Fast algorithms for placing large entries along the diagonal of a sparse matrix

Solving a sparse system of linear equations A x = b is one of the most fundamental operations inside any circuit simulator. The equations/rows in the matrix A are often rearranged/permuted before factorization and applying direct or iterative methods to obtain the solution. Permuting the rows of the...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 235; no. 3; pp. 756 - 764
Main Authors: Kundeti, Vamsi, Rajasekaran, Sanguthevar
Format: Journal Article
Language:English
Published: Kidlington Elsevier B.V 01.12.2010
Elsevier
Subjects:
Algebra
Algorithms
Asymptotic properties
Bipartite matching
Combinatorial optimization
Computer simulation
Diagonally dominant sparse matrix
Exact sciences and technology
Finite differences and functional equations
Gaussian elimination
Linear and multilinear algebra, matrix theory
Linear equations
Marking
Matching
Mathematical analysis
Mathematical models
Mathematics
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Pre ordering sparse matrix
Sciences and techniques of general use
Sparse matrix algorithms
Weighted matching algorithms
Bipartite matching
Pre ordering sparse matrix
Diagonally dominant sparse matrix
Combinatorial optimization
Sparse matrix algorithms
Weighted matching algorithms
Iterative method
Numerical method
Stochastic method
Convergence
Non linear equation
Direct method
Industry
Sparse matrix
Ordering
Fast algorithm
Matrix equation
Numerical stability
Transcendental equation
Jacobi method
Numerical linear algebra
Equation system
Experimental result
Numerical analysis
Linear system
Algorithm performance
Matrix inversion
Applied mathematics
Algebraic equation
Matrix method
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary:Solving a sparse system of linear equations A x = b is one of the most fundamental operations inside any circuit simulator. The equations/rows in the matrix A are often rearranged/permuted before factorization and applying direct or iterative methods to obtain the solution. Permuting the rows of the matrix A so that the entries with large absolute values lie on the diagonal has several advantages like better numerical stability for direct methods (e.g., Gaussian elimination) and faster convergence for indirect methods (such as the Jacobi method). Duff (2009) [3] has formulated this as a weighted bipartite matching problem (the MC64 algorithm). In this paper we improve the performance of the MC64 algorithm with a new labeling technique which improves the asymptotic complexity of updating dual variables from O ( | V | + | E | ) to O ( | V | ) , where | V | is the order of the matrix A and | E | is the number of non-zeros. Experimental results from using the new algorithm, when benchmarked with both industry benchmarks and UFL sparse matrix collection, are very promising. Our algorithm is more than 60 times faster (than Duff’s algorithm) for sparse matrices with at least a million non-zeros.
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ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2010.07.002