The Rayleigh quotient iteration and some generalizations for nonnormal matrices

Author:
B. N. Parlett

Journal:
Math. Comp. **28** (1974), 679-693

MSC:
Primary 65F15

MathSciNet review:
0405823

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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.

**[1]**Hendrik Jan Buurema,*A geometric proof of convergence for the 𝑄𝑅 method*, Rijksuniversiteit te Groningen, Groningen, 1970. Doctoral dissertation, University of Groningen. MR**0383717****[2]**S. H. Crandall,*Iterative procedures related to relaxation methods for eigenvalue problems*, Proc. Roy. Soc. London. Ser. A.**207**(1951), 416–423. MR**0042789****[3]**P. Lancaster,*A generalized Rayleigh quotient iteration for lambda-matrices*, Arch. Rational Mech. Anal.**8**(1961), 309–322. MR**0139262****[4]**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. 1, 233–241. MR**1553461**, 10.1007/BF00298007**[5a]**W. Kahan,*Inclusion Theorems for Clusters of Eigenvalues of Hermitian Matrices*, Technical Report, Dept. of Computer Sci., University of Toronto, 1967.**[5b]**B. N. Parlett and W. Kahan,*On the convergence of a practical 𝑄𝑅 algorithm. (With discussion)*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 114–118. MR**0255035****[5c]**B. Parlett,*Certain Matrix Eigenvalue Techniques Discussed from a Geometric Point of View*, AERE Report 7168, Theor. Physics Div., AERE, Berkshire, England.**[6]**Lord Rayleigh,*The Theory of Sound*, 2nd rev. ed., Macmillan, New York, 1937.**[7]**G. Temple,*The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems*, Proc. Roy. Soc. London. Ser. A.**211**(1952), 204–224. MR**0046141****[8]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422****[9]**J. H. Wilkinson,*Global convergence of tridiagonal 𝑄𝑅 algorithm with origin shifts*, Linear Algebra and Appl.**1**(1968), 409–420. MR**0234622****[10]**Marvin Marcus and Henryk Minc,*A survey of matrix theory and matrix inequalities*, Allyn and Bacon, Inc., Boston, Mass., 1964. MR**0162808****[11]**G. W. Stewart,*Error and perturbation bounds for subspaces associated with certain eigenvalue problems*, SIAM Rev.**15**(1973), 727–764. MR**0348988**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1974-0405823-3

Keywords:
Eigenvector,
eigenvalue,
iterative methods,
Rayleigh Quotient,
global convergence,
nonnormal matrix

Article copyright:
© Copyright 1974
American Mathematical Society