Browsing Mathematics & Statistics Faculty by Issue Date
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A note on the fixedpoint iteration for the matrix equations $X\pm A^*X^{1}A=I$
(Elsevier, 2008)The fixedpoint iteration is a simple method for finding the maximal Hermitian positive definite solutions of the matrix equations $X\pm A^*X^{1}A=I$ (the plus/minus equations). The convergence of this method ... 
Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case
(SIAM, 2009)In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical ... 
Detecting and solving hyperbolic quadratic eigenvalue problems
(SIAM, 2009)Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A + \lambda B + C$ are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with ... 
An Improved Arc Algorithm for Detecting Definite Hermitian Pairs
(SIAM, 2009)A 25year old and somewhat neglected algorithm of Crawford and Moon attempts to determine whether a given Hermitian matrix pair $(A,B)$ is definite by exploring the range of the function $f(x) = x^*(A + iB)x/x^*(A + ... 
Solving a structured quadratic eigenvalue problem by a structurepreserving doubling algorithm
(SIAM, 2010)In studying the vibration of fast trains, we encounter a palindromic quadratic eigenvalue problem (QEP) $(\lambda^2 A^T + \lambda Q + A)z = 0$, where $A, Q \in \mathbb{C}^{n \times n}$ and $Q^T = Q$. Moreover, the matrix ... 
The matrix equation $X+A^TX^{1}A=Q$ and its application in nano research
(SIAM, 2010)The matrix equation $X+A^TX^{1}A=Q$ has been studied extensively when $A$ and $Q$ are real square matrices and $Q$ is symmetric positive definite. The equation has positive definite solutions under suitable conditions, ... 
On Newton's method and Halley's method for the principal $p$th root of a matrix
(Elsevier, 2010)If $A$ is a matrix with no negative real eigenvalues and all zero eigenvalues of $A$ are semisimple, the principal $p$th root of $A$ can be computed by Newton's method or Halley's method, with a preprocessing procedure ... 
Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory
(Elsevier, 2010)We determine and compare the convergence rates of various fixedpoint iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory. 
Complex symmetric stabilizing solution of the matrix equation $X+A^{T}X^{1}A=Q$
(Elsevier, 2011)We study the matrix equation $X+A^{T}X^{1}A=Q$, where $A$ is a complex square matrix and $Q$ is complex symmetric. Special cases of this equation appear in Green's function calculation in nano research and also in the ... 
On a nonlinear matrix equation arising in nano research
(SIAM, 2012)The matrix equation $X+A^{T}X^{1}A=Q$ arises in Green's function calculations in nano research, where $A$ is a real square matrix and $Q$ is a real symmetric matrix dependent on a parameter and is usually indefinite. ... 
Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research
(Elsevier, 2012)The Green's function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form $X+(C^*+{\rm i} \eta D^*)X^{1}(C+{\rm i} \eta D)=R+{\rm i}\eta P$, where $R$ ... 
Note on NordhausGaddum problems for Colin de Verdiere type parameters
(Public Knowledge Network, 2013)We establish the bounds 4 on the Nordhaus Gaddum sum upper bound multipliers for all graphs G, in connections with certain Colin de Verdi ere type graph parameters. The NordhausGaddum sum lower bound is conjectured ... 
Parameters Related to TreeWidth, Zero Forcing, and Maximum Nullity of a Graph
(Wiley Periodicals, Inc., 2013)Treewidth, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the ... 
Monotone convergence of Newtonlike methods for Mmatrix algebraic Riccati equations
(Springer, 2013)For the algebraic Riccati equation whose four coefficient matrices form a nonsingular $M$matrix or an irreducible singular $M$matrix $K$, the minimal nonnegative solution can be found by Newton's method and the doubling ... 
On algebraic Riccati equations associated with Mmatrices
(Elsevier, 2013)We consider the algebraic Riccati equation for which the four coefficient matrices form an $M$matrix $K$. When $K$ is a nonsingular $M$matrix or an irreducible singular $M$matrix, the Riccati equation is known to have ... 
Iterative methods for a linearly perturbed algebraic matrix Riccati equation arising in stochastic control
(Taylor & Francis, 2013)We start with a discussion of coupled algebraic Riccati equations arising in the study of linearquadratic optimal control problem for Markov jump linear systems. Under suitable assumptions, this system of equations has ... 
Colin de Verdiere parameters of chordal graphs
(International Linear Algebra Society, 201301)The Colin de Verdi`ere parameters mu and nu are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal ... 
On the null space struture associted with trees and cycles
(The Charles Babbage Research Centre, 201302)In this work, we study the structure of the null spaces of matrices associated with graphs. Our primary tool is utilizing Schur complements based on certain collections of independent vertices. This idea is applied in ... 
Minimum number of distinct eigenvalues of graphs
(International Linear Algebra Society, 201309)The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to ... 
On the relationship between zero forcing number and certain graph coverings
(2014)The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced ...