The Rayleigh quotient iteration and some generalizations for nonnormal matrices
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- by B. N. Parlett PDF
- Math. Comp. 28 (1974), 679-693 Request permission
Abstract:
The Rayleigh Quotient Iteration (RQI) was developed for real symmetric matrices. Its rapid local convergence is due to the stationarity of the Rayleigh Quotient at an eigenvector. Its excellent global properties are due to the monotonic decrease in the norms of the residuals. These facts are established for normal matrices. Both properties fail for nonnormal matrices and no generalization of the iteration has recaptured both of them. We examine methods which employ either one or the other of them.References
- Hendrik Jan Buurema, A geometric proof of convergence for the $QR$ method, Rijksuniversiteit te Groningen, Groningen, 1970. Doctoral dissertation, University of Groningen. MR 0383717
- S. H. Crandall, Iterative procedures related to relaxation methods for eigenvalue problems, Proc. Roy. Soc. London Ser. A 207 (1951), 416β423. MR 42789, DOI 10.1098/rspa.1951.0129
- P. Lancaster, A generalized Rayleigh quotient iteration for lambda-matrices, Arch. Rational Mech. Anal. 8 (1961), 309β322. MR 139262, DOI 10.1007/BF00277446
- A. M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, Arch. Rational Mech. Anal. 1 (1957), no.Β 1, 233β241. MR 1553461, DOI 10.1007/BF00298007 W. Kahan, Inclusion Theorems for Clusters of Eigenvalues of Hermitian Matrices, Technical Report, Dept. of Computer Sci., University of Toronto, 1967.
- B. N. Parlett and W. Kahan, On the convergence of a practical $\textrm {QR}$ algorithm. (With discussion), Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp.Β 114β118. MR 0255035 B. Parlett, Certain Matrix Eigenvalue Techniques Discussed from a Geometric Point of View, AERE Report 7168, Theor. Physics Div., AERE, Berkshire, England. Lord Rayleigh, The Theory of Sound, 2nd rev. ed., Macmillan, New York, 1937.
- G. Temple, The accuracy of Rayleighβs method of calculating the natural frequencies of vibrating systems, Proc. Roy. Soc. London Ser. A 211 (1952), 204β224. MR 46141, DOI 10.1098/rspa.1952.0034
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
- J. H. Wilkinson, Global convergence of tridiagonal $\textrm {QR}$ algorithm with origin shifts, Linear Algebra Appl. 1 (1968), 409β420. MR 234622, DOI 10.1016/0024-3795(68)90017-7
- Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964. MR 0162808
- G. W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev. 15 (1973), 727β764. MR 348988, DOI 10.1137/1015095
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Math. Comp. 28 (1974), 679-693
- MSC: Primary 65F15
- DOI: https://doi.org/10.1090/S0025-5718-1974-0405823-3
- MathSciNet review: 0405823