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Chun-Hua Guo

Publications and source records attributed to Chun-Hua Guo.

7 recordsLinked to original sources
arXiv2024

Comments on finite termination of the generalized Newton method for absolute value equations

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) $Ax-|x|=b$. The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever $ρ(|A^{-1}|)<1/3$. We also present new results for the case where $A-I$ is a nonsingular $M$-matrix or an irreducible singular $M$-matrix. When $A-I$ is an irreducible singular $M$-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.

math.NA↗
arXiv2022

Computing the extremal nonnegative solutions of the M-tensor equation with a nonnegative right side vector

We consider the tensor equation whose coefficient tensor is a nonsingular M-tensor and whose right side vector is nonnegative. Such a tensor equation may have a large number of nonnegative solutions. It is already known that the tensor equation has a maximal nonnegative solution and a minimal nonnegative solution (called extremal solutions collectively). However, the existing proofs do not show how the extremal solutions can be computed. The existing numerical methods can find one of the nonnegative solutions, without knowing whether the computed solution is an extremal solution. In this paper, we present new proofs for the existence of extremal solutions. Our proofs are much shorter than existing ones and more importantly they give numerical methods that can compute the extremal solutions. Linear convergence of these numerical methods is also proved under mild assumptions. Some of our discussions also allow the coefficient tensor to be a Z-tensor or allow the right side vector to have some negative elements.

math.NA↗
arXiv2018

A study of Schröder's method for the matrix $p$th root using power series expansions

When $A$ is a matrix with all eigenvalues in the disk $|z-1|<1$, the principal $p$th root of $A$ can be computed by Schröder's method, among many other methods. In this paper we present a further study of Schröder's method for the matrix $p$th root, through an examination of power series expansions of some sequences of scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schröder's method, a monotonic convergence result when $A$ is a nonsingular $M$-matrix, and a structure preserving result when $A$ is a nonsingular $M$-matrix or a real nonsingular $H$-matrix with positive diagonal entries.

math.NA↗
arXiv2017

A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.

math.NA↗
arXiv2012

On algebraic Riccati equations associated with M-Matrices

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 a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.

math.NA↗