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A parallel algorithm for circulant tridiagonal linear systems

1998
*
Information Processing Letters
*

In this paper, based upon the divide-and-conquer strategy, we propose a

doi:10.1016/s0020-0190(98)00018-0
fatcat:pzrjmyervnbl5iiq2geiz6vjcm
*parallel**algorithm**for*solving the circulant*tridiagonal**systems*, which is simpler than the two previous*algorithms*proposed by Agui ... Our*algorithm*can be easily generalized to solve the*tridiagonal**systems*, the block-*tridiagonal**systems*, and the circulant block-*tridiagonal**systems*. @ 1998 Elsevier Science B.V. ... Agui and Jimenez [ 1 ] proposed a*parallel**algorithm*to solve the*tridiagonal**systems*. And, Chung et al. [4] proposed a*parallel**algorithm*to solve the circulant block-*tridiagonal**systems*. ...##
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Parallel Tridiagonal Equation Solvers

1975
*
ACM Transactions on Mathematical Software
*

This paper compares three

doi:10.1145/355656.355657
fatcat:dcegxoxuajhllhqsig6yz75ota
*parallel**algorithms**for*the direct solution of*tridiagonal*linear*systems*of equations. The*algorithms*are suitable*for*computers such as ILLIAC IV and CDC STAR. ... The quadratic convergence is superior to the linear convergence reported by Traub*for*several*parallel*iterative*tridiagonal*solvers.*Parallel**Tridiagonal*Equation Solvers by Harold S. Stone ... Gene Golub of Stanford University*for*many discussions, comments, and suggestions regarding this paper. We also acknowledge helpful suggestions and conversations with Prof. ...##
###
A recursive doubling algorithm for solution of tridiagonal systems on hypercube multiprocessors

1989
*
Journal of Computational and Applied Mathematics
*

In this paper, we give a limited processor version of the

doi:10.1016/0377-0427(89)90362-2
fatcat:kxuexu4sarftpfdodseo6b5gye
*recursive*doubling*algorithm**for*the solution of*tridiagonal*linear*systems*using O(n/p +log p)*parallel*arithmetic steps on a*parallel*computer ... The*recursive*doubling*algorithm*as developed by Stone can be used to solve a*tridiagonal*linear*system*of size n on a*parallel*computer with n processors using O(log n)*parallel*arithmetic steps. ... We are currently implementing*parallel**algorithms**for*the solutions of general recurrence relations,*tridiagonal*, block*tridiagonal*, and banded linear*systems*on the Intel hypercube, and investigating ...##
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A Parallel Method for Tridiagonal Equations

1981
*
ACM Transactions on Mathematical Software
*

A new (partition) method

doi:10.1145/355945.355947
fatcat:pf46qvc4nfgk3ma5vb6n5aw76i
*for*solving a tndiagonal*system*of lmear equations is presented in this paper The method is suitable*for*both*parallel*and vector computers. ... smaller than that of the*recursive*doubling. ... They also gave conditions*for*stability of the*recursive*doubling*algorithm*as applied to the LU decomposition of a*tridiagonal**system*. ...##
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An iteration-based hybrid parallel algorithm for tridiagonal systems of equations on multi-core architectures

2015
*
Concurrency and Computation
*

The present paper will describe several

doi:10.1002/cpe.3511
fatcat:fcmtcrbwvrhrdnjaahoucjpztq
*parallel**tridiagonal*solvers and present a new hybrid*parallel**algorithm**for*the*tridiagonal**systems*. ... The Thomas*algorithm*, a specific Gaussian elimination method, is one of the first*algorithms*considered*for**tridiagonal*linear*system*. ... Thanks are also due to Qinghong Yang and Yulu Pan*for*the discussions on efficient multithreading implementation of the*parallel**algorithms*. ...##
###
Toward an Efficient Parallel Eigensolver for Dense Symmetric Matrices

1998
*
SIAM Journal on Scientific Computing
*

We describe a

doi:10.1137/s1064827596300681
fatcat:xmdfnmoakzbh3kankiszjfxrom
*parallel**algorithm**for*finding the eigenvalues and eigenvectors of a dense symmetric matrix. ... Since the different steps have different*algorithmic*characteristics, this problem serves as an perfect vehicle*for*exploring some issues associated with*parallel*linear algebra calculations, In particular ... llidiagonzdiza tion*Algorithm*. Our*parallel**algorithm**for*Householder*tridiagonalization**for*processor q is depicted in Fig. 6 . ...##
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Analysis of non-strict functional implementations of the Dongarra-Sorensen eigensolver

1994
*
Proceedings of the 8th international conference on Supercomputing - ICS '94
*

We study the producer-consumer

doi:10.1145/181181.181574
dblp:conf/ics/SurB94
fatcat:742uwp2fvjhhzgk25bs4lathsq
*parallelism*of Eigensolvers composed of a*tridiagonalization*function, a*tridiagonal*solver, and a matrix multiplication, written in the non-strict functional programming ... We show that in a non-strict functional execution model, the Dongarra-Sorensen*algorithm*can run completely in*parallel*with the Householder function. ...*For*the*tridiagonal*solver we h a v e studied implementations of the Dongarra-Sorensen*algorithm*and compared them with the more traditional QL*algorithm*. ...##
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A time-optimal algorithm for solving (block-)tridiagonal linear systems of dimension N on a distributed computer of N nodes
[article]

2018
*
arXiv
*
pre-print

Our paper presents two major results: First, we provide an

arXiv:1801.09840v2
fatcat:pgync2sfandk7eslvrh6u3y6d4
*algorithm*that achieves the optimal*parallel*time complexity*for*solving a*tridiagonal*linear*system*and thin-banded linear*systems*. ... We are concerned with the fastest possible direct numerical solution*algorithm**for*a thin-banded or*tridiagonal*linear*system*of dimension N on a distributed computing network of N nodes that is connected ... Conclusions We presented an*algorithm**for*the efficient*parallel*solution of (block-)*tridiagonal*linear*systems*. ...##
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Generalized scans and tridiagonal systems

2001
*
Theoretical Computer Science
*

We also describe a variety of

doi:10.1016/s0304-3975(99)00311-4
fatcat:4t3vu2u4wngelbkejjtbyc654i
*parallel**algorithms*including some that are well known*for**tridiagonal**systems*and some that are much better suited to distributed computation. ... Motivated by the analysis of known*parallel*techniques*for*the solution of linear*tridiagonal**system*, we i n troduce generalized scans, a class of*recursively*de ned lengthpreserving, sequence-to-sequence ...*for*its bearing on the solution of banded*systems*naturally viewed as block*tridiagonal**systems*. ...##
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Implementation of parallel tridiagonal solvers for a heterogeneous computing environment

2016
*
ANZIAM Journal
*

*Algorithms*such as cyclic reduction and spike reduce a single large

*tridiagonal*

*system*into multiple small independent

*systems*which are solved in

*parallel*. ... The standard Thomas

*algorithm*

*for*solving such

*systems*is inherently serial, forming a bottleneck in computation. ... A special thanks to the hpc group

*for*their support, particularly in providing access to specialist fpga and gpu resources. ...

##
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Using cyclic reduction on a parallel computer to improve the performance of an underwater sound implicit finite difference model

1991
*
Computers and Mathematics with Applications
*

The numerical

doi:10.1016/0898-1221(91)90218-s
fatcat:z2yrbeootrfjbb7ejxz52meaoe
*algorithm*used in subroutine lrid of the IFD code,*for*solving a*tridiagonal**system*of equations is based on Gaussian elimination and is*recursive*. ... The cyclic reduction*algorithm*of Hockney and Jesshope [10] is an*algorithm**for*solving*tridiagonal**systems*of equations that can be vectorized and that can utilize the unique characteristics of a*parallel*...##
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Cluster Computation for Flood Simulations1
[chapter]

2001
*
Lecture Notes in Computer Science
*

Simulation of the water flood problems often leads to solving of large sparse

doi:10.1007/3-540-48228-8_43
fatcat:v7slsobodnblfeeizftl2nlvl4
*systems*of partial differential equations.*For*such*systems*, the numerical method is very CPU-time consuming. ... Therefore, the*parallel*simulation is essential*for*water flood study with satisfactory accuracy. ... The cyclic reduction*algorithm*was chosen because of its possibility*for**parallelism*and better computational runtime than the*recursive*doubling*algorithm*. ...##
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Generalized scans and tri-diagonal systems
[chapter]

1995
*
Lecture Notes in Computer Science
*

We also describe a variety of

doi:10.1007/3-540-59042-0_71
fatcat:vdb23x2p5veh5op35oyt2s2voi
*parallel**algorithms*including some that are well known*for**tridiagonal**systems*and some that are much better suited to distributed computation. ... Motivated by the analysis of known*parallel*techniques*for*the solution of linear*tridiagonal**system*, we i n troduce generalized scans, a class of*recursively*de ned lengthpreserving, sequence-to-sequence ...*tridiagonal**system*6 with one less equation, to be solved formally via*recursion*, and a single equation*for*y 3 which is solved subsequently. ...##
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A PARALLEL PREFIX ALGORITHM FOR ALMOST TOEPLITZ TRIDIAGONAL SYSTEMS

1995
*
International journal of high speed computing
*

Experimental results show that the simple

doi:10.1142/s0129053395000300
fatcat:2ymegf55g5a6vpa7b75qxerjsm
*parallel*prex*algorithm*is a good*algorithm**for*symmetric, almost symmetric Toeplitz*tridiagonal**systems*and*for*the compact scheme on high-performance computers ... However, the resulting*systems*from compact schemes are*tridiagonal**systems*that are dicult to solve eciently on*parallel*computers. ... Carter of the Scalable Computing Laboratory at Ames Laboratory*for*providing the access and general help on the Ames 16K PEs MarPar*parallel*computer. ...##
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Page 6012 of Mathematical Reviews Vol. , Issue 2002H
[page]

2002
*
Mathematical Reviews
*

The authors present an algebraic abstraction of various well- established techniques

*for**parallel*decomposition of*tridiagonal**systems*of linear equations. ... There is no requirement that the*recursion*be applied by centering on each record in sequential order, allowing the approach to cover a wide variety of deflationary*recursive**algorithms*amenable to*parallel*...
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