Hamilton and Jacobi come full circle: Jacobi algorithms for structured Hamiltonian eigenproblems

We develop Jacobi algorithms for solving the complete eigenproblem for Hami ltonian and skew-Hamiltonian matrices that are also symmetric or skew-symme tric. Based on the direct solution of 4 x 4, and in one case, 8 x 8 subprob lems, these structure preserving algorithms produce symplectic orthogonal b ases for the invariant subspaces associated with a matrix in any one of the four classes under consideration. The key step in the construction of the algorithms is a quaternion characterization of the 4 x 4 symplectic orthogo nal group, and the subspaces of 4 x 4 Hamiltonian, skew-Hamiltonian, symmet ric and skew-symmetric matrices. In addition to preserving structure, these algorithms are inherently parallelizable, numerically stable, and show asy mptotic quadratic convergence. (C) 2001 Elsevier Science Inc. All rights re served.