On numerical methods of finding an approximate solution of multi-parameters eigenvalue problems

2017;
: pp. 197-205
https://doi.org/10.23939/mmc2017.02.197
Received: October 04, 2017
1
Lviv Polytechnic National University
2
Ivan Franko National University of Lviv

This survey is intended to give an overview of the existing numerical methods and techniques for solving a multi-parameter eigenvalue problem.  It also collects some well-known examples of multi-parameter spectral problem and its practical applications in different scientific areas.

  1. Saad Y.  Numerical methods for large eigenvalue problems. Second edition.  SIAM (2011).
  2. Hochstenbach M. E., Plestenjak B. Harmonic Rayleigh-Ritz for the Multiparameter eigenvalue problem.  Department of Mathematics and Computing Science, Eindhoven University of Technology (2006).
  3. Atkinson F. V.  Multiparameter spectral theory.   Bull. Amer. Math. Soc. 74 (1), 1--27 (1968).
  4. Müller R. E.  Numerical solution of multiparameter eigenvalue problems.  ZAMM. 62 (12), 681–686 (1982).
  5. Molzahn D.  Power system models formulated as eigenvalue problems and properties of their solutions --– A thesis for the degree of master of science. University of Wisconsin-Madison (2010).
  6. Dai H.  Numerical methods for solving multiparameter eigenvalue problems.  International Journal of Computer Mathematics. 72, 331--347 (1999).
  7. Blum E. K., Curtis A. R.  A convergent gradient method for matrix eigenvector-eigentuple problems.  Numer. Math. 31 (3), 247--263 (1978).
  8. Slivnik T., Tomšič G.  A numerical method for the solution of two-parameter eigenvalue problem.  J. Comp. Appl. Math. 15 (1), 109--115 (1986).
  9. Sleeman B. D.  Multiparameter spectral theory in Hilbert space.  London, San Francisco, Melbourne, Pitnam Press (1978).
  10. Volkmer H.  Multiparameter eigenvalue problems and expansion theorem.  Lect. Notes Math. 1336 (1988).
  11. Atkinson F. V.  Multiparameter Eigenvalue Problems. Matrices and Compact Operators. Vol. 1.  New York, London, Academic Press (1972).
  12. Fox L., Hayes L., Mayers D. F.  The double eigenvalue problem.  John J. H. Miller (Ed.), Topics in Numerical Analysis, Proceedings of the Royal Irish Academy Conference on Numerical Analysis, 1972, Academic Press, New York (1973), pp. 93-112.
  13. Ji X.  Numerical solution of joint eigenpairs of a family of commutative matrices.  Appl. Math. Lett. 4 (3), 57--60 (1991).
  14. Ji X.  A two-dimensional bisection method for solving two-parameter eigenvalue problems.  SIAM J. Matrix Anal. Appl. 13 (4), 1085–1093 (1992).
  15. Shimasaki M.  Homotopy algorithm for two-parameter eigenvalue problems. Trans Japan SIAM. 5, 121--129 (1995).
  16. Shimasaki M.  Numerical method based on homotopy algorithm for two-parameter problems.  Trans Japan SIAM. 6, (1996).
  17. Plestenjak B. A.  A continuation method for a right definite two-parameter eigenvalue problem.  SIAM J. Matrix. Anal. Appl. 21 (4), 1163--1184 (2000).
Math. Model. Comput. Vol.4, No.2, pp.197-205 (2017)