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

We develop Jacobi algorithms for solving the complete eigenproblem for Hamiltonian and skew-Hamiltonian matrices that are also symmetric or skew-symmetric. Based on the direct solution of 4×4, and in one case, 8×8 subproblems, these structure preserving algorithms produce symplectic orthogonal bases 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 4×4 symplectic orthogonal group, and the subspaces of 4×4 Hamiltonian, skew-Hamiltonian, symmetric and skew-symmetric matrices. In addition to preserving structure, these algorithms are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.