Linzhang Lu

Newton iterations for a non‐symmetric algebraic Riccati equation

The computation of the minimal positive solution of a non-symmetric algebraic Riccati equation arising in transport theory is considered. It was shown in (SIAM J Matrix Anal Appl, submitted) that this can be done via only computing the minimal positive solution of a vector equation, which is derived from special form of the solutions of the Riccati equation and by exploitation of the special structure of the coefficient matrices of the Riccati equation. In this paper, the Newton method is developed for the vector equation. The Newton method is more simple and efficient than the corresponding Newton method directly for original Riccati equation and can preserve the form that any solution of the Riccati equation must satisfy. Combination of the simple iteration and the Newton iteration is also considered. Numerical examples are given. Copyright

PROPERTIES OF A QUADRATIC MATRIX EQUATION AND THE SOLUTION OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION

Abstract We discuss some properties of a quadratic matrix equation with some restrictions, then use these results on the algebraic Riccati equation to get a new algorithm. The algorithm sufficiently takes account of the structure of the associated matrix; hence it is very effective.

Perron complement and Perron root

For a nonnegative irreducible matrix A, this paper is concerned with the estimation and determination of the unique Perron root or spectral radius of A. We present a new method that utilizes the relation between Perron roots of the nonnegative matrix and its (generalized) Perron complement. Several numerical examples are given to show that our method is effective, at least, for some classes of nonnegative matrices.

An iterative algorithm for the solution of the discrete-time algebraic Riccati equation

Abstract The discrete-time algebraic Riccati equation is solved in this study by an iterative algorithm for the square root of a squared Hamiltonian matrix, which is obtained from the S + −1 transformation of the symplectic pencil associated with the Riccati equation. The symplectic Givens and n × n block-diagonal orthogonal transformations are used before the iterative process so that the iteration is structure-preserving and can achieve on average 60% reduction of computation time compared with the QZ algorithm. A formal analysis for roundoff errors and some numerical examples are also given.

On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem

Summary.In this paper, we study the inverse eigenvalue problem of a specially structured Jacobi matrix, which arises from the discretization of the differential equation governing the axial of a rod with varying cross section (Ram and Elhay 1998 Commum. Numer. Methods Engng. 14 597-608). We give a sufficient and some necessary conditions for such inverse eigenvalue problem to have solutions. Based on these results, a simple method for the reconstruction of a Jacobi matrix from eigenvalues is developed. Numerical examples are given to demonstrate our results.

Implicit numerical approximation scheme for the fractional Fokker-Planck equation

In this paper, we consider an anomalous subdiffusion process, governed by fractional Fokker-Planck equation. An effective numerical method for approximating Fokker-Planck equation in a bounded domain is presented. The stability and convergence of the numerical method are analyzed. Some numerical examples are presented to show the application of the present technique. The numerical results exhibit the good performance of our theoretical analysis.

Spectral properties of a class of matrix splitting preconditioners for saddle point problems

Based on the accelerated Hermitian and skew-Hermitian splitting iteration scheme (Bai and Golub, 2007), we propose a new two-parameter matrix splitting preconditioner in this paper. Spectral properties of the preconditioned matrix are analyzed in detail. Furthermore, based on this preconditioner, an improved version of matrix splitting preconditioner is presented and analyzed. Finally, performance of the preconditioners is compared by using GMRES( m ) as an iterative solver on linear systems arising from the discretization of Stokes and Navier-Stokes equations.

The inverse eigenvalue problem of generalized reflexive matrices and its approximation

A real symmetric unipotent matrix P is said to be generalized reflection matrix. A real matrix A is said to be a generalized reflexive matrix with respect to generalized reflection matrix dual (P,Q) if A = PAQ. This paper involves related inverse eigenvalue problems of generalized reflexive matrices and their optimal approximation. Necessary and sufficient conditions for the solvability of the problem are derived,the general expression of the solution is given. The optimal approximate solution is also provided.

A shift-splitting preconditioner for a class of block two-by-two linear systems☆

Abstract In this paper, we construct a shift-splitting preconditioner for a class of block two-by-two linear systems. The proposed preconditioner is extracted from a stationary iterative method which is unconditionally convergent. Moreover, the eigenvalue distribution of the corresponding preconditioned matrix is studied. Numerical experiments are presented to show that our new preconditioner can be quite competitive when used to precondition Krylov subspace iterative methods such as GMRES.

Alternating-directional PMHSS iteration method for a class of two-by-two block linear systems ☆

Abstract In Bai et al. (2013), a preconditioned modified HSS (PMHSS) method was proposed for a class of two-by-two block systems of linear equations. In this paper, the PMHSS method is modified by adding one more parameter in the iteration. Convergence of the modified PMHSS method is guaranteed. Theoretic analysis and numerical experiment show that the modification improves the PMHSS method.