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A Chart of Numerical Methods for Structured Eigenvalue Problems

We consider eigenvalue problems for real and complex matrices with two of the following algebraic properties: symmetric, Hermitian, skew symmetric, skew Hermitian, symplectic, conjugate symplectic, J-symmetric, J-Hermitian, J-skew symmetric, J-skew Hermitian. In the complex case we found numerically stable algorithms that preserve and exploit both structures in 24 out of the 44 nontrivial cases with such a twofold structure. Of the remaining 20, we found algorithms that preserve part of the structure of 9 pairs. In the real case we found algorithms 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’ QR algorithm, the Quaternion QR-algorithm and structure revealing, symplectic, unitary similarity transformations.