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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)
Software:
LAPACK
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