Abstract
This paper deals with the development of spectrumdichotomy methods for matrices with large norms. Such matrices often result from discretizations of differential operators. The results of some numerical experiments, including an investigation of the stability of plane-parallel Poiseuille flow, are given.
Similar content being viewed by others
References
Godunov, S.K., Lektsii po sovremennym aspektam lineinoi algebry (Lectures on Modern Aspects of Linear Algebra), Novosibirsk: Nauch. Kniga, 2002.
Godunov, S.K., Problem of the Dichotomy of the Spectrum of a Matrix, Sib. Mat. J., 1986, vol. 27, no. 5, pp. 24–37.
Godunov, S.K., Modern Aspects of Linear Algebra, vol. 175, AMS, 1998.
Godunov, S.K. and Sadkane, M., Elliptic Dichotomy of a Matrix Spectrum, Lin. Alg. Appl., 1996, vol. 248, pp. 205–232.
Bulgakov, A.Ya. and Godunov, S.K., Circular Dichotomy of the Spectrum of a Matrix, Sib. Mat. J., 1988, vol. 29, no. 5, pp. 59–70.
Bulgakov, A.Ya., Justification of Guaranteed Accuracy in the Calculation of Invariant Subspaces of Non-Self-Adjoint Matrices, Tr. Inst. Mat. SB USSR Acad. Sci., 1989, vol. 15, pp. 12–92.
Malyshev, A.N., Vvedenie v vychislitel’nuyu lineinuyu algebru (Introduction to Numerical Linear Algebra), Novosibirsk: Nauka, 1991.
Demidenko, G., On a Functional Approach to Spectral Problems of Linear Algebra, Selc¸ uk J. Appl. Math., 2001, vol. 2, no. 2, pp. 39–52.
Sayed Ali, M. and Sadkane, M., On a Lyapunov Type Equation Related to Parabolic Spectral Dichotomy, El. J. Lin. Alg., 2006, vol. 15, pp. 134–142.
Boiko, A.V., Grek, G.R., Dovgal, A.V., and Kozlov, V.V., Physical Mechanisms of Transition to Turbulence in Open Flows, Izhevsk: Institute of Computer Research, 2006.
Biberdorf, E.A., A Criterion for the Dichotomy of Roots of a Polynomial on the Unit Circle, Sib. Zh. Ind. Mat., 2000, vol. 3, no. 1, pp. 16–32.
Godunov, S.K. and Sadkane, M., Some New Algorithms for the Spectral Dichotomy Methods, Lin. Alg. Its Appl., 2003, vol. 383, pp. 173–194.
Biberdorf, E.A. and Popova, N.I., Garantirovannaya tochnost’ sovremennykh algoritmov lineinoi algebry (Guaranteed Accuracy ofModern Algorithms of Linear Algebra), Novosibirsk: SB RAS, 2006.
Biberdorf, E.A., Garantirovannaya tochnost’ v prikladnykh zadachakh lineinoi algebry (Guaranteed Accuracy in Applied Problems of Linear Algebra), Novosibirsk: Novosibirsk State Univ., 2008.
Blinova, M.A., ANumerical Investigation of the Stability of Some Flows Using a Method of Matrix Spectrum Dichotomy, Bachelor’s Degree Graduation Project, Novosibirsk: Novosibirsk State Univ., 2016.
Acknowledgments
This work was supported by the Russian Foundation for Basic Research, project no. 17-01-00791. The authors would like to thank Academician S.K. Godunov for comprehensive discussions on the subject of this work and A.N. Kudryavtsev for the formulation and discussions of the problem of stability of plane-parallel flow.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.A. Biberdorf, M.A. Blinova, N.I. Popova, 2018, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2018, Vol. 21, No. 2, pp. 139–154.
Rights and permissions
About this article
Cite this article
Biberdorf, E.A., Blinova, M.A. & Popova, N.I. Some Modifications of the Method of Matrix Spectrum Dichotomy and Their Applications to Stability Problems. Numer. Analys. Appl. 11, 108–120 (2018). https://doi.org/10.1134/S1995423918020027
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995423918020027