On designing optimal parallel triangular solvers

This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent o...

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Bibliographic Details
Published in:Information and computation Vol. 161; no. 2; pp. 172 - 210
Main Author: Santos, Eunice E.
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 2000
Elsevier
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science; control theory; systems
distributed-memory
Exact sciences and technology
LogP model
Mathematics
matrix computation
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
numerical methods
parallel algorithmsand complexity
Sciences and techniques of general use
Theoretical computing
triangular solvers
matrix computation
distributed-memory
LogP model
numerical methods
triangular solvers
parallel algorithmsand complexity
Triangular matrix
Parallel algorithm
Triangular linear system
Numerical method
Forward substitution scheme
Complexity
Latency
Equation system
Distributed memory
Distributed memory multiprocessor system
Linear system
Computer program
Matrix calculus
Backward substitution scheme
Time complexity
Message transmission
ISSN:0890-5401, 1090-2651
Online Access:Get full text
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Description
Summary:This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent of the data layout, lower bounds for data layouts in which the number of data items per processor is bounded, and lower bounds for specific data layouts commonly used in designing parallel algorithms for this problem are presented in this paper. Furthermore, algorithms are provided which have running times within a constant factor of the lower bounds described. One interesting result is that the popular two-dimensional block matrix layout necessarily results in significantly longer running times than simpler one-dimensional schemes. Finally, a generalization of the lower bounds to banded triangular linear systems is presented.
ISSN:0890-5401
1090-2651
DOI:10.1006/inco.2000.2866