The Shifting Technique for Solving a Nonsymmetric Algebraic Riccati Equation
For researchers solving Riccati equations in transport theory, this work provides a more reliable and faster algorithm, though it is an incremental improvement over existing SDA methods.
This paper addresses the slow convergence of fixed-point iteration for solving nonsymmetric algebraic Riccati equations from transport theory. By applying a shifting technique to the structure-preserving doubling algorithm (SDA), they guarantee quadratic convergence without breakdown, and also modify the simple iteration for critical cases, significantly reducing computational steps.
This paper analyzes a special instance of nonsymmetric algebraic matrix Riccati equations arising from transport theory. Traditional approaches for finding the minimal nonnegative solution of the matrix Riccati equations are based on the fixed point iteration and the speed of the convergence is linear. Relying on simultaneously matrix computation, a structure-preserving doubling algorithm (SDA) with quadratic convergence is designed for improving the speed of convergence. The difficulty is that the double algorithm with quadratic convergence cannot guarantee to work all the time. Our main trust in this work is to show that applied with a suitable shifted technique, the SDA is guaranteed to converge quadratically with no breakdown. Also, we modify the conventional simple iteration algorithm in the critical case to dramatically improve the speed of convergence. Numerical experiments strongly suggest that the total number of computational steps can be significantly reduced via the shifting procedure.