The shift-inverted \(J\)-Lanczos algorithm for the numerical solutions of large sparse algebraic Riccati equations
A shift invert Lanczos algorithm applicable to compute the eigenvalues of a large and sparse Hamiltonian matrix is described. It computes a symplectic basis two columns at a time, in which the matrix is represented by a 2 by 2 block matrix with 3 diagonal and one tridiagonal block. It is shown that convergence occurs, and symplecticity gets lost, when a quantity computed from the eigenvectors of the small block matrix and its last nondiagonal elements gets small, in a way very similar to the symmetric case. A numerical example is described, and different shift and invert transformations are discussed.
large sparse algebraic Riccati equations
scientific article; zbMATH DE number 1067178