Some properties of generalized K-centrosymmetric H-matrices

Every n×n$n\times n$ generalized K-centrosymmetric matrix A   can be reduced into a 2×2$2\times 2$ block diagonal matrix (see [Z. Liu, H. Cao, H. Chen, A note on computing matrix–vector products with generalized centrosymmetric (centrohermitian) matrices, Appl. Math. Comput. 169 (2) (2005) 1332–1345]). This block diagonal matrix is called the reduced form of the matrix A. In this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices. Finally exploiting these properties, the development of structure-preserving algorithms for certain computations for generalized K-centrosymmetric H-matrices is discussed.     

Block tridiagonal reduction of perturbed normal and rank structured matrices

It is well known that if a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ solves the matrix equation f(A,A^H)=0$f(A,A^H)=0$, where f(x,y)$f(x,y)$ is a linear bivariate polynomial, then A is normal; A   and A^H$A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.     

Improved Newton’s method with exact line searches to solve quadratic matrix equation

In this paper, we study the matrix equation AX²+BX+C=0$A{X}^2+BX+C=0$, where A,B$A,B$ and C$C$ are square matrices. We give two improved algorithms which are better than Newton’s method with exact line searches to calculate the solution. Some numerical examples are reported to illustrate our algorithms.     

Total least squares fitting Michaelis–Menten enzyme kinetic model function

The Michaelis–Menten enzyme kinetic model f(x;a,b)=ax/(b+x)$f(x;a,b)=\mathit{ax}/(b+x)$, a,b>0$a,b>0$, is widely used in biochemistry, pharmacology, biology and medical research.     

Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f=f(y)$f=f(y)$ and g=g(y)$g=g(y)$ arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S^*(f,g)$S^{*}(f,g)$ of the Sylvester resultant matrix S(f,g)$S(f,g)$. In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S(f,g)$S(f,g)$, and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S^*(f,g)$S^{*}(f,g)$, and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented.     

A new algorithm for linear systems of the Pascal type

In this paper, we give an algorithm for solving linear systems of the Pascal matrices. The method is based on the explicit factorization of the Pascal matrices. The algorithm costs no multiplications and O(n²)$O\left(n^2\right)$ additions. The linear systems of the generalized Pascal matrices are also considered. Some examples are given.     

A modified harmonic block Arnoldi algorithm with adaptive shifts for large interior eigenproblems

The harmonic block Arnoldi method can be used to find interior eigenpairs of large matrices. Given a target point or shift τ$\tau$ to which the needed interior eigenvalues are close, the desired interior eigenpairs are the eigenvalues nearest τ$\tau$ and the associated eigenvectors. However, it has been shown that the harmonic Ritz vectors may converge erratically and even may fail to do so. To do a better job, a modified harmonic block Arnoldi method is coined that replaces the harmonic Ritz vectors by some modified harmonic Ritz vectors. The relationships between the modified harmonic block Arnoldi method and the original one are analyzed. Moreover, how to adaptively adjust shifts during iterations so as to improve convergence is also discussed. Numerical results on the efficiency of the new algorithm are reported.     

Improvements of preconditioned AOR iterative method for L-matrices

In this paper, we improve the preconditioned AOR method of linear systems considered by Evans et al. [The AOR iterative method for new preconditioned linear systems, Comput. Appl. Math. 132 (2001) 461–466]. In Evans’ paper, the coefficient matrix of linear system have to be an L  -matrix with _ai,i+1ai+1,i>0$a_{i,i+1}{a}_{i+1,i}>0$, i=1,…,n-1$i=1,\dots ,n-1$ and 0<_a1nan1<1$0<a_{1n}{a}_{n1}<1$. When _ai,i+1ai+1,i=0$a_{i,i+1}{a}_{i+1,i}=0$ for some i∈N={1,…,n-1}$i\in N=\{1,\dots ,n-1\}$, the preconditioned method is invalid. In order to solve the above problem, a new preconditioner is presented. Meanwhile, some recent results are improved.     

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H$H$-matrices. A truth is found that the class of conjugate generalized H$H$-matrices is a subclass of the class of generalized H$H$-matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H$H$-matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H$H$-matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.     

Equidistributed error mesh for problems with exponential boundary layers

In this paper we present a new piecewise-linear finite element mesh suitable for the discretization of the one-dimensional convection–diffusion equation -εu^″-bu^′=0$-\epsilon u^{″}-{\mathit{bu}}^{\prime }=0$, u(0)=0$u(0)=0$, u(1)=1$u(1)=1$. The solution to this equation exhibits an exponential boundary layer which occurs also in more complicated convection–diffusion problems of the form -εΔu-b∂u/∂x+cu=f$-\epsilon \mathrm{\Delta }u-b\partial u/\partial x+\mathit{cu}=f$. The new mesh is based on the equidistribution of the interpolation error and it takes into account finite computer arithmetic. It is demonstrated numerically that for the above problem, the new mesh has remarkably better convergence properties than the well-known Shishkin and Bakhvalov meshes.     

The RKGL method for the numerical solution of initial-value problems

We introduce the RKGL method for the numerical solution of initial-value problems of the form y^′=f(x,y)$y^{\prime }=f(x,y)$, y(a)=α$y(a)=\alpha$. The method is a straightforward modification of a classical explicit Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f(x,y)$f(x,y)$ needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ah^r+1+Bh^2m${\mathit{Ah}}^{r+1}+{\mathit{Bh}}^{2m}$, where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.     

On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm _ξk≥1${\xi }_k\ge 1$ of the inner linear system at outer iteration k+1$k+1$ and can be considerably weaker than the condition _ξk≤ξ<1${\xi }_k\le \xi <1$ with ξ$\xi$ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖_rk‖$\Vert r_k\Vert$ be the residual norm of approximating eigenpair at outer iteration k$k$. Then all the available cubic and quadratic convergence results require _ξk=O(‖_rk‖)${\xi }_k=O\left(\Vert r_k\Vert \right)$ and _ξk≤ξ${\xi }_k\le \xi$ with a fixed ξ$\xi$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that _ξk≤ξ${\xi }_k\le \xi$ with a constant ξ<1$\xi <1$ not near one, _ξk=1−O(‖_rk‖)${\xi }_k=1-O\left(\Vert r_k\Vert \right)$ and _ξk=1−O(‖_rk‖²)${\xi }_k=1-O\left({\Vert r_k\Vert }^2\right)$, respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.     

Necessary and sufficient conditions of some strong optimal solutions to the interval linear programming

This paper considers optimal solutions of general interval linear programming problems. Necessary and sufficient conditions of (A,b)$(\mathbf{A},\mathbf{b})$-strong and (A,b,c)$(\mathbf{A},\mathbf{b},\mathbf{c})$-strong optimal solutions to the interval linear programming with inequality constraints are proposed. The features of the proposed methods are illustrated by some examples.     

A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type method to compute selected eigenpairs of the product eigenvalue problem _Am?_A1x=λx,$A_m?A_{1}x=\lambda x,$ where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.