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.

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.

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.

scholarly journals Software for Solving Fully Fuzzy Linear Systems with Rectangular Matrix

2020 ◽  
Vol 10 (1) ◽  
pp. 129-139
Author(s):  
A.V. Panteleev ◽  
V.S. Saveleva
Keyword(s):  
Linear Systems ◽  
Linear Equations ◽  
Linear System Of Equations ◽  
Software Application ◽  
Rectangular Matrix ◽  
Fuzzy Linear System ◽  
Triangular Numbers ◽  
Systems Of Linear Equations ◽  
The Mean ◽  
Fuzzy Linear Systems

The article discusses the problem of solving a fully fuzzy linear system of equations with a fuzzy rectangular matrix and a fuzzy right-hand side described by fuzzy triangular numbers in a form of deviations from the mean. A solution algorithm based on finding pseudo-solutions of systems of linear equations and corresponding software is formed. Various examples of created software application for arbitrary fuzzy linear systems are given.

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.

The use of negative penalty functions in linear systems of equations

2006 ◽  
Vol 462 (2074) ◽  
pp. 2965-2975 ◽  
Author(s):  
Harm Askes ◽  
Sinniah Ilanko
Keyword(s):  
Exact Solution ◽  
Linear Systems ◽  
Vibration Analysis ◽  
Linear Equations ◽  
Penalty Functions ◽  
Penalty Parameter ◽  
Systems Of Equations ◽  
Mathematical Proofs ◽  
Systems Of Linear Equations ◽  
Linear Systems Of Equations

Contrary to what is commonly thought, it is possible to obtain convergent results with negative (rather than positive) penalty functions. This has been shown and proven on various occasions for vibration analysis, but in this contribution it will also be shown and proven for systems of linear equations subjected to one or more constraints. As a key ingredient in the developed arguments, a pseudo-force is identified as the derivative of the constrained degree of freedom with respect to the inverse of the penalty parameter. Since this pseudo-force can be proven to be constant for large absolute values of the penalty parameter, it follows that the exact solution is bounded by the results obtained with negative and positive penalty parameters. The mathematical proofs are presented and two examples are shown to illustrate the principles.

scholarly journals “It Just Works Better”: Introducing the 2:1 Model of Co-Teaching in Teacher Preparation

2018 ◽  
Vol 36 (2) ◽  
Author(s):  
Christina M. Tschida ◽  
Judith J. Smith ◽  
Elizabeth A. Fogarty
Keyword(s):  
Teacher Preparation ◽  
Student Teachers ◽  
Student Teaching ◽  
Teacher Candidates ◽  
Cooperating Teachers ◽  
Teaching Experience ◽  
First Year ◽  
Cooperating Teacher ◽  
Teaching Standards ◽  
Professional Teaching

Many issues influence reform in teacher preparation including national accountability efforts, professional teaching standards, and local or regional factors. This study examines a rurally-located teacher education program’s efforts to reform clinical preparation through co-teaching. Researchers argue that their adaption of the typical one-to-one (1:1) model of co-teaching to a two-to-one (2:1) model, where two teacher candidates work collaboratively with one cooperating teacher, greatly enhances the student teaching experience. This phenomenological research describes the first year of implementation. Despite cooperating teacher concerns about teacher candidates being prepared for their own classrooms, student teachers learned valuable lessons in collaboration and co-planning, built strong relationships with peers and cooperating teachers, and greatly impacted K-6 student learning. Implications suggest a 2:1 co-teaching model of student teaching allows for fewer placements, which ultimately allows selection of quality cooperating teachers who mentor teacher candidates in powerful ways.  

scholarly journals More on Generalizations and Modifications of Iterative Methods for Solving Large Sparse Indefinite Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Jen-Yuan Chen ◽  
David R. Kincaid ◽  
Yu-Chien Li
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Linear Equations ◽  
Indefinite Systems ◽  
Systems Of Linear Equations ◽  
Indefinite Linear Systems

Continuing from the works of Li et al. (2014), Li (2007), and Kincaid et al. (2000), we present more generalizations and modifications of iterative methods for solving large sparse symmetric and nonsymmetric indefinite systems of linear equations. We discuss a variety of iterative methods such as GMRES, MGMRES, MINRES, LQ-MINRES, QR MINRES, MMINRES, MGRES, and others.

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