Advertisement

Eigenvalue Problems

  • J. Stoer
  • R. Bulirsch
Part of the Texts in Applied Mathematics book series (TAM, volume 12)

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
$$F(x;\lambda ): \equiv \left[ {\begin{array}{*{20}{c}} {{f_1}({\xi _1}, \ldots ,{\xi _n};\lambda )} \\ { \ldots \ldots \ldots \ldots \ldots \ldots \ldots } \\ {{f_{n + 1}}({\xi _1}, \ldots ,{\xi _n};\lambda )} \end{array}} \right] = 0,$$
(6.0.1)
in which the functions f i also depend on an additional parameter λ. Usually, (6.0.1) has a solution x = [ξ 1,..., ξ n ] T only for specific values λ = λ i , i = 1, 2, ..., of this parameter. These values λ i are called eigenvalues of the eigenvalue problem (6.0.1), and a corresponding solution x = x(λ i ) of (6.0.1) eigensolution belonging to the eigenvalue λ i .

Keywords

Eigenvalue Problem Characteristic Polynomial Minimal Polynomial Tridiagonal Matrix Elementary Divisor 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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 I1/5 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., Stoer, J., Witzgall, C.: Absolute and monotonic norms. Numer. Math. 3, 257–264 (1961).MathSciNetMATHCrossRefGoogle Scholar
  4. Bowdler, H., Martin, R. S., Wilkinson, J. H.: The QR and QL algorithms for symmetric matrices. Contribution II/3 in Wilkinson and Reinsch (1971).Google Scholar
  5. Bunse, W., Bunse-Gerstner, A.: Numerische Lineare Algebra. Stuttgart: Teubner (1985).MATHCrossRefGoogle Scholar
  6. Cullum, J., Willoughby, R. A.: Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Vol. 1: Theory, Vol. II: Programs. Progress in Scientific Computing, Vol. 3, 4. Basel: Birkhäuser (1985).Google Scholar
  7. 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
  8. 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.,332–345 (1961/62).Google Scholar
  9. Garbow, B. S., et al.: Matrix Eigensystem Routines—EISPACK Guide Extension. Lecture Notes in Computer Science 51. Berlin, Heidelberg, New York: Springer-Verlag (1977).Google Scholar
  10. Givens, J. W.: Numerical computation of the characteristic values of a real symmetric matrix. Oak Ridge National Laboratory Report ORNL-1574 (1954).Google Scholar
  11. Golub, G. H., Reinsch, C.: Singular value decomposition and least squares solution. Contribution I/10 in Wilkinson and Reinsch (1971).Google Scholar
  12. Golub, G. H., Van Loan, C. F.: Matrix Computations. Baltimore: The Johns-Hopkins University Press (1983).MATHGoogle Scholar
  13. Golub, G. H., Wilkinson, J. H.: Ill-conditioned eigensystems and the computation of the Jordan canonical form. SIAM Review 18, 578–619 (1976).MathSciNetMATHCrossRefGoogle Scholar
  14. Householder, A. S.: The Theory of Matrices in Numerical Analysis. New York: Blaisdell (1964).MATHGoogle Scholar
  15. Kaniel, S.: Estimates for some computational techniques in linear algebra. Math. Comp. 20, 369–378 (1966).MathSciNetMATHCrossRefGoogle Scholar
  16. Kielbasinski, A., Schwetlick, H.: Numerische Lineare Algebra. Thun, Frankfurt/M.: Deutsch (1988).MATHGoogle Scholar
  17. Kublanovskaya, V. N.: On some algorithms for the solution of the complete eigen-value problem. Z. Vycisl. Mat. i Mat. Fiz. 1, 555–570 (1961).MathSciNetGoogle Scholar
  18. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. 45, 255–282 (1950).MathSciNetCrossRefGoogle Scholar
  19. Martin, R. S., Peters, G., Wilkinson, J. H.: The QR algorithm for real Hessenberg matrices. Contribution I1/14 in Wilkinson and Reinsch (1971).Google Scholar
  20. Martin, R. S., Reinsch, C., Wilkinson. J. H.: Householder’s tridiagonalization of a symmetric matrix. Contribution 1/2 in Wilkinson and Reinsch (1971).Google Scholar
  21. Martin, R. S., Wilkinson, J. H.: Reduction of the symmetric eigenproblem Ax = 2Bx and related problems to standard form. Contribution II/10 in Wilkinson and Reinsch (1971).Google Scholar
  22. Martin, R. S., Wilkinson, J. H.: Similarity reduction of a general matrix to Hessenberg form. Contribution II/13 in Wilkinson and Reinsch (1971).Google Scholar
  23. Moler, C. B., Stewart, G. W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256 (1973).MathSciNetMATHCrossRefGoogle Scholar
  24. Paige, C. C.: The computation of eigenvalues and eigenvectors of very large sparse matrices. Ph.D. thesis, London University (1971).Google Scholar
  25. Parlett, B. N.: Convergence of the QR algorithm. Numer. Math. 7, 187–193 (1965) (corr. in 10 163–164 (1967)).Google Scholar
  26. Parlett, B. N.: The Symmetric Eigenvalue Problem. Englewood Cliffs, N.J.: Prentice-Hall (1980).MATHGoogle Scholar
  27. Parlett, B. N., Poole, W. G.: A geometric theory for the QR, LU and power iterations. SIAM J. Numer. Anal. 10. 389–412 (1973).MathSciNetMATHCrossRefGoogle Scholar
  28. Parlett, B. N., Scott, D. S.: The Lanczos algorithm with selective orthogonalization. Math Comp. 33, 217–238 (1979).MathSciNetMATHCrossRefGoogle Scholar
  29. Peters, G., Wilkinson J. H.: Ax =.1Bx and the generalized eigenproblem. SIAM J. Numer. Anal. 7, 479–492 (1970).MathSciNetMATHCrossRefGoogle Scholar
  30. 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
  31. Peters, G., Wilkinson J. H.: The calculation of specified eigenvectors by inverse iteration. Contribution II/18 in Wilkinson and Reinsch (1971).Google Scholar
  32. Rutishauser, H.: Solution of eigenvalue problems with the LR-transformation. Nat. Bur. Standards Appl. Math. Ser. 49, 47–81 (1958).MathSciNetGoogle Scholar
  33. Rutishauser, H.: The Jacobi method for real symmetric matrices. Contribution II/1 in Wilkinson and Reinsch (1971).Google Scholar
  34. Saad, Y.: On the rates of convergence of the Lanczos and the block Lanczos methods. SIAM J. Num. Anal. 17, 687–706 (1980).MathSciNetMATHCrossRefGoogle Scholar
  35. Schwarz, H. R., Rutishauser, H., Stiefel, E.: Numerik symmetrischer Matrizen. 2d ed. Stuttgart: Teubner (1972). (English translation: Englewood Cliffs, N.J.: Prentice-Hall (1973.))Google Scholar
  36. Smith, B. T., et al.: Matrix Eigensystems Routines—EtsPAcK Guide. Lecture Notes in Computer Science 6, 2d ed. Berlin, Heidelberg, New York: Springer-Verlag (1976).Google Scholar
  37. Stewart, G. W.: Introduction to matrix computations. New York, London: Academic Press (1973).MATHGoogle Scholar
  38. Wilkinson, J. H.: Note on the quadratic convergence of the cyclic Jacobi process. Numer. Math. 4, 296–300 (1962).MathSciNetMATHCrossRefGoogle Scholar
  39. Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press (1965).MATHGoogle Scholar
  40. Wilkinson, J. H.: Global convergence of tridiagonal QR algorithm with origin shifts. Linear Algebra and Appl. 1, 409–420 (1968).MathSciNetMATHCrossRefGoogle Scholar
  41. 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 1993

Authors and Affiliations

  • J. Stoer
    • 1
  • R. Bulirsch
    • 2
  1. 1.Institut für Angewandte MathematikUniversität WürzburgWürzburgGermany
  2. 2.Institut für MathematikTechnische UniversitätMünchenGermany

Personalised recommendations