A Structure-Preserving Curve for Symplectic Pairs and Its Applications

Abstract

The main purpose of this paper is the study of numerical methods for the maximal solution of the matrix equation $X+A^*X^{-1}A = Q$, where $Q$ is Hermitian positive definite. We construct a smooth curve parameterized by $t\ge 1$ of symplectic pairs with a special structure, in which the curve passes through all iteration points generated by the known numerical methods, including the fixed-point iteration, the structure-preserving doubling algorithm (SDA), and Newton's method provided that $A^*Q^{-1}A=AQ^{-1}A^*$. In the theoretical section, we give a necessary and sufficient condition for the existence of this structure-preserving curve for each parameter $t\ge1$. We also study the monotonicity and boundedness properties of this curve. In the application section, we use this curve to measure the convergence rates of those numerical methods. Numerical results illustrating these solutions are also presented.