A method for separating nearly multiple eigenvalues for Hermitian matrix. (English) Zbl 1112.65032
The following problem is discussed: Given a symmetric matrix and assume that by one of the methods of verified computation a small interval has been determined which encloses \(k\) of its eigenvalues. Verify that they are not a k-fold eigenvalue. An algorithm using verified computation, e.g. using INTLAB, is given, which can decide this question. Two examples with \(k=2\) (the infamous Wilkinson matrix) and \(k=3\) illustrate the method.
Reviewer: Ludwig Elsner (Bielefeld)
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 65G20 | Algorithms with automatic result verification |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
Keywords:
multiple eigenvalue; symmetric matrix; verified computation; numerical examples; eigenvalue enclosure; algorithm; Wilkinson matrixSoftware:
INTLABReferences:
| [1] | Rump, S. M., INTLAB-INTerval LABoratory, (Cesndes, Tibor, Developments in Reliable Computing (1999), Kluwer Academic Publishers: Kluwer Academic Publishers The Netherlands), 77-104 · Zbl 0949.65046 |
| [2] | Rump, S. M., Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl., 324, 209-226 (2001) · Zbl 0986.65031 |
| [3] | Wilkinson, J. H., The Algebric Eigenvalue Problem (1965), Oxford University Press: Oxford University Press London · Zbl 0258.65037 |
| [4] | Yamamoto, N., A simple method for error bounds of eigenvalues of symmetric matrices, Linear Algebra Appl., 324, 227-234 (2001) · Zbl 0981.65042 |
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