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.

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.

Enhancing Mathematical Modeling Activities in Classroom Instruction

2019 ◽  
pp. 275-280
Author(s):  
Luckson Muganyizi Kaino
Keyword(s):  
Mathematical Modeling ◽  
Linear Equations ◽  
Real Life ◽  
Thinking Skills ◽  
School Level ◽  
Useful Knowledge ◽  
Life Problems ◽  
Inconsistent Systems ◽  
Secondary School Level ◽  
Systems Of Linear Equations

The ability of students in mathematical modeling was enhanced through activities that involved systems of linear equations with two variables. Students involved were in form four, at the final year of the ordinary secondary school level where they were expected to have mastered the knowledge on systems of linear equations with two variables. Students’ knowledge on ill-conditioned linear systems was explored as well as their knowledge on practical problems in linear equations. Then after, mathematics subject teachers guided students to identify practical problems in linear equations of two variables. Students were put into groups to think of problems in real life and come up with solutions. The solutions were related to the real situations in the environment and each group had to make a presentation in the class. Problems in transportation, manufacturing, production and diet were identified by students and the results presented for discussion. It came out clearly that students acquired knowledge on solving real life problems at the end of the activities. Before these activities, students had theoretical knowledge on solving problems with two unknowns without relating these to real life problems. While knowledge on independent and inconsistent systems was known to students, enthusiasm was noted among students at the end of the activities when they got involved in real aspects of solutions obtained. It was concluded that with more time availed in the school curricula, students can acquire useful knowledge on mathematical modeling to achieve problem-thinking skills that involve real life situations.

The Back Page: My Favorite Lesson: Unfolding the Solution of Linear Systems

2010 ◽  
Vol 104 (2) ◽  
pp. 160
Author(s):  
Sarah B. Bush
Keyword(s):  
Linear Systems ◽  
Student Teaching ◽  
Linear Equations ◽  
Teaching Experience ◽  
Correct Solution ◽  
First Year ◽  
Know How ◽  
Big Picture ◽  
Systems Of Linear Equations ◽  
Algebra Class

I often think back to a vivid memory from my student-teaching experience. Then, I naively believed that the weeks spent with my first-year algebra class discussing and practicing the art of solving systems of linear equations by graphing, substitution, and elimination was a success. But just at that point the students started asking revealing questions such as “How do you know which method to pick so that you get the correct solution?” and “Which systems go with which methods?” I then realized that my instruction had failed to guide my students toward conceptualizing the big picture of linear systems and instead had left them with a procedure they did not know how to apply. At that juncture I decided to try this discovery-oriented lesson.

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

scholarly journals Accelerating advanced preconditioning methods on hybrid architectures

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Ernesto Dufrechou
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Krylov Subspace ◽  
Linear Equations ◽  
Memory Systems ◽  
Sparse Linear Systems ◽  
Data Parallel ◽  
Systems Of Linear Equations ◽  
Sparse Systems ◽  
Computational Bottleneck

Many problems, in diverse areas of science and engineering, involve the solution of largescale sparse systems of linear equations. In most of these scenarios, they are also a computational bottleneck, and therefore their efficient solution on parallel architectureshas motivated a tremendous volume of research.This dissertation targets the use of GPUs to enhance the performance of the solution of sparse linear systems using iterative methods complemented with state-of-the-art preconditioned techniques. In particular, we study ILUPACK, a package for the solution of sparse linear systems via Krylov subspace methods that relies on a modern inverse-based multilevel ILU (incomplete LU) preconditioning technique.We present new data-parallel versions of the preconditioner and the most important solvers contained in the package that significantly improve its performance without affecting its accuracy. Additionally we enhance existing task-parallel versions of ILUPACK for shared- and distributed-memory systems with the inclusion of GPU acceleration. The results obtained show a sensible reduction in the runtime of the methods, as well as the possibility of addressing large-scale problems efficiently.

PARALLEL ITERATIVE REGULARIZATION ALGORITHMS FOR LARGE OVERDETERMINED LINEAR SYSTEMS

2010 ◽  
Vol 07 (04) ◽  
pp. 525-537 ◽  
Author(s):  
PHAM KY ANH ◽  
VU TIEN DUNG
Keyword(s):  
Linear Systems ◽  
Linear Equations ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Overdetermined Systems ◽  
Overdetermined Linear Systems ◽  
Systems Of Linear Equations ◽  
Parallel Iterative

In this paper, we study the performance of some parallel iterative regularization methods for solving large overdetermined systems of linear equations.

A Short Review on Fuzzy System of Linear Equations Applications

2019 ◽  
pp. 75-87
Author(s):  
Hale Gonce Kocken ◽  
Inci Albayrak
Keyword(s):  
Fuzzy System ◽  
Operational Research ◽  
Linear Equations ◽  
Real Life ◽  
Short Review ◽  
System Of Linear Equations ◽  
Life Problems ◽  
Solution Methods ◽  
The Common ◽  
Common Application

Fuzzy system of linear equations (FSLE) plays a major role in various areas such as operational research, physics, statistics, economics, engineering, and social sciences since the parameters of FSLE are not always exactly known and stable in real-life problems. This effect may follow the lack of exact information, changeable economic conditions, etc. Although there exist many review papers on the solution methods for FSLE, they are not based on the applications. This chapter has attempted to provide a short review on real-life applications of FSLE. In addition, for the common application areas, the fundamental models and the solution methods are presented considering the most cited and leading papers in the literature.

SEBSM-Based Iterative Method for Solving Large Systems of Linear Equations and Its Applications in Engineering Computation

2016 ◽  
Vol 13 (05) ◽  
pp. 1650024 ◽  
Author(s):  
Jin-Xiu Hu ◽  
Xiao-Wei Gao ◽  
Zhi-Chao Yuan ◽  
Jian Liu ◽  
Shi-Zhang Huang
Keyword(s):  
Iterative Method ◽  
Direct Method ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Relaxation Factor ◽  
Banded Matrix ◽  
Speed Up ◽  
Systems Of Linear Equations ◽  
Matrix Formula ◽  
Large Systems

In this paper, a new iterative method, for solving sparse nonsymmetrical systems of linear equations is proposed based on the Simultaneous Elimination and Back-Substitution Method (SEBSM), and the method is applied to solve systems resulted in engineering problems solved using Finite Element Method (FEM). First, SEBSM is introduced for solving general linear systems using the direct method. And, then an iterative method based on SEBSM is presented. In the method, the coefficient matrix [Formula: see text] is split into lower, diagonally banded and upper matrices. The iterative convergence can be controlled by selecting a suitable bandwidth of the diagonally banded matrix. And the size of the working array needing to be stored in iteration is as small as the bandwidth of the diagonally banded matrix. Finally, an accelerating strategy for this iterative method is proposed by introducing a relaxation factor, which can speed up the convergence effectively if an optimal relaxation factor is chosen. Two numerical examples are given to demonstrate the behavior of the proposed method.

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