Matthew M. Lin

Chun-Yueh Chiang, Matthew M. Lin

The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google's PageRank. The shift strategy arises from the concept investigated by Brauer [1] for changing the value of an eigenvalue of a matrix to the desired one, while keeping the remaining eigenvalues and the original eigenvectors unchanged. The idea of shifting distinct eigenvalues can easily be generalized by Brauer's idea. However, shifting an eigenvalue with multiple multiplicities is a challenge issue and worthy of our investigation. In this work, we propose a new way for updating an eigenvalue with multiple multiplicities and thoroughly analyze its corresponding Jordan canonical form after the update procedure.

Matthew M. Lin, Chun-Yueh Chiang

Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to apply the fixed-point iteration with usually only the linear convergence rate. To advance the existing methods, we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order. We realize our way by starting with examples of solving the scalar equations and, also, connect this method with some well-known equations including, but not limited to, the Stein matrix equation, the generalized eigenvalue problem, the generalized nonlinear matrix equation, the discrete-time algebraic Riccati equations to express the capacity of this method.

Matthew M. Lin, Chun-Yueh Chiang

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a discrete-type flow depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.

Matthew M. Lin

The study of solving the inverse eigenvalue problem for nonnegative matrices has been around for decades. It is clear that an inverse eigenvalue problem is trivial if the desirable matrix is not restricted to a certain structure. Provided with the real spectrum, this paper presents a numerical procedure, based on the induction principle, to solve two kinds of inverse eigenvalue problems, one for nonnegative matrices and another for symmetric nonnegative matrices. As an immediate application, our approach can offer not only the sufficient condition for solving inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices, but also a quick numerical way to solve inverse eigenvalue problem for stochastic matrices. Numerical examples are presented for problems of relatively larger size.

Chun-Yueh Chiang, Matthew M. Lin

We propose a new way for speeding up the search of the maximal solution $X_+$ of $X + A^\top X^{-1} A = Q$. It is known that the speed of convergence of traditional approaches for solving this problem depends highly on the spectral radius $ρ(X_+^{-1}A)$. If $ρ(X_+^{-1}A)$ is close to one or equal to one, the iterations of traditional approaches converges very slowly or does not converge. Our goal is to come up with a shifting tactic to remove the singularities embedded in $ρ(X_+^{-1}A)$. Finally, an example is used to demonstrate the capacity of our method.

Matthew M. Lin, Chun-Yueh Chiang

We consider the $\top$-Stein equation $X = AX^\top B + C$, where the operator $(\cdot)^\top$ denotes the transpose ($\top$) of a matrix. In the first part of this paper, we analyze necessary and sufficient conditions for the existence and uniqueness of the solution $X$. In the second part, a numerical algorithm for solving $\top$-Stein equation is given under the solvability conditions.

Chun-Yueh Chiang, Matthew M. Lin

This paper analyzes a special instance of nonsymmetric algebraic matrix Riccati equations arising from transport theory. Traditional approaches for finding the minimal nonnegative solution of the matrix Riccati equations are based on the fixed point iteration and the speed of the convergence is linear. Relying on simultaneously matrix computation, a structure-preserving doubling algorithm (SDA) with quadratic convergence is designed for improving the speed of convergence. The difficulty is that the double algorithm with quadratic convergence cannot guarantee to work all the time. Our main trust in this work is to show that applied with a suitable shifted technique, the SDA is guaranteed to converge quadratically with no breakdown. Also, we modify the conventional simple iteration algorithm in the critical case to dramatically improve the speed of convergence. Numerical experiments strongly suggest that the total number of computational steps can be significantly reduced via the shifting procedure.

Chun-Yueh Chiang, Matthew M. Lin, Xiao-Qing Jin

Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method. {Having theoretical interest, we provide in the appendix a necessary and sufficient condition for the existence of a $2\times 2$ real matrix, or even a nonnegative matrix, with prescribed eigenvalues, singular values, and main diagonal entries.

Matthew M. Lin, Chun-Yueh Chiang

Algebraic Riccati equations are encountered in many applications of control and engineering problems, e.g., LQG problems and $H^\infty$ control theory. In this work, we study the properties of one type of discrete-time algebraic Riccati equations. Our contribution is twofold. First, we present sufficient conditions for the existence of a unique positive definite solution. Second, we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order. Numerical experiments strongly support that our approach performs extremely well even in the almost critical case. As a byproduct, we provide show that this method is capable of computing the unique negative definite solution, once it exists.

Bo Dong, Matthew M. Lin, Haesun Park

Integer data sets frequently appear in many applications in sciences and technology. To analyze these, integer low rank approximation has received much attention due to its capacity of representing the results in integers preserving the meaning of the original data sets. To our knowledge, none of previously proposed techniques developed for real numbers can be successfully applied, since integers are discrete in nature. In this work, we start with a thorough review of algorithms for solving integer least squares problems, {and} then develop a block coordinate descent method based on the integer least squares estimation to obtain the integer low rank approximation of integer matrices. The numerical application on association analysis and numerical experiments on random integer matrices are presented. Our computed results seem to suggest that our method can find a more accurate solution than other existing methods for continuous data sets.

Zhi Zhao, Xiao-Qing Jin, Matthew M. Lin

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of space-time fractional advection-diffusion equations. To start with, an implicit method based on two-sided Grünwald formulae is proposed with a discussion of the stability and consistency. Then, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual ({preconditioned} CGNR) method, with an easily constructed preconditioner, are developed. Importantly, because the resulting systems are Topelitz-like, the fast Fourier transform can be applied to significantly reduce the computational cost. Numerical experiments are implemented to show the efficiency of our preconditioner, even with cases of variable coefficients.

Matthew M. Lin, Chun-Yueh Chiang

This work is to provide a comprehensive treatment of the relationship between the theory of the generalized (palindromic) eigenvalue problem and the theory of the Sylvester-type equations. Under a regularity assumption for a specific matrix pencil, we show that the solution of the $\star$-Sylvester matrix equation is uniquely determined and can be obtained by considering its corresponding deflating subspace. We also propose an iterative method with quadratic convergence to compute the stabilizing solution of the $\star$-Sylvester matrix equation via the well-developed palindromic doubling algorithm. We believe that our discussion is the first which implements the tactic of the deflating subspace for solving Sylvester equations and could give rise to the possibility of developing an advanced and effective solver for different types of matrix equations.