Fixed-Point Iteration Schemes to Solve Symmetric Algebraic Riccati Equation
Pages 3516 - 3532
Abstract
Nonlinear matrix equations have important applications in optimal control problems. This study introduces parametric iterative methods aimed at determining solutions for the symmetric algebraic Riccati equation (SARE) expressed as:These methods leverage fixed-point and weight-splitting schemes for computation. The study demonstrates the convergence of the proposed schemes to nonnegative solutions of the SARE in specific situations. Additionally, various numerical examples illustrate the effectiveness of these schemes in solving several instances of SAREs.
References
[1]
W.F. Arnold III, Numerical Solution of Algebraic Matrix Riccati Equations (Technical report, Naval Weapons Center China Lake CA, 1984)
[2]
Bai Z-Z The convergence of the two-stage iterative method for Hermitian positive definite linear systems Appl. Math. Lett. 1998 11 2 1-5
[3]
Bai Z-Z Splitting iteration methods for non-Hermitian positive definite systems of linear equations Hokkaido Math. J. 2007 36 4 801-814
[4]
Bai Z-Z, Sun J-C, and Wang D-R A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations Comput. Math. Appl. 1996 32 12 51-76
[5]
Bai Z-Z, Golub GH, and Pan J-Y Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems Numer. Math. 2004 98 1 1-32
[6]
A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. SIAM (1994)
[7]
S. Bhagat, S.D. Joshi, Quantification of differential information using matrix pencil and its applications. Circuits Syst. Signal Process. pp. 1–24 (2022)
[8]
R. Byers, Hamiltonian and Symplectic Algorithms for the Algebraic Riccati Equation. Cornell University (1983)
[9]
J.E. Dennis, Jr. R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM (1996)
[10]
R. Erfanifar, M. Hajarian, Splitting iteration methods to solve non-symmetric algebraic Riccati matrix equation . Numer. Algorithms. pp. 1–26 (2023)
[11]
R. Erfanifar, K. Sayevand, M. Hajarian, Solving system of nonlinear matrix equations over Hermitian positive definite matrices. Linear Multilinear A pp. 1–34 (2022)
[12]
Erfanifar R, Sayevand K, and Esmaeili H A novel iterative method for the solution of a nonlinear matrix equation Appl. Numer. Math. 2020 153 503-518
[13]
Erfanifar R, Sayevand K, and Hajarian M An efficient inversion-free method for solving the nonlinear matrix equation J. Franklin Inst. 2022 359 7 3071-3089
[14]
Fiedler M and Ptak V On matrices with non-positive off-diagonal elements and positive principal minors Czechoslov. Math. J. 1962 12 3 382-400
[15]
Guan J and Lu L New alternately linearized implicit iteration for M-matrix algebraic Riccati equations J. Mathe. Study 2017 50 1 54-64
[16]
Guo C-H Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices SIAM J. Matrix Anal. Appl. 2001 23 1 225-242
[17]
Hajarian M A finite iterative method for solving the general coupled discrete-time periodic matrix equations Circuits Syst. Signal Process. 2015 34 1 105-125
[18]
Hernández-Verón MA and Romero N Solving symmetric algebraic Riccati equations with high order iterative schemes Mediterr. J. Math. 2018 15 2 51
[19]
Hernández-Verón MA and Romero N An improvement of the Newton method for solving symmetric algebraic Riccati equations Mediterr. J. Math. 2023 20 5 261
[20]
P. Lancaster, M, Tismenetsky, The Theory of Matrices: with Applications. Elsevier (1985)
[21]
Rehman A and Kyrchei II Solving and algorithm to system of Quaternion Sylvester-type matrix equations with -Hermicity Adv. Appl. Clifford Algebras 2022 32 4 1-17
[22]
Rehman A, Kyrchei II, Khan IA, Nasir M, and Ali I Closed-form formula for a classical system of matrix equations Arab. J. Basic Appl. Sci 2022 29 1 258-268
[23]
R.S. Varga, Iterative Analysis. Springer (1962)
[24]
Ye F, Chen J, Gao L, Nie W, and Sun Q A mixing matrix estimation algorithm for the time-delayed mixing model of the underdetermined blind source separation problem Circuits Syst. Signal Process. 2019 38 4 1889-1906
[25]
Zeng L, Zhang X, Chen L, Cao T, and Yang J Deterministic construction of Toeplitzed structurally chaotic matrix for compressed sensing Circuits Syst. Signal Process. 2015 34 3 797-813
Recommendations
On inexact Newton methods based on doubling iteration scheme for symmetric algebraic Riccati equations
Numerical methods for solving the symmetric algebraic Riccati equation by using Newton's method are considered in this paper. Instead of direct methods, a fast doubling iteration scheme is applied to inexactly solve the Lyapunov equations arising in ...
Splitting iteration methods to solve non-symmetric algebraic Riccati matrix equation
AbstractThe non-symmetric algebraic Riccati equation (NARE) occurs in several areas of applied mathematics and engineering such as spectral factorizations of rational matrix functions, constructive rational matrix functions, transport theory, optimal ...
On modified HSS iteration methods for continuous Sylvester equations
We introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semidefinite matrices. It is found ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
Publisher
Birkhauser Boston Inc.
United States
Publication History
Published: 29 March 2024
Accepted: 23 February 2024
Revision received: 22 February 2024
Received: 19 January 2023
Author Tags
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 06 Jan 2025