TY - JOUR
T1 - On the semigroup property for some structured iterations
AU - Lin, Matthew M.
AU - Chiang, Chun Yueh
N1 - Funding Information:
The first author was supported by the Ministry of Science and Technology of Taiwan under grants 107-2115-M-006-007-MY2 and 108-2636-M-006-006.The second author was supported by the Ministry of Science and Technology of Taiwan under grant 108-2115-M-150-002.The authors would like to thank the editor and anonymous referees for their highly valuable comments and suggestions which helped to improve the manuscript. This research work is partially supported by the Ministry of Science and Technology, Taiwan and the National Center for Theoretical Sciences in Taiwan . The corresponding author (Chun-Yueh Chiang) would like to thank Prof. Shinya Miyajima in Iwate University for his invitation to give a talk about some related works at the NMSP2019 workshop and the support from the the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Funding Information:
The authors would like to thank the editor and anonymous referees for their highly valuable comments and suggestions which helped to improve the manuscript. This research work is partially supported by the Ministry of Science and Technology, Taiwan and the National Center for Theoretical Sciences in Taiwan . The corresponding author (Chun-Yueh Chiang) would like to thank Prof. Shinya Miyajima in Iwate University for his invitation to give a talk about some related works at the NMSP2019 workshop and the support from the the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/8/15
Y1 - 2020/8/15
N2 - 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.
AB - 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.
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U2 - 10.1016/j.cam.2020.112768
DO - 10.1016/j.cam.2020.112768
M3 - Article
AN - SCOPUS:85079322994
SN - 0377-0427
VL - 374
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
M1 - 112768
ER -