An Optimization Approach to the Shift Selection Problem in Alternating Direction Implicit Iteration

The ADI (alternating direction implicit) iteration is an efficient numerical method for solving the large Lyapunov equations. In this paper, we study the selection of shift parameters for the ADI method which involves the solution of min-max problem-ellipse method. Based on a pair of Arnoldi processes, we first obtain the ellipse which contains the spectrum of the coefficient matrix of Lyapunov equation. After converting the min-max problem into the edge of ellipse, we use the Lingo software which can handle enormous numerical problems to solve the parameter selection problem. The numerical experiments show the efficiency of the ADI parameter selection.