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A Hamiltonian QR algorithm. (English) Zbl 0611.65026

It is shown first that an algebraic Riccati equation: \(G+A^*X+XA- XFX=0_ n\) may be solved by reducing the Hamiltonian matrix \(H=\left[ \begin{matrix} A^*\quad G\\ F\quad -A\end{matrix} \right]\) to a triangular form: \(Q^*HQ=\left[ \begin{matrix} T^*_ 1\\ 0_ n\end{matrix} \begin{matrix} T_ 2\\ - T_ 1\end{matrix} \right]\) where \(T_ 1\) is lower triangular, \(T^*_ 2=T_ 2\) and Q is unitary and symplectic. The so called QR algorithm [J. Francis, Comput. J. 4, 332-345 (1962; Zbl 0104.343)] requiring the construction at each step of a QR (unitary-triangular) factorization is modified such that one takes full advantage of the special structure of the Hamiltonian matrices in the case \(rank(F)=1\). It is shown that the proposed Hamiltonian QR algorithm preserves numerical stability and the Hamiltonian structure requiring significantly less work and storage for problems of size greater than about 20 than the general QR algorithm.
Reviewer: S.Mirica

MSC:

65F30 Other matrix algorithms (MSC2010)
65K10 Numerical optimization and variational techniques
15A24 Matrix equations and identities
93C15 Control/observation systems governed by ordinary differential equations

Citations:

Zbl 0104.343
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