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Eigenvalue Problems

  • J. Stoer
  • R. Bulirsch

Abstract

Many practical problems in engineering and physics lead to eigenvalue problems. Typically, in all these problems, an overdetermined system of equations is given, say n + 1 equations for n unknowns ξ 1 ... , ξ n of the form.

Keywords

Eigenvalue Problem Characteristic Polynomial Hermitian Matrix Minimal Polynomial Tridiagonal Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References for Chapter 6

  1. Barth, W., Martin, R. S., Wilkinson, J. H.: Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection. Contribution II/5 in Wilkinson and Reinsch (1971).Google Scholar
  2. Bauer, F. L., Fike, C. T.: Norms and exclusion theorems. Numer. Math. 2, 137–141 (1960).MathSciNetMATHCrossRefGoogle Scholar
  3. Bauer, F. L., Fike, C. T., Stoer, J., Witzgall, C.: Absolute and monotonie norms. Numer. Math. 3, 257–264 (1961).MathSciNetMATHCrossRefGoogle Scholar
  4. Bowdler, H., Martin, R. S., Reinsch, C., Wilkinson, J. H.: The QR and QL algorithms for symmetric matrices. Contribution II/3 in Wilkinson and Reinsch (1971).Google Scholar
  5. Eberlein, P. J.: Solution to the complex eigenproblem by a norm reducing Jacobi type method. Contribution II/17 in Wilkinson and Reinsch (1971).Google Scholar
  6. Francis, J. F. G.: The QR transformation. A unitary analogue to the LR transformation. I. Computer J. 4, 265–271 (1961/62). The QR transformation. II. ibid., 1 332–345 (1961/62).MathSciNetGoogle Scholar
  7. Garbow, B. S., et al.: Matrix Eigensystem Routines—eispack Guide Extension. Lecture Notes in Computer Science 51. Berlin, Heidelberg, New York: Springer-Verlag 1977.MATHCrossRefGoogle Scholar
  8. Givens, J. W.: Numerical computation of the characteristic values of a real symmetric matrix, Oak Ridge National Laboratory Report ORNL-1574 (1954).MATHGoogle Scholar
  9. Golub, G. H., Reinsch, C.: Singular value decomposition and least squares solution. Contribution I/10 in Wilkinson and Reinsch (1971).Google Scholar
  10. Golub, G. H., Reinsch, C., Wilkinson, J. H.: Ill-conditioned eigensystems and the computation of the Jordan canonical form. SIAM Review 18, 578–619 (1976).MathSciNetMATHCrossRefGoogle Scholar
  11. Householder, A. S.: The Theory of Matrices in Numerical Analysis. New York: Blaisdell 1964.MATHGoogle Scholar
  12. Martin, R. S., Reinsch, C., Wilkinson, J. H.: Householder’s tridiagonalization of a symmetric matrix. Contribution II/2 in Wilkinson and Reinsch (1971).Google Scholar
  13. Martin, R. S., Reinsch, C., Wilkinson, J. H.: Similarity reduction of a general matrix to Hessenberg form. Contribution II/13 in Wilkinson and Reinsch (1971).Google Scholar
  14. Martin, R. S., Reinsch, C., Peters, G., Wilkinson, J. H.: The QR algorithm for real Hessenberg matrices. Contribution II/14 in Wilkinson and Reinsch (1971).Google Scholar
  15. Martin, R. S., Reinsch, C., Wilkinson, J. H.: Reduction of the symmetric eigenproblem Ax = λ B x and related problems to standard form. Contribution II/10 in Wilkinson and Reinsch (1971).Google Scholar
  16. Moler, C. B., Stewart, G. W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256 (1973).MathSciNetMATHCrossRefGoogle Scholar
  17. Peters, G., Wilkinson, J. H.: Ax = λ B x and the generalized eigenproblem. SIAM J. Numer. Anal. 7, 479–492 (1970).MathSciNetMATHCrossRefGoogle Scholar
  18. Peters, G., Wilkinson, J. H.: The calculation of specified eigenvectors by inverse iteration. Contribution II/18 in Wilkinson and Reinsch (1971).Google Scholar
  19. Peters, G., Wilkinson, J. H.: Eigenvectors of real and complex matrices by LR and QR triangularizations. Contribution II/15 in Wilkinson and Reinsch (1971).Google Scholar
  20. Rutishauser, H.: Solution of eigenvalue problems with the LR-transformation. Nat. Bur. Standards Appl. Math. Ser. 49, 47–81 (1958).MathSciNetGoogle Scholar
  21. Rutishauser, H.: The Jacobi method for real symmetric matrices. Contribution II/1 in Wilkinson and Reinsch (1971).Google Scholar
  22. Schwarz, H. R., Rutishauser, H., Stiefel, E.: Numerik symmetrischer Matrizen. 2nd ed. Stuttgart: Teubner 1972. (English translation: Englewood Cliffs, N.J.: Prentice-Hall 1973.)MATHGoogle Scholar
  23. Smith, B. T. et al.: Matrix Eigensystems Routines—eispack Guide. Lecture Notes in Computer Science 6, 2nd ed. Berlin, Heidelberg, New York: Springer-Verlag 1976.CrossRefGoogle Scholar
  24. Stewart, G. W.: Introduction to Matrix Computations. New York, London: Academic Press 1973.MATHGoogle Scholar
  25. Wilkinson, J. H.: Note on the quadratic convergence of the cyclic Jacobi process. Numer. Math. 4, 296–300 (1962).MathSciNetMATHCrossRefGoogle Scholar
  26. Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press (1965).MATHGoogle Scholar
  27. Wilkinson, J. H.: Global convergence of tridiagonal QR algorithm with origin shifts. Linear Algebra and Appl 1, 409–420 (1968).MathSciNetMATHCrossRefGoogle Scholar
  28. Wilkinson, J. H., Reinsch, C.: Linear algebra, Handbook for Automatic Computation, Vol. II. Berlin, Heidelberg, New York: Springer-Verlag (1971).Google Scholar

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • J. Stoer
    • 1
  • R. Bulirsch
    • 2
  1. 1.Institut für Angewandte MathematikUniversität Würzburg am HublandWürzburgFederal Republic of Germany
  2. 2.Institut für MathematikTechnische UniversitätMünchenFederal Republic of Germany

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