The Rayleigh quotient iteration and some generalizations for nonnormal matrices
Author:
B. N. Parlett
Journal:
Math. Comp. 28 (1974), 679693
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]
H. J. Buurema, A Geometric Proof of Convergence for the QR Method, Report TW62, Mathematisch Institut, Groningen, North East Netherlands, 1968. MR 0383717 (52:4597)
 [2]
S. H. Crandall, "Iterative procedures related to relaxation methods for eigenvalue problems," Proc. Roy. Soc. London Ser. A, v. 207, 1951, pp. 416423. MR 13, 163. MR 0042789 (13:163b)
 [3]
P. Lancaster, "A generalized Rayleigh quotient iteration for lambdamatrices," Arch. Rational Mech. Anal., v. 8, 1961, pp. 309322. MR 25 #2697. MR 0139262 (25:2697)
 [4]
A. M. Ostrowski, "On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors. IVI," Arch. Rational Mech. Anal., v. 14, 1958/59, pp. 233241, 423428, 325340, 341347, 472481, 153165. MR 21 #427; #4541a, b; #6691; 22 #8654. MR 1553461
 [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 & W. Kahan, "On the convergence of a practical QR algorithm. (With discussion)," Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), vol. I: Mathematics, Software, NorthHolland, Amsterdam, 1969, pp. 114118. MR 40 #8242. MR 0255035 (40:8242)
 [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, v. 211, 1952, pp. 204224. MR 13, 691. MR 0046141 (13:691h)
 [8]
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. MR 0184422 (32:1894)
 [9]
J. H. Wilkinson, "Global convergence of tridiagonal QR algorithm with origin shifts," Linear Algebra and Appl., v. 1, 1968, pp. 409420. MR 38 #2938. MR 0234622 (38:2938)
 [10]
M. Marcus & H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, Mass., 1964. MR 29 #112. MR 0162808 (29:112)
 [11]
G. W. Stewart, "Error and perturbation bounds for subspaces associated with certain eigenvalue problems," SIAM J. Numer. Anal. (To appear.) MR 0348988 (50:1482)
 [1]
 H. J. Buurema, A Geometric Proof of Convergence for the QR Method, Report TW62, Mathematisch Institut, Groningen, North East Netherlands, 1968. MR 0383717 (52:4597)
 [2]
 S. H. Crandall, "Iterative procedures related to relaxation methods for eigenvalue problems," Proc. Roy. Soc. London Ser. A, v. 207, 1951, pp. 416423. MR 13, 163. MR 0042789 (13:163b)
 [3]
 P. Lancaster, "A generalized Rayleigh quotient iteration for lambdamatrices," Arch. Rational Mech. Anal., v. 8, 1961, pp. 309322. MR 25 #2697. MR 0139262 (25:2697)
 [4]
 A. M. Ostrowski, "On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors. IVI," Arch. Rational Mech. Anal., v. 14, 1958/59, pp. 233241, 423428, 325340, 341347, 472481, 153165. MR 21 #427; #4541a, b; #6691; 22 #8654. MR 1553461
 [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 & W. Kahan, "On the convergence of a practical QR algorithm. (With discussion)," Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), vol. I: Mathematics, Software, NorthHolland, Amsterdam, 1969, pp. 114118. MR 40 #8242. MR 0255035 (40:8242)
 [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, v. 211, 1952, pp. 204224. MR 13, 691. MR 0046141 (13:691h)
 [8]
 J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. MR 0184422 (32:1894)
 [9]
 J. H. Wilkinson, "Global convergence of tridiagonal QR algorithm with origin shifts," Linear Algebra and Appl., v. 1, 1968, pp. 409420. MR 38 #2938. MR 0234622 (38:2938)
 [10]
 M. Marcus & H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, Mass., 1964. MR 29 #112. MR 0162808 (29:112)
 [11]
 G. W. Stewart, "Error and perturbation bounds for subspaces associated with certain eigenvalue problems," SIAM J. Numer. Anal. (To appear.) MR 0348988 (50:1482)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197404058233
PII:
S 00255718(1974)04058233
Keywords:
Eigenvector,
eigenvalue,
iterative methods,
Rayleigh Quotient,
global convergence,
nonnormal matrix
Article copyright:
© Copyright 1974
American Mathematical Society
