ESTIMATING THE CONVERGENCE RATE OF FUNCTIONAL ITERATIONS FOR SOLVING QUADRATIC MATRIX EQUATIONS ARISING IN HYPERBOLIC QUADRATIC EIGENVALUE PROBLEMS (Study on Nonlinear Analysis and Convex Analysis)

Abstract

We consider Bernoulli's method for solving quadratic matrix equations (QMEs) having form Q(X) = AX^2 +BX+ C = 0 arising in hyperbolic quadratic eigenvalue problems (QEPs) and quasi-birth-death problems (QBDs) where A, B, C ∈ R^[m×m] satisfy Esenfeld's condition [8]. First, we analyze the exsistence of a solution and the convergence of the methods. Second, we sharpen bounds of the rates of convergence. Finally, in numerical experimentations, we show that the modified bounds give appropriate estimations of the numbers of iterations.