A Chart of Numerical Methods for Structured Eigenvalue Problems - Bunse-Gerstner, Angelika; Byers, Ralph; Mehrmann, Volker (SIAM journal on matrix analysis and applications, 1992)
It is common in applied mathematics to encounter matrices that are symmetric, Hermitian, skew symmetric, skew Hermitian, symplectic, conjugate symplectic, $J$-symmetric, $J$-Hermitian, $J$-skew symmetric, or $J$-skew Hermitian. Eigenvalue algorithms for real and complex matrices that have at least two such algebraic structures are considered. In the complex case numerically stable algorithms were found that preserve and exploit both structures of 40 out of the 66 pairs studied. Of the remaining 26, algorithms were found that preserve part of the structure of 12 pairs. In the real case algorithms were found for all pairs studied. The algorithms are constructed from a small set of numerical tools, including orthogonal reduction to Hessenberg form, simultaneous diagonalization of commuting normal matrices, Francis's QR algorithm, the quaternion QR-algorithm, and structure revealing, symplectic, unitary similarity transformations.