On nonlinear matrix equations from the first standard form

Ren-Cang Li (Department of Mathematics, University of Texas, Arlington, Tx., U.S.A.; and Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong)

Abstract

In numerically solving nonlinear matrix equations, including algebraic Riccati equations, that are associated with the eigenspaces of certain regular matrix pencils by the doubling algorithms, the matrix pencils must first be brought into one of the two standard forms. Conversely, each standard form leads to a kind of nonlinear matrix equations, which are of interest in their own right. In this paper, we are concerned with the nonlinear matrix equations associated with the first standard form (SF1). Under the nonnegativeness assumption, we investigate solution existence and the convergence of the doubling algorithm. We obtain several results that resemble the ones for SF1 derived from an $M$-matrix algebraic Riccati equation.

Keywords

first standard form, SF1, nonnegative matrix, $M$-matrix, minimal nonnegative solution, doubling algorithm

2010 Mathematics Subject Classification

Primary 15A24, 65F30. Secondary 65H10.

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J. Xue was supported in part by the NNSFC, China 12171101 and Laboratory of Mathematics for Nonlinear Science, Fudan University, China.

R.-C. Li was supported in part by NSF grants NSF DMS-1719620 and DMS-2009689.

Received 22 February 2022

Accepted 25 March 2022

Published 12 September 2022