On the semigroup property for some structured iterations
This provides a new method for accelerating convergence in solving nonlinear matrix equations, which are fundamental in science and engineering, though the approach is demonstrated only on examples without empirical validation.
The authors propose a technique leveraging a semigroup property to solve nonlinear matrix equations with arbitrarily high-order convergence, demonstrated on scalar equations and several matrix equations including Stein, Riccati, and generalized eigenvalue problems.
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.