Skip to main content

A Review of Numerical Methods for Eigenvalue Problems Nonlinear in the Parameter

  • Chapter
Numerik und Anwendungen von Eigenwertaufgaben und Verzweigungsproblemen

Abstract

Let X be a complex Banach-space and B(X) the algebra of bounded linear operators on X. When X is of finite dimension we shall make this explicit by writing X n for X.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P.M. Anselone and L.B. Rail, The solution of characteristic value-vector problems by Newton’s method, Numer. Math 11 (1968), 38–45.

    Article  Google Scholar 

  2. E.M. Barston, Eigenvalue problems for Lagrangian systems, J. Math. Phys. 13 (1972), 720–725.

    Article  Google Scholar 

  3. H. Bart, Voles of the resolvent of an operator function, Proc. Roy. Irish Acad. 74A (1974), 169–184.

    Google Scholar 

  4. H. Bart, M.A. Kaashoek and D.C. Lay, Stability properties of finite meromorphic operator functions, I, II and III, Indag. Math. 36 (1974), 217–259.

    Google Scholar 

  5. F.L. Bauer, Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischer Eigenwertprobleme, Z.A.M.P. 8, (1975), 214–235.

    Article  Google Scholar 

  6. H. Baumgärtel, Endlichdimensionale Analytische Störangsthéorie, Akad. Verlag, Berlin, 1972.

    Google Scholar 

  7. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Internat. Ser. Mon. on Physics, Clarendon Press, Oxford, 1961.

    Google Scholar 

  8. G.W. Cross, Square roots of linear transformations, Ph.D. Thesis, Dept. of Math., Univ. of Calgary, 1975.

    Google Scholar 

  9. G.W. Cross and P. Lancaster, Square roots of complex matrices, Lin. and Multilin. Alg. 1 (1974), 289–293.

    Article  Google Scholar 

  10. J.E. Denis, J.F. Traub and R.P. Weber, The algebraic theory of matrix polynomials, SIAM. J. Numer. Anal. (To appear).

    Google Scholar 

  11. J.E. Denis, J.F. Traub and R.P. Weber, Algorithms for solvents of matrix polynomials, SIAM J. Numer. Anal. (To appear).

    Google Scholar 

  12. A. Friedman and M. Shinbrot, Nonlinear eigenvalue problems, Acta Math. 121 (1968), 77–125.

    Article  Google Scholar 

  13. F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York, 1960.

    Google Scholar 

  14. I. Gohberg, M.A. Kaashoek and D.C. Lay, Equivalence, linearisation and decomposition of holomorphic operator functions (To appear).

    Google Scholar 

  15. I.C. Gohberg and M.G. Krein, Introduction to the theory of Linear Nonselfadjoint Operators, Amer. Math. Soc, Providence, 1969. (Translation from Russian).

    Google Scholar 

  16. I.C. Gohberg, P. Lancaster and L. Rodman, Spectral analysis of matrix polynomials I. Canonical forms and divisors, J. Lin. Alg. Applies. (To appear).

    Google Scholar 

  17. I. Gohberg, P. Lancaster and L. Rodman, Spectral analysis of matrix polynomials, II. Resolvent forms and spectral divisors, J. Lin. Alg. Applies. (To appear).

    Google Scholar 

  18. M.A. Golberg, A generalized Rayleigh quotient iteration for eigenvalue problems nonlinear in the parameter, Jour. Optim. Theory and Appl. 11. (1973), 146–158.

    Article  Google Scholar 

  19. B. Gramsch, Spektraleigenschaften analytischer Operatorfunktionen, Math. Z. 101 (1967), 165–181.

    Article  Google Scholar 

  20. R.D. Grigorieff, Diskrete Approximation von Eigenwertproblemen I. Qualitative Konvergenz, Numer. Math. 24 (1975), 355–374.

    Article  Google Scholar 

  21. R.D. Grigorieff, Diskrete Approximation von Eigenwertproblemen II. Konvergenzordnung, Numer. Math. 24 (1975), 415–433.

    Article  Google Scholar 

  22. K.P. Hadeler, Eigenwerte von Operatorpolynomen, Arch. Rat. Mech. Anal. 20 (1965), 72–80.

    Article  Google Scholar 

  23. P.F. Hennion, Algorithm 170. Reduction of a matrix containing polynomial elements, Comm. Assoc. Comp. Mach. 6 (1963), 165–166.

    Google Scholar 

  24. S. Hildebrandt, Über die Losung nichtlinearer Eigenwertaufgaben mit dem Galerkinverfahren, Math. Zeit. 101 (1967), 255–264.

    Article  Google Scholar 

  25. E. Hughes, On a quadratic eigenvalue problem, SIAM J. Math. Anal. 5, (1974), 319–326.

    Article  Google Scholar 

  26. G.A. Isaev, The linear factorization of polynomial operator pencils, Math. Notes 13 (1973), 333–338 (Transl. from Russian)

    Article  Google Scholar 

  27. H. Jeggle, Über die Approximation von linearen Gleichungenzweiter Art und Eigenwertproblemen in Banach-Räumen, Math. Z. 124 (1972), 319–342.

    Article  Google Scholar 

  28. L. Kaufman, The LZ-algorithm to solve the generalized eigenvalue problem, SIAM J. Numer. Anal. 11 (1974), 997–1024.

    Article  Google Scholar 

  29. M.V. Keldys, On the completeness of the eigenfunctions of some classes of nonself adjoint linear operators, Russian Math. Surveys 26 (1971), 15–44.

    Article  Google Scholar 

  30. M.G. Krein and H. Langer, On some mathematical principles of linear theory of damped vibrations of continua, Proc. Int’l. Symp. in Appl. of the Theory of Functions in Cont. Mechanics, Moscow, 1965, pp. 283-322.

    Google Scholar 

  31. V.N. Kublanovskaya, On an application of Newton’s method to the determination of eigenvalues of λ-matrices, SIAM J. Num. Anal. 7, (1970), 532–537.

    Article  Google Scholar 

  32. H. Kummer, Zur praktischen Behandlung nichtlinearer Eigenwertaufgaben abgeschlossener linearer Operatoren, Mitt. Math. Sem. Giessen, 1964.

    Google Scholar 

  33. P. Lancaster, A generalised Rayleigh-quotient iteration for lambda matrices, Arch. Rat. Mech. Anal. 6 (1961), 309–322.

    Article  Google Scholar 

  34. P. Lancaster, A Igorithns for lambda-matrices, Numer. Math. 6, (1964), 388–394.

    Article  Google Scholar 

  35. P. Lancaster, Lambda-matrices and Vibrating Systems, Pergamon Press, Oxford, 1966.

    Google Scholar 

  36. P. Lancaster, A fundamental theorem on lambda-matrices with applications, I: Ordinary Differential Equations with constant coefficients, J. Lin. Alg. and Appl. (To appear).

    Google Scholar 

  37. P. Lancaster, A fundamental theorem on lambda-matrices with applications, II: Difference equations with constant coefficients, J. Lin. Alg. and Appl. (To appear).

    Google Scholar 

  38. P. Lancaster and J.G. Rokne, Solutions of nonlinear operator equations, SIAM J. Math. Anal. (To appear).

    Google Scholar 

  39. C.E. Langenhop, A now reduction of λ-matrices, Lin. Alg. and Appl. 9 (1974), 185–198.

    Article  Google Scholar 

  40. M.I. Mavlyanova, Solution of a particular eigenvalue problem for a polynomial matrix, Seminars in Math., Steklov Inst., Vol. 18, (1972), 65–70. (Transln. by Consultants Bureau, New York).

    Google Scholar 

  41. M.I. Mavlyanova, On a method for constructing the matrix solution for a polynomial matrix, Ibid. pp. 71-79.

    Google Scholar 

  42. C.B. Moler and G.W. Stewart, An algorithm for the generalized matrix eigenvalue problem, SIAM J. Numer. Anal. 10 (1973), 241–256.

    Article  Google Scholar 

  43. D.E. Müller, A method for solving algebraic equations using an automatic computer, Math. Tab. 10 (1956), 208–215.

    Google Scholar 

  44. P.H. Müller, Eine neue Methode zur Behandlung Bichtlinearer Eigenwertaufgaben, Math. Z. 70 (1959), 381–406.

    Article  Google Scholar 

  45. P.H. Müller and H. Kummer, Zur Praktischen Bestimmung nichtlinear auftretender Eigenwerte Anwendung des Verfahrens auf eine Stabilitätsuntersuchung, Z.A.M.M. 40, (1960), 136–143.

    Google Scholar 

  46. M.R. Osborne, A new method for the solution of eigenvalue problems, Computer J. 7 (1964), 228–232.

    Article  Google Scholar 

  47. M.R. Osborne, and S. Michaelson, The numerical solution of eigenvalue problems in which the eigenvalue parameter appears nonlinearly, with an application to differential equations, Computer J. 7 (1964), 58–65.

    Article  Google Scholar 

  48. I.S. Pace and S. Barnett, Efficient algorithms for linear system calculations I, Smith form and common divisor of polynomial matrices, Int. J. Systems Sci. 5 (1974), 403–411.

    Article  Google Scholar 

  49. M.V. Pattabhiraman, The generalized Rayleigh quotient, Canad. Math. Bull. 17 (1974), 251–256.

    Article  Google Scholar 

  50. G. Peters and J.H. Wilkinson, Ax =λBx and the generalized eigenproblem, SIAM J. Numer. Anal. 7, (1970), 479–492.

    Article  Google Scholar 

  51. W.R. Richert, Zur Behandlung von Eigenwertaufgaben mit nichtlinear auftretendem Parameter, Numer. Math. 22 (1974), 275–287.

    Article  Google Scholar 

  52. W.R. Richert, Zur Fehlerab Schätzung für Eigenwertaufgaben vom Typ2 I-λA-B)x = 0. Numer. Math. 22 (1974), 233–239.

    Article  Google Scholar 

  53. A. Ruhe, Algorithms for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 10, (1973), 674–689.

    Article  Google Scholar 

  54. H. Rutishauser, Solution of the eigenvalue problem, with the LR transformation, N.B.S. Appl. Math. Series 49, Washington, D.C., 1958.

    Google Scholar 

  55. F. Stummel, Diskrete Konvergenz linearer Operatoren II, Math. Z. 120, (1971), 231–264.

    Article  Google Scholar 

  56. J. Terray and P. Lancaster, A boundary value problem from the study of heat transfer, I.S.N.M. 27 (1975), 303–308.

    Google Scholar 

  57. J. Terray and P. Lancaster, On the numerical calculation of eigenvalues and eigenvectors of operator polynomials, J. Math. Anal. Appl. (To appear).

    Google Scholar 

  58. J.F. Traub, A class of globally convergent iteration functions for the solution of polynomial equations, Math. Comp. 20 (1966), 113–138.

    Article  Google Scholar 

  59. F.G. Tricomi, Integral Equations, Interscience, New York, 1957.

    Google Scholar 

  60. J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford, 1965.

    Google Scholar 

  61. A.C. Zaanen, Linear Analysis, North-Holland, Amsterdam, 1956.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1977 Springer Basel AG

About this chapter

Cite this chapter

Lancaster, P. (1977). A Review of Numerical Methods for Eigenvalue Problems Nonlinear in the Parameter. In: Bohl, E., Collatz, L., Hadeler, K.P. (eds) Numerik und Anwendungen von Eigenwertaufgaben und Verzweigungsproblemen. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’analyse Numérique, vol 38. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5579-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-5579-2_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-0938-1

  • Online ISBN: 978-3-0348-5579-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics