Introduction to Iterative Methods for the Solution of Linear Systems

Multipreconditioned GMRES for simulating stochastic automata networks

Open Mathematics ◽

10.1515/math-2018-0083 ◽

2018 ◽

Vol 16 (1) ◽

pp. 986-998

Author(s):

Chun Wen ◽

Ting-Zhu Huang ◽

Xian-Ming Gu ◽

Zhao-Li Shen ◽

Hong-Fan Zhang ◽

...

Keyword(s):

Steady State ◽

Probability Distribution ◽

Linear Systems ◽

Iterative Methods ◽

Communication Systems ◽

Queueing Systems ◽

Steady State Probability ◽

Stochastic Automata ◽

Automata Networks ◽

Generator Matrices

AbstractStochastic Automata Networks (SANs) have a large amount of applications in modelling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seidel are inefficient due to the huge size of the generator matrices. In this paper, the multipreconditioned GMRES (MPGMRES) is considered by using two or more preconditioners simultaneously. Meanwhile, a selective version of the MPGMRES is presented to overcome the rapid increase of the storage requirements and make it practical. Numerical results on two models of SANs are reported to illustrate the effectiveness of these proposed methods.

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Preconditioned iterative methods for multi-linear systems based on the majorization matrix

Linear and Multilinear Algebra ◽

10.1080/03081087.2021.1931654 ◽

2021 ◽

pp. 1-20

Author(s):

Fatemeh Panjeh Ali Beik ◽

Mehdi Najafi-Kalyani ◽

Khalide Jbilou

Keyword(s):

Linear Systems ◽

Iterative Methods ◽

Preconditioned Iterative ◽

Preconditioned Iterative Methods

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Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

Scientific Programming ◽

10.1155/2012/243875 ◽

2012 ◽

Vol 20 (3) ◽

pp. 241-255 ◽

Cited By ~ 18

Author(s):

Eric Bavier ◽

Mark Hoemmen ◽

Sivasankaran Rajamanickam ◽

Heidi Thornquist

Keyword(s):

Linear Systems ◽

Iterative Methods ◽

Sparse Matrix ◽

Sparse Matrices ◽

Direct Methods ◽

Iterative Solvers ◽

Software Project ◽

Sparse Linear Systems ◽

Scalar Type ◽

Mixed Precision

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

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