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Dynamics of the QR-flow for upper Hessenberg real matrices
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain |
We investigate the main phase space properties of the QR-flow when restricted to upper Hessenberg matrices. A complete description of the linear behavior of the equilibrium matrices is given. The main result classifies the possible $ \alpha $- and $ \omega $-limits of the orbits for this system. Furthermore, we characterize the set of initial matrices for which there is convergence towards an equilibrium matrix. Several numerical examples show the different limit behavior of the orbits and illustrate the theory.
References:
[1] |
A. Bostan and P. Dumas,
Wronskians and linear independence, The American Mathematical Monthly, 117 (2010), 722-727.
doi: 10.4169/000298910x515785. |
[2] |
M. P. Calvo, A. Iserles and A. Zanna,
Numerical solution of isospectral flows, Math. Comp., 66 (1997), 1461-1486.
doi: 10.1090/S0025-5718-97-00902-2. |
[3] |
M. P. Calvo, A. Iserles and A. Zanna,
Semi-explicit methods for isospectral flows, Advances in Computational Mathematics, 14 (2001), 1-24.
doi: 10.1023/A:1016635812817. |
[4] |
M. T. Chu,
The generalized Toda flow, the QR algorithm and the center manifold theory, SIAM J. Alg. Disc. Meth., 5 (1984), 187-201.
doi: 10.1137/0605020. |
[5] |
M. T. Chu,
Matrix differential equations: A continuous realization process for linear algebra problems, Nonlinear Anal., 18 (1992), 1125-1146.
doi: 10.1016/0362-546X(92)90157-A. |
[6] |
M. T. Chu,
Linear algebra algorithms as dynamical systems, Acta Numer., 17 (2008), 1-86.
doi: 10.1017/S0962492906340019. |
[7] |
M. T. Chu and L. K. Norris,
Isospectral flows and abstract matrix factorizations, SIAM J. Numer. Anal., 25 (1988), 1383-1391.
doi: 10.1137/0725080. |
[8] |
M. J. Colbrook and A. C. Hansen,
On the infinite-dimensional QR algorithm, Numerische Mathematik, 143 (2019), 17-83.
doi: 10.1007/s00211-019-01047-5. |
[9] |
P. Deift, L. C. Li, T. Nanda and C. Tomei,
The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math., 39 (1986), 183-232.
doi: 10.1002/cpa.3160390203. |
[10] |
P. Deift, L. C. Li and C. Tomei,
Matrix factorizations and integrable systems, Comm. Pure Appl. Math., 42 (1989), 443-521.
doi: 10.1002/cpa.3160420405. |
[11] |
P. Deift, T. Nanda and C. Tomei,
Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal., 20 (1983), 1-22.
doi: 10.1137/0720001. |
[12] |
P. J. Eberlein and C. P. Huang,
Global convergence of the QR algorithm for unitary matrices with some results for normal matrices, SIAM J. Numer. Anal., 12 (1975), 97-104.
doi: 10.1137/0712009. |
[13] |
H. Flaschka,
The Toda lattice. Ⅰ. Existence of integrals, Phys. Rev. B (3), 9 (1974), 1924-1925.
doi: 10.1103/PhysRevB.9.1924. |
[14] |
J. G. F. Francis,
The $QR$ transformation: A unitary analogue to the $LR$ transformation. Ⅰ, Comput. J., 4 (1961/62), 265-271.
doi: 10.1093/comjnl/4.3.265. |
[15] |
G. H. Golub and C. F. Van Loan, Matrix Computations, Fourth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. |
[16] |
R. T. Gregory and D. L. Karney, A Collection of Matrices for Testing Computational Algorithms, Corrected reprint of the 1969 edition. Robert E. Krieger Publishing Co., Huntington, N.Y., 1978. |
[17] |
J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. |
[18] |
A. C. Hansen, On the Approximation of Spectra of Linear Hilbert Space Operators, PhD thesis, University of Cambridge, 2008. |
[19] |
M. Hénon,
Integrals of the Toda lattice, Phys. Rev. B (3), 9 (1974), 1921-1923.
doi: 10.1103/PhysRevB.9.1921. |
[20] |
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. |
[21] |
À. Jorba and M. R. Zou,
A software package for the numerical integration of ODEs by means of high-order Taylor methods, Experiment. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[22] |
Y. Kodama and B. A. Shipman, Fifty years of the finite nonperiodic Toda lattice: A geometric and topological viewpoint, J. Phys. A, 51 (2018), 353001, 39 pp.
doi: 10.1088/1751-8121/aacecf. |
[23] |
J. Moser,
Finitely many mass points on the line under the influence of an exponential potential - an integrable system, Dynamical Systems, Theory and Applications, Lecture Notes in Phys., Springer, Berlin, 38 (1975), 467-497.
|
[24] |
T. Nanda,
Differential equations and the $QR$ algorithm, SIAM J. Numer. Anal., 22 (1985), 310-321.
doi: 10.1137/0722019. |
[25] |
B. Parlett,
Canonical decomposition of hessenberg matrices, Math. Comp., 21 (1967), 223-227.
doi: 10.1090/S0025-5718-1967-0228519-6. |
[26] |
B. Parlett,
Global convergence of the basic QR algorithm on Hessenberg matrices, Math. Comp., 22 (1968), 803-817.
doi: 10.2307/2004579. |
[27] |
W. W. Symes,
Hamiltonian group actions and integrable systems, Phys. D, 1 (1980), 339-374.
doi: 10.1016/0167-2789(80)90017-2. |
[28] |
W. W. Symes,
The $QR$ algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D, 4 (1981/82), 275-280.
doi: 10.1016/0167-2789(82)90069-0. |
[29] |
M. Toda, Theory of Nonlinear Lattices, Second edition, Springer Series in Solid-State Sciences, 20. Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-83219-2. |
[30] |
C. Tomei,
The Toda lattice, old and new, J. Geom. Mech., 5 (2013), 511-530.
doi: 10.3934/jgm.2013.5.511. |
[31] |
M. Webb, Isospectral Algorithms, Toeplitz Matrices and Orthogonal Polynomials, PhD thesis, University of Cambridge, 2017. |
[32] |
F. Z. Zhang, Matrix Theory. Basic Results and Techniques, Second edition, Universitext. Springer, New York, 2011.
doi: 10.1007/978-1-4614-1099-7. |
show all references
References:
[1] |
A. Bostan and P. Dumas,
Wronskians and linear independence, The American Mathematical Monthly, 117 (2010), 722-727.
doi: 10.4169/000298910x515785. |
[2] |
M. P. Calvo, A. Iserles and A. Zanna,
Numerical solution of isospectral flows, Math. Comp., 66 (1997), 1461-1486.
doi: 10.1090/S0025-5718-97-00902-2. |
[3] |
M. P. Calvo, A. Iserles and A. Zanna,
Semi-explicit methods for isospectral flows, Advances in Computational Mathematics, 14 (2001), 1-24.
doi: 10.1023/A:1016635812817. |
[4] |
M. T. Chu,
The generalized Toda flow, the QR algorithm and the center manifold theory, SIAM J. Alg. Disc. Meth., 5 (1984), 187-201.
doi: 10.1137/0605020. |
[5] |
M. T. Chu,
Matrix differential equations: A continuous realization process for linear algebra problems, Nonlinear Anal., 18 (1992), 1125-1146.
doi: 10.1016/0362-546X(92)90157-A. |
[6] |
M. T. Chu,
Linear algebra algorithms as dynamical systems, Acta Numer., 17 (2008), 1-86.
doi: 10.1017/S0962492906340019. |
[7] |
M. T. Chu and L. K. Norris,
Isospectral flows and abstract matrix factorizations, SIAM J. Numer. Anal., 25 (1988), 1383-1391.
doi: 10.1137/0725080. |
[8] |
M. J. Colbrook and A. C. Hansen,
On the infinite-dimensional QR algorithm, Numerische Mathematik, 143 (2019), 17-83.
doi: 10.1007/s00211-019-01047-5. |
[9] |
P. Deift, L. C. Li, T. Nanda and C. Tomei,
The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math., 39 (1986), 183-232.
doi: 10.1002/cpa.3160390203. |
[10] |
P. Deift, L. C. Li and C. Tomei,
Matrix factorizations and integrable systems, Comm. Pure Appl. Math., 42 (1989), 443-521.
doi: 10.1002/cpa.3160420405. |
[11] |
P. Deift, T. Nanda and C. Tomei,
Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal., 20 (1983), 1-22.
doi: 10.1137/0720001. |
[12] |
P. J. Eberlein and C. P. Huang,
Global convergence of the QR algorithm for unitary matrices with some results for normal matrices, SIAM J. Numer. Anal., 12 (1975), 97-104.
doi: 10.1137/0712009. |
[13] |
H. Flaschka,
The Toda lattice. Ⅰ. Existence of integrals, Phys. Rev. B (3), 9 (1974), 1924-1925.
doi: 10.1103/PhysRevB.9.1924. |
[14] |
J. G. F. Francis,
The $QR$ transformation: A unitary analogue to the $LR$ transformation. Ⅰ, Comput. J., 4 (1961/62), 265-271.
doi: 10.1093/comjnl/4.3.265. |
[15] |
G. H. Golub and C. F. Van Loan, Matrix Computations, Fourth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. |
[16] |
R. T. Gregory and D. L. Karney, A Collection of Matrices for Testing Computational Algorithms, Corrected reprint of the 1969 edition. Robert E. Krieger Publishing Co., Huntington, N.Y., 1978. |
[17] |
J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. |
[18] |
A. C. Hansen, On the Approximation of Spectra of Linear Hilbert Space Operators, PhD thesis, University of Cambridge, 2008. |
[19] |
M. Hénon,
Integrals of the Toda lattice, Phys. Rev. B (3), 9 (1974), 1921-1923.
doi: 10.1103/PhysRevB.9.1921. |
[20] |
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. |
[21] |
À. Jorba and M. R. Zou,
A software package for the numerical integration of ODEs by means of high-order Taylor methods, Experiment. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[22] |
Y. Kodama and B. A. Shipman, Fifty years of the finite nonperiodic Toda lattice: A geometric and topological viewpoint, J. Phys. A, 51 (2018), 353001, 39 pp.
doi: 10.1088/1751-8121/aacecf. |
[23] |
J. Moser,
Finitely many mass points on the line under the influence of an exponential potential - an integrable system, Dynamical Systems, Theory and Applications, Lecture Notes in Phys., Springer, Berlin, 38 (1975), 467-497.
|
[24] |
T. Nanda,
Differential equations and the $QR$ algorithm, SIAM J. Numer. Anal., 22 (1985), 310-321.
doi: 10.1137/0722019. |
[25] |
B. Parlett,
Canonical decomposition of hessenberg matrices, Math. Comp., 21 (1967), 223-227.
doi: 10.1090/S0025-5718-1967-0228519-6. |
[26] |
B. Parlett,
Global convergence of the basic QR algorithm on Hessenberg matrices, Math. Comp., 22 (1968), 803-817.
doi: 10.2307/2004579. |
[27] |
W. W. Symes,
Hamiltonian group actions and integrable systems, Phys. D, 1 (1980), 339-374.
doi: 10.1016/0167-2789(80)90017-2. |
[28] |
W. W. Symes,
The $QR$ algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D, 4 (1981/82), 275-280.
doi: 10.1016/0167-2789(82)90069-0. |
[29] |
M. Toda, Theory of Nonlinear Lattices, Second edition, Springer Series in Solid-State Sciences, 20. Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-83219-2. |
[30] |
C. Tomei,
The Toda lattice, old and new, J. Geom. Mech., 5 (2013), 511-530.
doi: 10.3934/jgm.2013.5.511. |
[31] |
M. Webb, Isospectral Algorithms, Toeplitz Matrices and Orthogonal Polynomials, PhD thesis, University of Cambridge, 2017. |
[32] |
F. Z. Zhang, Matrix Theory. Basic Results and Techniques, Second edition, Universitext. Springer, New York, 2011.
doi: 10.1007/978-1-4614-1099-7. |








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