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

Stability of Structured Hamiltonian Eigensolvers

“…Therefore structure-preserving algorithms for symmetric or skew-symmetric Hamiltonian or skew-Hamiltonian eigenproblems have to use real symplectic orthogonal transformations, that is, matrices U ∈ R 2n×2n satisfying U T JU = J, U T U = I. As in [10], we denote by SpO(2n) the group of real symplectic orthogonal matrices. Any U ∈ SpO(2n) can be written as U = […”

“…The Hamiltonian Jacobi algorithm based on symplectic Givens rotations and symplectic double Jacobi rotations of the form J ⊗ I 2n , where J is a 2 × 2 Jacobi rotation, preserves the Hamiltonian structure. This Jacobi algorithm, when it converges, builds a Hamiltonian Schur decomposition [7 Recently, Faßbender, Mackey, and Mackey [10] developed Jacobi algorithms for structured Hamiltonian eigenproblems that preserve the structure and produce a complete basis of symplectic orthogonal eigenvectors for the two symmetric cases and a symplectic orthogonal basis for all the real invariant subspaces for the two skewsymmetric cases. These Jacobi algorithms are based on the direct solution of 4 × 4, and in one case 8 × 8, subproblems using appropriate transformations.…”

“…Immediately, we see that the difficult part in deriving these algorithms is to define the appropriate symplectic orthogonal transformation S that computes the canonical form of the restriction to the (i, j, i + n, j + n) plane of H. Faßbender, Mackey, and Mackey [10] show that by using a quaternion representation of the 4 × 4 symplectic orthogonal group, as well as 4 × 4 Hamiltonian and skew-Hamiltonian matrices in the tensor square of the quaternion algebra, we can define and construct 4 × 4 symplectic orthogonal matrices R that do the job. These transformations are based on rotations of P ∼ = R 3 , the subspace of pure quaternions.…”

“…We start by defining two types of quaternion rotations. This enables us to encode the formulas in [10] into one. Let e s = e 1 be a standard basis vector of R 4 and p ∈ R 4 such that p = 0, e T 1 p = 0 (p is a pure quaternion), and p/ p 2 = e s .…”

“…The motivation for this work comes from recently developed Jacobi algorithms for structured Hamiltonian eigenproblems [10]. These algorithms are structure-preserving, inherently parallelizable, and hence attractive for solving large-scale eigenvalue problems.…”