Alternating-directional doubling algorithm for \(M\)-matrix algebraic Riccati equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.65045
DOI: 10.1137/110835463
author: Wei-Chao Wang
author: Ren-Cang Li
author: Weiguo Wang
published in: SIAM Journal on Matrix Analysis and Applications
publication date: 2012-08-23T00:00:00Z
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 6070723
zbMATH Keywords: matrix Riccati equation
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: minimal nonnegative solution
zbMATH Keywords: doubling algorithm
zbMATH Keywords: bilinear transformation
zbMATH Keywords: Cayley transforms
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1137/110835463
OpenAlex ID: W2012272511
Recommended article: Highly accurate doubling algorithms for \(M\)-matrix algebraic Riccati equations
Recommended article: A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation
Recommended article: On the Doubling Algorithm for a (Shifted) Nonsymmetric Algebraic Riccati Equation
Recommended article: Doubling algorithm for continuous-time algebraic Riccati equation
Recommended article: ALGORITHMS FOR THE RICCATI EQUATION WITH A NONSINGULAR M-MATRIX
Recommended article: A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations
Recommended article: Doubling Algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation
Recommended article: A new two‐phase structure‐preserving doubling algorithm for critically singular M‐matrix algebraic Riccati equations
Recommended article: Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction
Recommended article: New Alternately Linearized Implicit Iteration for M-matrix Algebraic Riccati Equations
title: Alternating-directional doubling algorithm for \(M\)-matrix algebraic Riccati equations
review text: A new doubling algorithm for computing the minimal nonnegative solution of an \(M\)-matrix algebraic Riccati equation \(XDX -AX-XB+ C = 0\) is presented and analysed. The matrices \(A\), \(B\), \(C\), and \(D\) are the blocks of the matrix NEWLINE\[NEWLINE W = \begin{pmatrix} B & -D \\ -C & A \end{pmatrix} \, , NEWLINE\]NEWLINE where \(W\) is a nonsingular or an irreducible singular \(M\)-matrix. At first, some known results on \(M\)-matrices are summarized and a new result on optimizing the product of two spectral radii of the generalized Cayley transforms of two \(M\)-matrices is proved. Then, the new alternating-directional doubling algorithm is described and analysed. The difference to known algorithms is in the initial setup. In the new algorithm two parameters are introduced to get a faster algorithm when the matrices \(A\) and \(B\) differ in magnitude. Furthermore, the convergence rates of the new algorithm and known algorithms are compared. It is shown theoretically and by numerical examples that the new doubling algorithm is faster than the doubling algorithms proposed by \textit{X.-X. Guo, W.-W. Lin} and \textit{S.-F. Xu} [Numer. Math. 103, No. 3, 393--412 (2006; Zbl 1097.65055)] and by \textit{D. A. Bini, B. Meini} and \textit{F. Poloni} [Numer. Math. 116, No. 4, 553--578 (2010; Zbl 1202.15016)].
reviewed by: Michael Jung
description: scientific article; zbMATH DE number 6070723