The QR-algorithm and generalized Toda flows. (English. Russian original) Zbl 0699.58057
Ukr. Math. J. 41, No. 7, 806-813 (1989); translation from Ukr. Mat. Zh. 41, No. 7, 944-952 (1989).
The classical QR-algorithm is formulated as follows: the arbitrary matrix from GL(n,C) should be transformed by a unitary transformation to the upper-diagonal form. In the work an abstract group-theoretical QR- algorithm is analyzed for an arbitrary Lie group G.
The generalized Toda flow is determined by solutions of a particular non- stationary dynamical system on a Lie algebra L of the group G. The close connection between generalized Toda flows and abstract QR-algorithms was discovered by Symes.
In the work the asymptotical behaviour of the Toda flow corresponding to the classical QR-algorithm is investigated in detail. The phase portraits are described for this class of dynamical systems. Asymptotically the underdiagonal matrix elements are vanishing. This allows to determine the spectra of different classes of matrices by numerical integration of Toda flows. It is shown that the phase portraits of Toda flows corresponding to normal matrices are structurally stable.
The generalized Toda flow is determined by solutions of a particular non- stationary dynamical system on a Lie algebra L of the group G. The close connection between generalized Toda flows and abstract QR-algorithms was discovered by Symes.
In the work the asymptotical behaviour of the Toda flow corresponding to the classical QR-algorithm is investigated in detail. The phase portraits are described for this class of dynamical systems. Asymptotically the underdiagonal matrix elements are vanishing. This allows to determine the spectra of different classes of matrices by numerical integration of Toda flows. It is shown that the phase portraits of Toda flows corresponding to normal matrices are structurally stable.
Reviewer: P.Prešnajder
Keywords:
phase portrait; unitary transformation; upper-diagonal form; Toda flow; asymptotical behaviourReferences:
| [1] | W. W. Symes, ?The WR-algorithm for the nonperiodic Toda lattice,? Physica,4D, 275-280 (1982). |
| [2] | P. Deift, T. Nanda, and Fomei, ?Differential equations for the symmetric eigenvalue problems,? SIAM J. Numer. Anal.,20, 1-22 (1983). · Zbl 0526.65032 · doi:10.1137/0720001 |
| [3] | M. T. Chu, ?The generalized Toda flow, the QR-algorithm and the center manifold theory,? SIAM J. Alg. Discr. Math.,5, No. 2, 187-201 (1984). · Zbl 0539.65015 · doi:10.1137/0605020 |
| [4] | D. S. Watkins, ?Isospectral flow,? SIAM Rev.,26, No. 3, 379-391 (1984). · Zbl 0559.65018 · doi:10.1137/1026075 |
| [5] | M. Shub and A. T. Vasquez, ?Some linear induced Morse-Smale systems, QR-algorithm and the Toda lattice,? Contemp. Math.,64, 181-194 (1987). · Zbl 0616.58023 |
| [6] | M. A. Semenov-Tyan’-Shanskii, ?What is a classical R-matrix?? Funkts. Analiz i Ego Prilozhen.,17, No. 4, 17-33 (1983). |
| [7] | V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Calculations [in Russian], Nauka, Moscow (1984). · Zbl 0537.65024 |
| [8] | M. Shayman, ?Phase portrait of the matrix Riccati equation,? SIAM J. Contr. Opt.,24, No. 1, 1-65 (1986). · Zbl 0594.34044 · doi:10.1137/0324001 |
| [9] | L. E. Faibusovich, ?The symplectic structure of the operator Riccati equation,? Kibernetika i Vychislit. Tekhnika,65, 62-68 (1985). |
| [10] | Ju. M. Berezansky, The Integration of Semi-Infinite Toda Chain by Means of Inverse Spectral Problem, Preprint 84.79, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev (1984). |
| [11] | P. Deift, L. C. Li, and C. Tomei, ?Toda flows with infinitely many variables,? J. Funct. Anal.,64, No. 3, 358-402 (1985). · Zbl 0615.58016 · doi:10.1016/0022-1236(85)90065-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
