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

1995
*
Mathematics of Computation
*

By using projections by a

doi:10.2307/2153451
fatcat:vftjxfl2vjhfjnybkonhgdmlym
*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 ...##
###
Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems
[chapter]

1993
*
Lecture Notes in Computer Science
*

By using projections by a

doi:10.1007/3-540-56686-4_44
fatcat:3i4shd2vxjdphfdjndcsh4jvne
*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 ...##
###
Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems

1995
*
Mathematics of Computation
*

By using projections by a

doi:10.1090/s0025-5718-1995-1270621-1
fatcat:lvcvkwpsrbhpbhkhhemdxri4i4
*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 ...##
###
Fast computation of linear generators for matrix sequences and application to the block Wiedemann algorithm

2001
*
Proceedings of the 2001 international symposium on Symbolic and algebraic computation - ISSAC '01
*

In this paper we describe how

doi:10.1145/384101.384145
dblp:conf/issac/Thome01
fatcat:nagxyxl6tncwzcynoldrpi25bq
*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. ...##
###
Solving Quadratic Equations with XL on Parallel Architectures
[chapter]

2012
*
Lecture Notes in Computer Science
*

This paper describes such an implementation

doi:10.1007/978-3-642-33027-8_21
fatcat:j5regqqc3zeallw6t5mcqaoepq
*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. ...##
###
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]

2016
*
Contemporary Developments in Finite Fields and Applications
*

This is achieved by modifying

doi:10.1142/9789814719261_0008
fatcat:hqh53xj32vcedbsg3v6zyu4oeu
*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. ...##
###
Process Scheduling in DSC and the Large Sparse Linear Systems Challenge

1995
*
Journal of symbolic computation
*

In

doi:10.1006/jsco.1995.1015
fatcat:ozsef3adbfeh3pfyrmbmt4cw5e
*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*. ...##
###
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)

1997
*
Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97
*

We analyse

doi:10.1145/258726.258742
fatcat:pzmbowxvpng23oa2vgwwlusbl4
*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. ...##
###
Distributed Matrix-Free Solution of Large Sparse Linear Systems over Finite Fields

1999
*
Algorithmica
*

We describe a coarse-grain

doi:10.1007/pl00008266
fatcat:57vywzc6qvgmhnclmkeve3frrm
*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. ...##
###
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]

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. ...

##
###
On randomized Lanczos algorithms

1997
*
Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97
*

These are compared to a similar randomized Lanczos

doi:10.1145/258726.258776
fatcat:fwis4smuojf3hfjqe4ghtblwzu
*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. ...##
###
Efficient parallel solution of sparse systems of linear diophantine equations

1997
*
Proceedings of the second international symposium on Parallel symbolic computation - PASCO '97
*

We then employ

doi:10.1145/266670.266678
dblp:conf/cap/Giesbrecht97
fatcat:wpcoaiwfibab7gxtthwpfk6aai
*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. ...
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