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Computing the block triangular form of a sparse matrix

Alex Pothen, Chin-Ju Fan
1990 ACM Transactions on Mathematical Software  
We consider 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.  ... 
doi:10.1145/98267.98287 fatcat:zc2y5kmesraunlihbofslyvymq

Towards a scalable hybrid sparse solver

Esmond G. Ng, Padma Raghavan
2000 Concurrency Practice and Experience  
Our goal is to develop 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  ... 
doi:10.1002/(sici)1096-9128(200002/03)12:2/3<53::aid-cpe473>3.0.co;2-b fatcat:kv7m4mb5lnh4jgigz6ygpmrkqy

GPU Preconditioning for Block Linear Systems Using Block Incomplete Sparse Approximate Inverses

Wenpeng Ma, Yiwen Hu, Wu Yuan, Xiazhen Liu, Hussein Abulkasim
2021 Mathematical Problems in Engineering  
In this study, 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.  ... 
doi:10.1155/2021/5558508 fatcat:kp2epjgernafpaa3qzerqu37yq

Sparse Matrices in MATLAB: Design and Implementation

John R. Gilbert, Cleve Moler, Robert Schreiber
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.  ... 
doi:10.1137/0613024 fatcat:uajvukqwnvhmvll6phefdx2znu

Towards a scalable hybrid sparse solver

Esmond G. Ng, Padma Raghavan
2000 Concurrency Practice and Experience  
Our goal is to develop 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  ... 
doi:10.1002/(sici)1096-9128(200002/03)12:2/3<53::aid-cpe473>3.3.co;2-2 fatcat:7bqxv5ygpbbvneb6yfxshyrske

Extended concept of stair-shape sparsity for the inverse of an asymmetric matrix

Jenn-Ching Luo
1993 Computers and Mathematics with Applications  
This paper extcmds 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.  ... 
doi:10.1016/0898-1221(93)90204-9 fatcat:5cz2dkf4uzcd7mm4y76ukqmil4

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

Sabine Le Borne
2008 Linear Algebra and its Applications  
In this paper, we propose 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.  ... 
doi:10.1016/j.laa.2007.11.025 fatcat:ykzck3flqrgxnldjii2by6kq5q

Technical Note: Improving the computational efficiency of sparse matrix multiplication in linear atmospheric inverse problems

Vineet Yadav, Anna M. Michalak
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.  ... 
doi:10.5194/gmd-2016-204 fatcat:u5qutmnqabbyrfbtk5dfol27se

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

Xiaolin Qin, Juan Tang, Yong Feng, Bernhard Bachmann, Peter Fritzson
2015 arXiv   pre-print
In general, 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.  ... 
arXiv:1506.03963v1 fatcat:j3n7vrsum5amna5o3pxn2e2rhu

Parallel algorithms for forward and back substitution in direct solution of sparse linear systems

Anshul Gupta, Vipin Kumar
1995 Proceedings of the 1995 ACM/IEEE conference on Supercomputing (CDROM) - Supercomputing '95  
We present 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.  ... 
doi:10.1145/224170.224471 dblp:conf/sc/GuptaK95 fatcat:nfqpljy7wredncqddhmq44s2xm

Sparse Matrix Methods for Circuit Simulation Problems [chapter]

Timothy A. Davis, E. Palamadai Natarajan
2011 Mathematics in Industry  
It relies on 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.  ... 
doi:10.1007/978-3-642-22453-9_1 fatcat:ehpfgtctovhwliylomtbwvsodu

Parallel Gaussian elimination for Gröbner bases computations in finite fields

Jean-Charles Faugère, Sylvain Lachartre
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.  ... 
doi:10.1145/1837210.1837225 dblp:conf/cap/FaugereL10 fatcat:yuocg24mprfa3d4jlateactfuy

Block Householder computation of sparse matrix singular values

Gary W. Howell
2010 Proceedings of the 2010 Spring Simulation Multiconference on - SpringSim '10  
This paper introduces 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.  ... 
doi:10.1145/1878537.1878630 fatcat:aug34khwa5h53o2b3wwatal6h4

Parallel Algorithms for Forward and Back Substitution in Linear Algebraic Equations of Finite Element Method

Sergiy Fialko
2019 Journal of Telecommunications and Information Technology  
This paper considers several algorithms for parallelizing 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.  ... 
doi:10.26636/jtit.2019.134919 fatcat:o6ctsvrbpzbcvgnyos6lhjiqia
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