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.

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.

A Convergence Analysis of the MINRES Method for Some Hermitian Indefinite Systems

2017 ◽  
Vol 7 (4) ◽  
pp. 827-836
Author(s):  
Ze-Jia Xie ◽  
Xiao-Qing Jin ◽  
Zhi Zhao
Keyword(s):  
Convergence Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Coefficient Matrix ◽  
Indefinite Systems ◽  
Indefinite Linear Systems ◽  
Theoretical Results

AbstractSome convergence bounds of the minimal residual (MINRES) method are studied when the method is applied for solving Hermitian indefinite linear systems. The matrices of these linear systems are supposed to have some properties so that their spectra are all clustered around ±1. New convergence bounds depending on the spectrum of the coefficient matrix are presented. Some numerical experiments are shown to demonstrate our theoretical results.

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.

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