Symplectic balancing of Hamiltonian matrices (Q2706473)

Balancing techniques are discussed for solving the eigenvalue problem of Hamiltonian matrices NEWLINE\[NEWLINE H=\left[\begin{matrix} A&G\cr Q&-A^T \end{matrix}\right]NEWLINE\]NEWLINE where \(A, G, Q\in{\mathbb R}^{n\times n}\) and \(G, Q\) are symmetric. Balancing a matrix for eigenvalue computation is aiming for improving accuracy of approximations. It consists of a permutation step and a scaling step. Permutations of a matrix are realized via similarity transformations to isolate eigenvalues. The scaling step adjusts the rows and columns of the permuted matrix so that its norm decreases. Based on symplectic transformations the author developed a nice balancing strategy for Hamiltonian matrices such that the Hamiltonian structure is preserved in the permutation and scaling processes. Efficiency of the proposed balancing strategies is illustrated in application to the algebraic Riccati equation and numerical examples.