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Pseudoeigenvalues, Spectral Portrait of a Matrix and their Connections with Different Criteria of Stability

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Error Control and Adaptivity in Scientific Computing

Part of the book series: NATO Science Series ((ASIC,volume 536))

Abstract

A new methodology for scientific computation aims at the design of problem solving environments with automatic result verification, providing full control over effects of the computational errors and the uncertainties in the data.

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References

  1. Kulisch, U.W. and Miranker, W.L. (1986) The arithmetic of the digital computer: A New Approach, SIAM Review 28, 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  2. Godunov, S.K., Antonov, A.G., Kiriluk, O.P. and Kostin V.I. (1993) Guaranteed Accuracy in Numerical Linear Algebra, Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  3. Akın, Ö. and Bulgak, H. (1998) Linear Difference Equations and Stability Theory, Selcuk University, Research Centre of Applied Mathematics, Konya (in Turkish).

    Google Scholar 

  4. Godunov, S.K. and Ryaben’kii, V.S. (1964) Theory of Difference Schemes: An Introduction, North-Holland, Amsterdam.

    MATH  Google Scholar 

  5. Trefethen, L.N. (1990) Approximation theory and numerical linear algebra, in J.C. Mason and M.G. Cox (eds.), Proceedings of the Second International Conference on Algorithms for Approximation, Chapman and Hall, London, pp. 336–359.

    Google Scholar 

  6. Kostin, V.I. and Razzakov, Sh.I. (1985) On the convergence of the orthogonal-power method for calculating the spectrum, Numerical methods of linear algebra, Nauka, Novosibirsk, pp. 55–84 (in Russian).

    Google Scholar 

  7. Trefethen, L.N. (1992) Pseudospectra of matrices, in D.F. Griffiths and G.A. Watson (eds.), Proceedings of the 14 Dundee Biennial Conference on Numerical Analysis, Longman Sci. Tech. Publ., Harlow, pp. 234–266.

    Google Scholar 

  8. Kostin, V.I. (1991) On definition of matrices’ spectra, High Perfomance Computing II.

    Google Scholar 

  9. Chaitin-Chatelin, F. and Fraysse, V. (1996) Lectures on Finite Precision Computations, SIAM, Philadelphiya.

    Book  MATH  Google Scholar 

  10. Godunov, S.K. (1997) Modern Aspects of the Linear Algebra, Nauchnaya Kniga, Novosibirsk (in Russian).

    Google Scholar 

  11. Wonham, W.M. (1979) Linear multivariable control: a geometric approach, Springer-Verlag, New-York.

    Book  MATH  Google Scholar 

  12. Bulgakov, A.Ya. (1980) An effectively calculable parameter for the stability quality of systems of linear differential equations with constant coefficients, Siberian Math J. 21, 339–347.

    Article  MathSciNet  Google Scholar 

  13. Godunov, S.K. (1990) The Problem of Guaranteed Precision in Numerical Methods of Linear Algebra, Amer. Math. Soc. Transl. (2) 147, pp. 65–73.

    Google Scholar 

  14. Bulgakov, A.Ya. (1991) The basis of guaranteed accuracy in the problem of separation of invariant subspaces for nonselfadjoint matrices, Siberian Advances in Mathematics, 1, 64–108 and 2, 1-56.

    MathSciNet  MATH  Google Scholar 

  15. Van Loan, C.F. (1984) How near is a stable matrix to unstable matrix? Linear algebra and its role in systems theory, Brunswick, Maine, 465–478.

    Google Scholar 

  16. Hinrichsen, D. and Pritchard, A.J. (1986) Stability radius for structured perturbations and the algebraic Riccati equations, Systems Control Letters 8, 105–113.

    Article  MathSciNet  MATH  Google Scholar 

  17. Davison, E.J. and Man, F.T. (1968) The numerical solution of A*Q+QA=-C, IEEE Trans. Aut. Control AC-13, 448–449.

    Article  MathSciNet  Google Scholar 

  18. Povzner, A.J. and Pavlov, B.V. (1973) On one method of the numerical integration of systems of ordinary differential equations, Zh. Vychislit. Mat. Mat. Fiz. 13, 256–259 (in Russian).

    MathSciNet  Google Scholar 

  19. Moler, C.B. and Van Loan, C.F. (1978) Nineteen dubious ways to compute the exponential of a matrix, SIAM Review 20, 801–836.

    Article  MathSciNet  MATH  Google Scholar 

  20. Ward, R.C. (1977) Numerical computation of the matrix exponential with accuracy estimate, SIAM J. Numerical Anal. 14, 600–610.

    Article  MATH  Google Scholar 

  21. Golub, G.H. and Van Loan, C.F. (1989) Matrix Computations, John Hopkins University Press, Baltimore.

    Google Scholar 

  22. Bulgakov, A.Ya. (1985) Computation of the exponential function of an asymptotically stable matrix, Numerical methods of linear algebra, Nauka, Novosibirsk, pp. 4–17 (in Russian).

    Google Scholar 

  23. Bulgakov, A.Ya. and Godunov, S.K. (1985) Calculation of positive definite solutions of Lyapunov’s equation, Numerical methods of linear algebra, Nauka, Novosibirsk, pp. 17–38 (in Russian).

    Google Scholar 

  24. Daletskii, Yu. L. and Krein, M.G. (1970) Stability of solutions to differential equations in Banach space, Nauka, Moscow (in Russian).

    Google Scholar 

  25. Bulgakov, A.Ya. (1995) The guaranteed accuracy for solving Riccati matrix equation, Siberian Advances in Mathematics 5, 1–49.

    MathSciNet  Google Scholar 

  26. Godunov, S.K. (1992) Norms of the solutions of Lourier-Riccati matrix equations as criteria of the quality of stabilizability and detectability, Siberian Advances in Mathematics 2, 135–157.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Research Center of Applied Mathematics, Selcuk University, Konya, Turkey

    H. Bulgak

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  1. H. Bulgak
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Editors and Affiliations

  1. Research Center of Applied Mathematics, Selcuk University, Konya, Turkey

    Haydar Bulgak

  2. Institut für Informatik, Technische Universität München, D-80290, München, Germany

    Christoph Zenger

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© 1999 Springer Science+Business Media Dordrecht

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Bulgak, H. (1999). Pseudoeigenvalues, Spectral Portrait of a Matrix and their Connections with Different Criteria of Stability. In: Bulgak, H., Zenger, C. (eds) Error Control and Adaptivity in Scientific Computing. NATO Science Series, vol 536. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4647-0_6

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  • DOI: https://doi.org/10.1007/978-94-011-4647-0_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-5809-1

  • Online ISBN: 978-94-011-4647-0

  • eBook Packages: Springer Book Archive

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