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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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
(52 #4597)
 [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
(13,163b)
 [3]
P.
Lancaster, A generalized Rayleigh quotient iteration for
lambdamatrices, Arch. Rational Mech. Anal. 8 (1961),
309–322. MR 0139262
(25 #2697)
 [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, http://dx.doi.org/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) NorthHolland, Amsterdam, 1969,
pp. 114–118. 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. 211 (1952), 204–224. MR 0046141
(13,691h)
 [8]
J.
H. Wilkinson, The algebraic eigenvalue problem, Clarendon
Press, Oxford, 1965. MR 0184422
(32 #1894)
 [9]
J.
H. Wilkinson, Global convergence of tridiagonal 𝑄𝑅
algorithm with origin shifts, Linear Algebra and Appl.
1 (1968), 409–420. MR 0234622
(38 #2938)
 [10]
Marvin
Marcus and Henryk
Minc, A survey of matrix theory and matrix inequalities, Allyn
and Bacon Inc., Boston, Mass., 1964. MR 0162808
(29 #112)
 [11]
G.
W. Stewart, Error and perturbation bounds for subspaces associated
with certain eigenvalue problems, SIAM Rev. 15
(1973), 727–764. 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)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65F15
Retrieve articles in all journals
with MSC:
65F15
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
