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Computing the block triangular form of a sparse matrix

1990
*
ACM Transactions on Mathematical Software
*

We consider

doi:10.1145/98267.98287
fatcat:zc2y5kmesraunlihbofslyvymq
*the*problem*of*permuting*the*rows and columns*of**a*rectangular or square, unsymmetric*sparse**matrix*to*compute*its*block**triangular**form*. ... We describe implementations*of*algorithms to*compute**the**block**triangular**form*and provide*computational*results on*sparse*matrices from test collections. ... ACKNOWLEDGMENTS We wish to thank Tom Coleman and John Gilbert for their help in improving*the*presentation*of**the*results in Section 2. ...##
###
Towards a scalable hybrid sparse solver

2000
*
Concurrency Practice and Experience
*

Our goal is to develop

doi:10.1002/(sici)1096-9128(200002/03)12:2/3<53::aid-cpe473>3.0.co;2-b
fatcat:kv7m4mb5lnh4jgigz6ygpmrkqy
*a*scalable and memory-e cient hybrid*of**the*two methods that can be implemented with high-e ciency on both serial and parallel*computers*and be suitable for*a*wide-range*of*problems ... Consider*the*solution*of*very large,*sparse*linear systems.*The*most popular techniques can be broadly classi ed as either direct" or iterative." ...*The**compute*-tree f o r 4 p r ocessors for*a**sparse**matrix*associated with*a*7X 7 grid is shown on*the*left; in*a*column oriented fan-in scheme this*compute*-tree r epresents operations on*blocks**of*columns ...##
###
GPU Preconditioning for Block Linear Systems Using Block Incomplete Sparse Approximate Inverses

2021
*
Mathematical Problems in Engineering
*

In this study,

doi:10.1155/2021/5558508
fatcat:kp2epjgernafpaa3qzerqu37yq
*the**block*version*of**the*incomplete*sparse*approximate inverses (ISAI) algorithm is studied, and*the**block*-ISAI is considered for preconditioning by proposing an efficient algorithm and ... Solving*sparse**triangular*systems is*the*building*block*for incomplete LU- (ILU-) based preconditioning, but parallel algorithms, such as*the*level-scheduling scheme, are sometimes limited by available ... e authors thank LetPub for its linguistic assistance during*the*preparation*of*this manuscript. ...##
###
Sparse Matrices in MATLAB: Design and Implementation

1992
*
SIAM Journal on Matrix Analysis and Applications
*

*The*

*sparse*data structure represents

*a*

*matrix*in space proportional to

*the*number

*of*nonzero entries, and most

*of*

*the*operations

*compute*

*sparse*results in time proportional t o t h e n umber

*of*arithmetic ... Dedicated to Gene Golub on

*the*occasion

*of*his 60th birthday. Abstract. We h

*a*ve extended

*the*

*matrix*

*computation*language and environment Matlab to include

*sparse*

*matrix*storage and operations. ...

*The*most common application

*of*

*block*

*triangular*

*form*is to solve

*a*reducible system

*of*linear equations by b l o c k b

*a*c k substitution, factoring only

*the*diagonal

*blocks*

*of*

*the*

*matrix*. ...

##
###
Towards a scalable hybrid sparse solver

2000
*
Concurrency Practice and Experience
*

Our goal is to develop

doi:10.1002/(sici)1096-9128(200002/03)12:2/3<53::aid-cpe473>3.3.co;2-2
fatcat:7bqxv5ygpbbvneb6yfxshyrske
*a*scalable and memory-e cient hybrid*of**the*two methods that can be implemented with high-e ciency on both serial and parallel*computers*and be suitable for*a*wide-range*of*problems ... Consider*the*solution*of*very large,*sparse*linear systems.*The*most popular techniques can be broadly classi ed as either direct" or iterative." ...*The**compute*-tree f o r 4 p r ocessors for*a**sparse**matrix*associated with*a*7X 7 grid is shown on*the*left; in*a*column oriented fan-in scheme this*compute*-tree r epresents operations on*blocks**of*columns ...##
###
Extended concept of stair-shape sparsity for the inverse of an asymmetric matrix

1993
*
Computers and Mathematics with Applications
*

This paper extcmds

doi:10.1016/0898-1221(93)90204-9
fatcat:5cz2dkf4uzcd7mm4y76ukqmil4
*the*concept*of*stair-shape spardty, introduced in*A*previous work*of**the*author for analysis*of**a*symmetric and*sparse**matrix*, to an asymmetric*matrix*. ...*The*e~ential idea first considers an asymmetric*matrix*as*a*combination*of*an upper*triangular**matrix*and*a*lower*triangular**matrix*, and then applies*the*stair-shape sparsity to*the*upp~ and lower*triangular*... Since*the*stair-shape sparsity deals with*a*lower*triangular**matrix*, this work will consider an asymmetric*matrix*[*A*] as*a*combination*of*an upper*triangular**matrix*and*a*lower*triangular**matrix*. ...##
###
Page 6155 of Mathematical Reviews Vol. , Issue 91K
[page]

1991
*
Mathematical Reviews
*

Summary: “We consider

*the*problem*of*permuting*the*rows and columns*of**a*rectangular or square, unsymmetric*sparse**matrix*to*compute*its*block**triangular**form*. ... Also,*a*relationship is established between*the*Gram-Schmidt and*the*basic Cholesky methods.” 91k:65075 65F50 Pothen, Alex (1-PAS-C); Fan, Chin-Ju (1-PAS-C)*Computing**the**block**triangular**form**of**a**sparse*...##
###
Block computation and representation of a sparse nullspace basis of a rectangular matrix

2008
*
Linear Algebra and its Applications
*

In this paper, we propose

doi:10.1016/j.laa.2007.11.025
fatcat:ykzck3flqrgxnldjii2by6kq5q
*a*new method to efficiently*compute**a*representation*of*an orthogonal basis*of**the*nullspace*of**a**sparse**matrix*operator B T with B ∈ R n×m , n > m. ... We will employ this observation to design an efficient*block*algorithm that*computes**a**sparse*representation*of**the*nullspace basis in almost optimal complexity. ... Since*the*Cholesky factor R 1 coincides with*the*upper*triangular**matrix*R 1*of**the*(reduced or full) QR-factorization, we can*compute**the**matrix**block*Y as Y = BR −1 1 , i.e. ...##
###
Technical Note: Improving the computational efficiency of sparse matrix multiplication in linear atmospheric inverse problems

2016
*
Geoscientific Model Development Discussions
*

*Matrix*multiplication

*of*two

*sparse*matrices is

*a*fundamental operation in linear Bayesian inverse problems for

*computing*covariance matrices

*of*observations and <i>

*a*posteriori</i> uncertainties. ... Two modifications

*of*this hybrid-parallel algorithm are also proposed for

*the*types

*of*operations typical

*of*atmospheric inverse problems, which further reduce

*the*cost

*of*

*sparse*

*matrix*multiplication ... Introduction

*Sparse*-

*Sparse*(SS)

*matrix*multiplication

*forms*

*the*

*computational*backbone

*of*scientific

*computation*in many fields. ...

##
###
Efficient algorithm for computing large scale systems of differential algebraic equations
[article]

2015
*
arXiv
*
pre-print

In general,

arXiv:1506.03963v1
fatcat:j3n7vrsum5amna5o3pxn2e2rhu
*the*feature*of*DAEs is*a**sparse*large scale system*of*fully nonlinear and high index. ... We exploit*the*shortest augmenting path algorithm for finding maximum value transversal (MVT) as well as*block**triangular**forms*(BTF). ...*The*problem is also closely related to*computing**the**block**triangular**form**of**a**sparse**matrix*and linear assignment problems over integer. ...##
###
Parallel algorithms for forward and back substitution in direct solution of sparse linear systems

1995
*
Proceedings of the 1995 ACM/IEEE conference on Supercomputing (CDROM) - Supercomputing '95
*

We present

doi:10.1145/224170.224471
dblp:conf/sc/GuptaK95
fatcat:nfqpljy7wredncqddhmq44s2xm
*a*detailed analysis*of*parallel complexity and scalability*of**the*best*of*these algorithms and*the*results*of*its implementation on up to 256 processors*of**the*Cray T3D parallel*computer*. ... We also show that for*a*wide class*of*problems,*the**sparse**triangular*solvers described in this paper are optimal and are asymptotically as scalable as*a*dense*triangular*solver. ... Solving*a**triangular*system corresponding to this supernode involves asymptotically*a**computation**of**the*same complexity as solving*the*entire*sparse**triangular*system. ...##
###
Sparse Matrix Methods for Circuit Simulation Problems
[chapter]

2011
*
Mathematics in Industry
*

It relies on

doi:10.1007/978-3-642-22453-9_1
fatcat:ehpfgtctovhwliylomtbwvsodu
*a*permutation to*block**triangular**form*(BTF), several methods for finding*a*fill-reducing ordering (variants*of*approximate minimum degree and nested dissection), and Gilbert/Peierls'*sparse*... They are permutable to*block**triangular**form*, which breaks*the**sparse*LU factorization problem into many smaller subproblems. ... Portions*of*this work were supported by*the*Department*of*Energy, and by National Science Foundation grants 0203270, 0620286, and 0619080. ...##
###
Parallel Gaussian elimination for Gröbner bases computations in finite fields

2010
*
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation - PASCO '10
*

*The*library is written in C and contains specific algorithms [11] to

*compute*Gaussian elimination as well as specific internal representation

*of*matrices (

*sparse*

*triangular*

*blocks*,

*sparse*rectangular

*blocks*... Polynomial system solving is one

*of*

*the*important area

*of*

*Computer*Algebra with many applications in Robotics, Cryptology,

*Computational*Geometry, etc. ... Acknowledgements:

*The*authors would like to thank Olivier Orcière for his helpful remarks. ...

##
###
Block Householder computation of sparse matrix singular values

2010
*
Proceedings of the 2010 Spring Simulation Multiconference on - SpringSim '10
*

This paper introduces

doi:10.1145/1878537.1878630
fatcat:aug34khwa5h53o2b3wwatal6h4
*block*Householder reduction*of**a*rectangular*sparse**matrix*to small band upper*triangular**form*. ...*The**computation*accesses*a**sparse**matrix*only for*sparse**matrix*dense*matrix*(SMDM) multiplications and for "just in time" extractions*of*row and column*blocks*. ...*The*author wishes to thank Jim Demmel, Gene Golub, Robert Funderlic, Jackie Hughes Oliver, and Franc Brglez for encouragement and advice. ...##
###
Parallel Algorithms for Forward and Back Substitution in Linear Algebraic Equations of Finite Element Method

2019
*
Journal of Telecommunications and Information Technology
*

This paper considers several algorithms for parallelizing

doi:10.26636/jtit.2019.134919
fatcat:o6ctsvrbpzbcvgnyos6lhjiqia
*the*procedure*of*forward and back substitution for high-order symmetric*sparse*matrices on multi-core*computers*with shared memory. ... It compares*the*proposed approaches for various finite-element problems*of*structural mechanics which generate*sparse*matrices*of*different structures. ... Acknowledgements*The*author is deeply grateful to SCAD Soft for providing support and for*the*collection*of*real-life problems involving design models created by SCAD Office users. ...
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