Solution of real and complex systems of linear equations

1966 ◽  
Vol 8 (3) ◽  
pp. 217-234 ◽  
Author(s):  
H. J. Bowdler ◽  
R. S. Martin ◽  
G. Peters ◽  
J. H. Wilkinson
Keyword(s):  
Complex Systems ◽  
Linear Equations ◽  
Systems Of Linear Equations

Accelerated GPMHSS Method for Solving Complex Systems of Linear Equations

2017 ◽  
Vol 7 (1) ◽  
pp. 143-155 ◽  
Author(s):  
Jing Wang ◽  
Xue-Ping Guo ◽  
Hong-Xiu Zhong
Keyword(s):  
Complex Systems ◽  
Linear Systems ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Linear Equations ◽  
Splitting Method ◽  
Complex Symmetric Linear Systems ◽  
Systems Of Linear Equations ◽  
Complex Symmetric ◽  
Symmetric Systems

AbstractPreconditioned modified Hermitian and skew-Hermitian splitting method (PMHSS) is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equations, and uses one parameter α. Adding another parameter β, the generalized PMHSS method (GPMHSS) is essentially a twoparameter iteration method. In order to accelerate the GPMHSS method, using an unexpected way, we propose an accelerated GPMHSS method (AGPMHSS) for large complex symmetric linear systems. Numerical experiments show the numerical behavior of our new method.

Solution of Real and Complex Systems of Linear Equations

1971 ◽  
pp. 93-110
Author(s):  
H. J. Bowdler ◽  
R. S. Martin ◽  
G. Peters ◽  
J. H. Wilkinson
Keyword(s):  
Complex Systems ◽  
Linear Equations ◽  
Systems Of Linear Equations

scholarly journals ABS-Based Direct Method for Solving Complex Systems of Linear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2527
Author(s):  
József Abaffy ◽  
Szabina Fodor
Keyword(s):  
Complex Systems ◽  
Linear Systems ◽  
Sparse Matrices ◽  
Direct Method ◽  
Linear Equations ◽  
Real Life ◽  
Computational Accuracy ◽  
Life Problems ◽  
Systems Of Linear Equations ◽  
Basic Features

Efficient solution of linear systems of equations is one of the central topics of numerical computation. Linear systems with complex coefficients arise from various physics and quantum chemistry problems. In this paper, we propose a novel ABS-based algorithm, which is able to solve complex systems of linear equations. Theoretical analysis is given to highlight the basic features of our new algorithm. Four variants of our algorithm were also implemented and intensively tested on randomly generated full and sparse matrices and real-life problems. The results of numerical experiments reveal that our ABS-based algorithm is able to compute the solution with high accuracy. The performance of our algorithm was compared with a commercially available software, Matlab’s mldivide (\) algorithm. Our algorithm outperformed the Matlab algorithm in most cases in terms of computational accuracy. These results expand the practical usefulness of our algorithm.

Solution of Real and Complex Systems of Linear Equations

1971 ◽  
pp. 93-110 ◽  
Author(s):  
H. J. Bowdler ◽  
R. S. Martin ◽  
G. Peters ◽  
J. H. Wilkinson
Keyword(s):  
Complex Systems ◽  
Linear Equations ◽  
Systems Of Linear Equations

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

scholarly journals A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhijun Luo ◽  
Lirong Wang
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Descent Direction ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Globally Convergent ◽  
Variable Distribution ◽  
Systems Of Linear Equations

A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

Mutual Embeddings

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550001
Author(s):  
ILKER NADI BOZKURT ◽  
HAI HUANG ◽  
BRUCE MAGGS ◽  
ANDRÉA RICHA ◽  
MAVERICK WOO
Keyword(s):  
Iterative Methods ◽  
Lower Bounds ◽  
Linear Equations ◽  
Graph Embedding ◽  
Open Problems ◽  
Linear Arrays ◽  
Systems Of Linear Equations

This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

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