Parallel Algorithms for the Solution of Toeplitz Systems of Linear Equations

Pentadiagonal Toeplitz systems of linear equations arise in many application areas. Because of the structure and many good properties of pentadiagonal Toeplitz matrices, they have been applied in Mechanical Control. Based on [1], in this paper, we present an improved fast algorithm for solving symmetric pentadiagonal Toeplitz systems.

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Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems

Advances in Numerical Analysis ◽

10.1155/2012/973407 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 5

Author(s):

N. Akhondi ◽

F. Toutounian

Keyword(s):

Convergence Property ◽

Linear Equations ◽

Coefficient Matrix ◽

Positive Definite ◽

Optimal Parameters ◽

Skew Circulant ◽

Systems Of Linear Equations ◽

Toeplitz Systems ◽

Two Parameters ◽

Two Parameter

We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrix . In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method.

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Vectorized Parallel Solver for Tridiagonal Toeplitz Systems of Linear Equations

Parallel Processing and Applied Mathematics - Lecture Notes in Computer Science ◽

10.1007/978-3-030-43229-4_9 ◽

2020 ◽

pp. 93-103

Author(s):

Beata Dmitruk ◽

Przemysław Stpiczyński

Keyword(s):

Linear Equations ◽

Parallel Solver ◽

Systems Of Linear Equations ◽

Toeplitz Systems

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A THREE-LEVEL PARALLELISATION SCHEME AND APPLICATION TO THE NELDER-MEAD ALGORITHM

Mathematical Modelling and Analysis ◽

10.3846/mma.2020.12139 ◽

2020 ◽

Vol 25 (4) ◽

pp. 584-607

Author(s):

Rima Kriauzienė ◽

Andrej Bugajev ◽

Raimondas Čiegis

Keyword(s):

Load Balancing ◽

Parallel Algorithms ◽

Linear Equations ◽

Workload Balancing ◽

Tridiagonal Matrices ◽

The Third ◽

Computational Data ◽

Systems Of Linear Equations ◽

Load Balancing Algorithm ◽

Nelder Mead Algorithm

We consider a three-level parallelisation scheme. The second and third levels define a classical two-level parallelisation scheme and some load balancing algorithm is used to distribute tasks among processes. It is well-known that for many applications the efficiency of parallel algorithms of these two levels starts to drop down after some critical parallelisation degree is reached. This weakness of the twolevel template is addressed by introduction of one additional parallelisation level. s an alternative to the basic solver some new or modified algorithms are considered on this level. The idea of the proposed methodology is to increase the parallelisation degree by using possibly less efficient algorithms in comparison with the basic solver. As an example we investigate two modified Nelder-Mead methods. For the selected application, a Schro¨dinger equation is solved numerically on the second level, and on the third level the parallel Wang’s algorithm is used to solve systems of linear equations with tridiagonal matrices. A greedy workload balancing heuristic is proposed, which is oriented to the case of a large number of available processors. The complexity estimates of the computational tasks are model-based, i.e. they use empirical computational data.

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