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A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems. (English) Zbl 0936.65045
Computation of eigenvalues of a $$(2n,2n)$$ skew-Hamiltonian matrix $$N$$ with complex elements is effected by transforming the $$(4n,4n)$$ block diagonal matrix with diagonal blocks $$N$$, $$\overline N$$ into a real skew-Hamiltonian matrix $${\mathcal N}$$ whose eigenvalues agree with those of $$N$$. The matrix $${\mathcal N}$$ may be put into a skew-Hamiltonian Schur from $${\mathcal R}$$ and the eigenvalues may be obtained by a method of C. F. Van Loan [Linear Algebra Appl. 61, 233-251 (1984; Zbl 0565.65018)]. The results apply to complex Hamiltonian matrices as well since if $$N$$ is skew-Hamiltonian then $$H= (-iN)$$ is Hamiltonian.
The eigenspaces associated with the positive and negative eigenvalues of $$H$$, $$\text{Inv}_+H$$, $$\text{Inv}_-H$$, are computed from the Hamiltonian Schur form $$(-i{\mathcal R})$$ assuming that $$H$$ has no purely imaginary eigenvalues. These eigenspaces are useful in certain control problems. A code ZHAEV has been written using the new numerical method and an error analysis of the method is presented. In two numerical tests ZHAEV is compared with ZGEEV, a general code for determining eigenvalues of nonsymmetric matrices. The results obtained show that the new code is faster and more accurate than the more general code.

MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 93B36 $$H^\infty$$-control 93B40 Computational methods in systems theory (MSC2010)
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