A Hamiltonian Krylov–Schur-type method based on the symplectic Lanczos process

Real Fast Structure-Preserving Algorithm for Eigenproblem of Complex Hermitian Matrices

“…The zero diagonals in follow from skew symmetry. It is easy to verify that the (2, 1) and (1,2) block of this matrix can be zeroed by applying a sequence of Householder and Givens symplectic similarity transformations. The first step is to zero 31 and 41 using a Householder symplectic 1 = (2, ).…”

“…According to Tables 1 and 2, it can be seen that the eigenvectors are different except the one corresponding to 1. In fact, by Table 1, the eigenvector corresponding to 7.4031 is (1) = (0.5522, 0.3534 + 0.4243 , −0.2650 + 0.5657 ) , and (2) = (−0.2343 − 0.5000 , 0.2343 − 0.5000 , 0.6247) by Table 2. It is easy to prove that rank ( (1) , (2) ) = 1, that is, (1) = (2) .…”

“…In fact, by Table 1, the eigenvector corresponding to 7.4031 is (1) = (0.5522, 0.3534 + 0.4243 , −0.2650 + 0.5657 ) , and (2) = (−0.2343 − 0.5000 , 0.2343 − 0.5000 , 0.6247) by Table 2. It is easy to prove that rank ( (1) , (2) ) = 1, that is, (1) = (2) . So the complex eigenvector after normalizing is not unique, which is the biggest difference from real eigenvector.…”

“…(6) Note that the matrix in (6) is not only real symmetry but also skew-Hamiltonian. Therefore, in the following discussion, inspired by the algorithms for the eigenproblem of Hamiltonian (skew-Hamiltonian) matrix [2][3][4][5][6][7][8][9][10][11], we devise an algorithm by taking full advantage of the special structure of (6).…”

<inline-formula><tex-math id="M1">\begin{document} $\mathcal{L}_{∞}$ \end{document}</tex-math></inline-formula>-norm computation for large-scale descriptor systems using structured iterative eigensolvers

“…A summary of this algorithm is presented in Subsection 3.2. Alternatively, one can use shiftand-invert variants of the Hamiltonian Lanczos algorithm [5,6]. In particular, the method discussed in [4] can be adapted here.…”