Journal Article10.1137/0730041
Parallel factorizations for tridiagonal matrices
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TL;DR: A wide class of efficient parallel solvers is derived by considering different parallel factorizations of partitioned matrices, and one of them derives a very efficient parallel method based on the cyclic reduction algorithm.
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Abstract: The authors analyze the problem of solving tridiagonal linear systems on parallel computers. A wide class of efficient parallel solvers is derived by considering different parallel factorizations of partitioned matrices. These solvers have a minimum requirement of data transmission. In fact, communication is only needed for solving a “reduced system,” whose dimension depends on the number of parallel processors used. Moreover, for a given partitioned tridiagonal matrix, the reduced system (which is again tridiagonal) is the same, and represents the only sequential part of the corresponding parallel solver.Three examples are discussed in more detail; one of them derives a very efficient parallel method based on the cyclic reduction algorithm.
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Citations
The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond
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TL;DR: Cyclic reduction is an algorithm invented by G. H. Golub and R. W. Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle.
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Almost block diagonal linear systems: sequential and parallel solution techniques, and applications
Pierluigi Amodio,Jeff Cash,George Roussos,R. W. Wright,Graeme Fairweather,Ian Gladwell,G. L. Kraut,Marcin Paprzycki +7 more
TL;DR: Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations.
The use of the factorization of five-diagonal matrices by tridiagonal Toeplitz matrices☆
Fasma Diele,Luciano Lopez +1 more
TL;DR: Either bounds for the inverse or numerical methods for solving linear systems may be derived in the factorization of five-diagonal matrices as the product of two Toeplitz tridiagonalMatrices.
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Two-Dimensional Vlasov Simulation of Raman Scattering and Plasma Beatwave Acceleration on Parallel Computers
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A Cyclic Reduction Approach to the Numerical Solution of Boundary Value ODEs
TL;DR: A modification of a cyclic reduction algorithm that takes advantage of the almost block diagonal structure of the linear system to solve linear systems that arise from the discretization of boundary value ODEs.
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Parallel Tridiagonal Equation Solvers
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