Abstract
We describe an algorithm with which one can verify solutions of an additive inverse matrix eigenvalue problem. The algorithm is based on Newton's method using a new criterion for terminating the iteration. In addition, it yields tight interval bounds for the solutions of the problem, thus guaranteeing most of their leading digits in a given floating point system.
Zusammenfassung
Wir geben einen Algorithmus an, mit dem man Lösungen eines additiven inversen Matrizen-Eigenwertproblems nachweisen kann. Der Algorithmus beruht auf dem Newton-Verfahren, für das ein neues Abbruchkriterium verwendet wird. Er liefert enge Schranken für die Lösungen des Problems und garantiert so die meisten ihrer führenden Ziffern in einem gegebenem Gleitpunktsystem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Alefeld, G.: Berechenbare Fehlerschranken für ein Eigenpaar unter Einschluß von Rundungsfehlern bei Verwendung des genauen Skalarprodukts. Z. angew. Math. Mech.67, 145–152 (1987).
Alefeld, G., Gienger, A., Potra, F. A.: Efficient numerical validation of solutions of nonlinear systems. SIAM J. Numer. Anal.31, 252–260 (1994).
Alefeld, G., Herzberger, J.: Introduction to interval computations. New York: Academic Press 1983.
Alefeld, G., Mayer, G.: A computer aided existence and uniquencess proof for an inverse matrix eigenvalue problem. Interval Computations (to appear).
Bohte, Z.: Numerical solution of the inverse algebraic eigenvalue problem. Comp. J.10, 385–388 (1967).
Friedland, S.: Inverse eigenvalue problems. Linear Algebra Appl.17, 15–51 (1977).
Friedland, S., Nocedal, J., Overton, M. L.: Four quadratically convergent methods for solving inverse eigenvalue problems. In: Griffiths, D. F., Watson, G. A. (eds.): Numerical analysis, pp. 47–65. Pitman Research Notes in Mathematics Series. Harlow: Longman 1986.
Friedland, S., Nocedal, J., Overton, M. L.: The formulation and analysis of numericla methods for inverse eigenvalue problems. SIAM J. Numer. Anal.24, 634–667 (1987).
Klatte, R., Kulisch, U., Neagá, M., Ratz, D., Ullrich, Ch.: PASCAL-XSC. Sprachbeschreibung mit Beispielen. Berlin Heidelberg New York Tokyo: Springer 1991.
Lohner, R.: Enclosing all eigenvalues of symmetric matrices. In: Ullrich, Ch., Wolff von Gudenberg, J. (eds.): Accurate numerical algorithms: a collection of research papers, Research Reports ESPRIT, Project 1072, DIAMOND, Vol. 1, pp. 87–103. Berlin Heidelberg New York Tokyo: Springer 1989.
Mayer, G.: Enclosures for eigenvalues and eigenvectors. In: Atanassova, L., Herzberger, J. (eds.): Computer arithmetic and enclosure methods, pp. 49–68. Amsterdam: Elsevier.
Mayer, G.: A unified approach to enclosure methods for eigenpairs. Z. angew. Math. Mech.74, 115–128 (1994).
Mayer, G.: Result verification for eigenvectors and eigenvalues. In: Herzberger, J. (ed.): Topics in validated computation. Amsterdam: Elsevier 1994.
Neher M.: Ein Einschließungsverfahren für das inverse Dirichletproblem. Doctorial Thesis., University of Karlsruhe, Karlsruhe, 1993.
Ortega, J. M.: Numerical analysis. A second course. Philadelphia: SIAM 1990.
Rump, S. M.: Solving algebraic problems with high accuracy. In: Kulisch, U. W., Miranker, W. L., (eds.): A new approach to scientific computation, pp. 51–120. New York: Academic Press 1983.
Stoer, J., Bulirsch, R.: Introduction to numerical analysis. Berlin Heidelberg New York: Springer 1980.
Author information
Authors and Affiliations
Additional information
We thank the Deutsche Forschungsgemeinschaft (‘DFG’) for supporting this work under grant Al 175/4-1 and Al 175/4-2.
Rights and permissions
About this article
Cite this article
Alefeld, G., Gienger, A. & Mayer, G. Numerical validation for an inverse matrix eigenvalue problem. Computing 53, 311–322 (1994). https://doi.org/10.1007/BF02307382
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02307382