Breaking Van Loan’s Curse: A Quest forStructure-Preserving Algorithms for Dense Structured Eigenvalue Problems

In 1981 Paige and Van Loan (Linear Algebra Appl 41:11–32, 1981) posed the open question to derive an \(\mathcal{O}(n^{3})\) numerically strongly backwards stable method to compute the real Hamiltonian Schur form of a Hamiltonian matrix. This problem is known as Van Loan’s curse. This chapter summarizes Volker Mehrmann’s work on dense structured eigenvalue problems, in particular, on Hamiltonian and symplectic eigenproblems. In the course of about 35 years working on and off on these problems the curse has been lifted by him and his co-workers. In particular, his work on SR methods and on URV-based methods for dense Hamiltonian and symplectic matrices and matrix pencils is reviewed. Moreover, his work on structure-preserving methods for other structured eigenproblems is discussed.

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