TY - JOUR

T1 - On the semigroup property for some structured iterations

AU - Lin, Matthew M.

AU - Chiang, Chun Yueh

N1 - 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.

UR - http://www.scopus.com/inward/record.url?scp=85079322994&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85079322994&partnerID=8YFLogxK

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 -