The shift-inverted J-Lanczos algorithm for the numerical solutions of large sparse algebraic Riccati equations

The goal of solving an algebraic Riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left-half plane. The purpose of this paper is to propose a structure-preserving Lanczos-type algorithm incorporated with shift and invert techniques, named shift-inverted J-Lanczos algorithm, for computing the stable invariant subspace for large sparse Hamiltonian matrices. The algorithm is based on the J-tridiagonalization procedure of a Hamiltonian matrix using symplectic similarity transformations. We give a detailed analysis on the convergence behavior of the J-Lanczos algorithm and present error bound analysis and Paige-type theorem. Numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high-speed vehicles are reported.