of Coppersmith’s block Wiedemann algorithm for the parallel solution of sparse linear systems  ...  Coppersmith (1991), who introduced parallelism in Wiedemann’s algorithm. The present work gives a probabilistic analysis of Cop-  ... 

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

G. Villard
1997 Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97  
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).  ...  Grateful thanks to Erich Kaltofen for his valuable questions.  ... 
doi:10.1145/258726.258742 fatcat:pzmbowxvpng23oa2vgwwlusbl4

Distributed Matrix-Free Solution of Large Sparse Linear Systems over Finite Fields

E. Kaltofen, A. Lobo
1999 Algorithmica  
We describe a coarse-grain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations.  ...  For example, we can solve a 252, 222 × 252, 222 system with about 11.04 million non-zero entries over the Galois field with 2 elements using 4 processors of an SP-2 multiprocessor, in about 26.5 hours  ...  Acknowledgement: The authors thank Charles Norton for valuable discussions and assistance with the use of the SP-2. Thanks also to David Hollinger and Nathan Schimke for technical support.  ... 
doi:10.1007/pl00008266 fatcat:57vywzc6qvgmhnclmkeve3frrm

Page 3473 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
solution of sparse linear systems (195-212); P.  ...  Wilson, Hyperplane sections of Fermat varieties in P® in char. 2 and some applications to cyclic codes (180-194); Erich Kaltofen, Analysis of Coppersmith’s block Wiedemann algorithm for the par- allel  ... 

Computation of Discrete Logarithms in $$ \mathbb{F}_{2^{607} } $$ [chapter]

Emmanuel Thomé
2001 Lecture Notes in Computer Science  
Coppersmith's algorithm.  ...  Although the computations have been carried out on fairly standard hardware, our opinion is that we are nearing the current limits of the manageable sizes for this algorithm, and that going substantially  ...  Linear algebra involved mostly resources from LIX, and also from UMS MEDICIS to the extent possible given the distribution constraints for this task.  ... 
doi:10.1007/3-540-45682-1_7 fatcat:eb5uh4zyuvepxau2aumu4nub64

On randomized Lanczos algorithms

Wayne Eberly, Erich Kaltofen
1997 Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97  
These are compared to a similar randomized Lanczos algorithm that has been used for integer factorization, and to the (provably reliable) algorithm of Wiedemann.  ...  Las Vegas algorithms that are based on Lanczos's method for solving symmetric linear systems are presented and analyzed.  ...  Further analysis of Coppersmith's block Wiedemann algorithm for the solution of sparse linear systems. In Proc. ISSAC '97 (1997). [15] Wiedemann, D. H.  ... 
doi:10.1145/258726.258776 fatcat:fwis4smuojf3hfjqe4ghtblwzu

Efficient parallel solution of sparse systems of linear diophantine equations

Mark Giesbrecht
1997 Proceedings of the second international symposium on Parallel symbolic computation - PASCO '97  
We then employ the Block-Wiedemann algorithm to solve these preconditioned systems efficiently in parallel. Solutions produced are small and space required is essentially linear in the output size.  ...  We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation.  ...  The need for such an error tolerance parameter is also present in the underlying Wiedemann and Block-Wiedemann algorithms for solving sparse singular systems over a field.  ... 
doi:10.1145/266670.266678 dblp:conf/cap/Giesbrecht97 fatcat:wpcoaiwfibab7gxtthwpfk6aai
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