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Problems and Results in Matrix Perturbation Theory

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Towards Intelligent Engineering and Information Technology

Part of the book series: Studies in Computational Intelligence ((SCI,volume 243))

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Abstract

The perturbation theory is important in applications and theoretical investigations as well. Here we investigate three groups of perturbation problems which are related to computational methods of importance. The first section is related to the solution of linear systems of equations and a posteriori error estimates of the computed solution. The second section gives optimal bounds for the perturbations of LU factorizations. The final section gives a sharp upper bound for the eigenvalue perturbation of general matrices, which is better than the classical result of Ostrowski. We also show two applications of this result. The first application gives a sharp perturbation bound for the zeros of polynomials. The second application is related to a result of Edelman and Murakami on the backward stability of companion matrix type polynomial solvers.

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References

  1. Auchmuty, G.: A Posteriori Error Estimates for Linear Equations. Numerische Mathematik 61, 1–6 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barrlund, A.: Perturbation Bounds for the LDL H and LU Decompositions. BIT 31, 358–363 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baumgartel, H.: Analytic Perturbation Theory for Matrices and Operators. Birkhauser Verlag, Basel (1985)

    Google Scholar 

  4. Beauzamy, B.: How the Roots of a Polynomial Vary with its Coefficients: a Local Quantitative Result. Canadian Mathematical Bulletin 42(1), 3–12 (1999)

    MATH  MathSciNet  Google Scholar 

  5. Bhatia, R.: Matrix Factorizations and their Perturbations. Linear Algebra and its Applications 197,198, 245–276 (1994)

    Article  MathSciNet  Google Scholar 

  6. Bhatia, R.: Perturbation Bounds for Matrix Eigenvalues. SIAM, Philadelphia (2007)

    MATH  Google Scholar 

  7. Bhatia, R., Elsner, L., Krause, G.: Bounds for the Variation of the Roots of a Polynomial and the Eigenvalues of a Matrix. Linear Algebra and its Applications 142, 195–209 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chang, X.-W., Paige, C.: On the Sensitivity of the LU Factorization. BIT 38, 486–501 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chang, X.-W., Paige, C.: Sensitivity Analyses for Factorizations of Sparse or Structured Matrices. Linear Algebra and its Applications 284, 53–71 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chang, X.-W., Paige, C., Stewart, G.W.: New Perturbation Analyses for the Cholesky Factorization. IMA J. Numer. Anal. 16, 457–484 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chu, E.K.: Generalization of the Bauer-Fike Theorem. Numerische Mathematik 49, 685–691 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  12. Demmel, J.W.: On Condition Numbers and the Distance to the Nearest Ill-posed Problem. Numerische Mathematik 51, 251–289 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Demmel, J., Diament, B., Malajovich, G.: On the Complexity of Computing Error Bounds. Foundations of Computational Mathematics 1, 101–125 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Drmač, Z., Omladič, M., Veselič, K.: On the Perturbation of the Cholesky Factorization. SIAM J. Matrix. Anal. Appl. 15, 1319–1332 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Edelman, A., Murakami, H.: Polynomial Roots from Companion Matrix Eigenvalues. Math. Comp. 64(210), 763–776 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Galántai, A.: Perturbation Theory for Full Rank Factorizations, Quaderni DMSIA, 1999/40. University of Bergamo, Bergamo (1999)

    Google Scholar 

  17. Galántai, A.: Componentwise perturbation bounds for the LU, LDU, and LDL T decompositions. Mathematical Notes, Miskolc 1, 109–118 (2000)

    MATH  Google Scholar 

  18. Galántai, A.: A Study of Auchmuty’s Error Estimate. Computers and Mathematics with Applications 42, 1093–1102 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Galántai, A.: Perturbations of Triangular Matrix Factorizations. Linear and Multilinear Algebra 51, 175–198 (2003)

    Article  MathSciNet  Google Scholar 

  20. Galántai, A.: Perturbation Bounds for Triangular and Full Rank Factorizations. Computers and Mathematics with Applications 50, 1061–1068 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Galántai, A., Hegedűs, C.J.: Perturbation Bounds for Polynomials. Numerische Mathematik 109, 77–100 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Galántai, A., Hegedűs, C.: Hyman’s Method Revisited. Journal of Computational Mathematics and Applied Mathematics 226, 246–258 (2009)

    Article  MATH  Google Scholar 

  23. Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1993)

    Google Scholar 

  24. Higham, N.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  25. Hogben, L.: Handbook of Linear Algebra. Chapman & Hall/CRC (2007)

    Google Scholar 

  26. Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  27. Kahan, W.M.: Numerical Linear Algebra. Canadian Mathematical Bulletin 9, 757–801 (1966)

    MATH  Google Scholar 

  28. Kahan, W.: Conserving Confluence Curbs Ill-Condition, technical report, AD-766 916, Computer Science, University of California, Berkeley (1972)

    Google Scholar 

  29. Kato, T.: Perturbation Theory for Linear Operators. Springer, Heidelberg (1966)

    MATH  Google Scholar 

  30. von Neumann, J., Goldstine, H.: Numerical Inverting of Matrices of High Order. Bull. Amer. Math. Soc. 53, 1021–1099 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  31. Nievergelt, Y.: Numerical Linear Algebra on the HP-28 or How to Lie with Supercalculators. American Mathematical Monthly, 539–544 (1991)

    Google Scholar 

  32. Ostrowski, A.: Recherches sur la méthode de Gräffe et les zeros des polynômes et des series de Laurent. Acta Math. 72, 99–257 (1940)

    Article  MathSciNet  Google Scholar 

  33. Ostrowski, A.: Über die Stetigkeit von charakteristischen Wurzeln in Abhängigkeit von den Matrizenelementen. Jahresber. deut. Mat.-Ver. 60, 40–42 (1957)

    MATH  MathSciNet  Google Scholar 

  34. Prasolov, V.V.: Problems and Theorems in Linear Algebra. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  35. Rump, S.M.: A Class of Arbitrary ill Conditioned Floating-Point Matrices. SIAM J. Matrix Anal. Appl. 12, 645–653 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  36. Stewart, G.W.: On the perturbation of LU, Cholesky and QR factorizations. SIAM J. Matrix. Anal. Appl. 14, 1141–1145 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  37. Stewart, G.W.: On the perturbation of LU and Cholesky factors. IMA J. Numer. Anal. 17, 1–6 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  38. Stewart, G., Sun, J.: Matrix Perturbation Theory. Academic Press, London (1990)

    MATH  Google Scholar 

  39. Sun, J.-G.: Perturbation Bounds for the Cholesky and QR Factorizations. BIT 31, 341–352 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  40. Sun, J.-G.: Componentwise Perturbation Bounds for some Matrix Decompositions. BIT 32, 702–714 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  41. Sun, J.-G.: Rounding-Error and Perturbation Bounds for the Cholesky and LDL T Factorizations. Linear Algebra and its Applications 173, 77–97 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  42. Toh, K., Trefethen, L.N.: Pseudozeros of Polynomials and Pseudospectra of Companion Matrices. Numerische Mathematik 68, 403–425 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  43. Turing, A.: Rounding-Off Errors in Matrix Processes. Quart. J. Mech. Appl. Math. 1, 287–308 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  44. Watkins, D.S.: Fundamentals of Matrix Computations. John Wiley & Sons, Chichester (1991)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Budapest Tech, Bécsi út 96/B, H-1034, Budapest, Hungary

    Aurél Galántai

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  1. Aurél Galántai
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Editors and Affiliations

  1. John von Neumann Fac. Informatics, Dept. Intelligent Engineering Systems, Budapest Tech., 1034, Budapest, Hungary

    Imre J. Rudas  & János Fodor  & 

  2. Systems Research Instiute, PAN Warszawa, Newelska 6, 01-447, Warszawa, Poland

    Janusz Kacprzyk

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Galántai, A. (2009). Problems and Results in Matrix Perturbation Theory. In: Rudas, I.J., Fodor, J., Kacprzyk, J. (eds) Towards Intelligent Engineering and Information Technology. Studies in Computational Intelligence, vol 243. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03737-5_3

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  • DOI: https://doi.org/10.1007/978-3-642-03737-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03736-8

  • Online ISBN: 978-3-642-03737-5

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