Convergence of the shifted algorithm for unitary Hessenberg matrices

Authors:
Tai-Lin Wang and William B. Gragg

Journal:
Math. Comp. **71** (2002), 1473-1496

MSC (2000):
Primary 65F15, 15A18

DOI:
https://doi.org/10.1090/S0025-5718-01-01387-4

Published electronically:
November 30, 2001

MathSciNet review:
1933041

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper shows that for unitary Hessenberg matrices the algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

**1.**H. Bowdler, R. S. Martin, C. Reinsch, and J. H. Wilkinson,*The and algorithms for symmetric matrices*, Numer. Math.**11**(1968), 293-306.**2.**H. J. Buurema,*A geometric proof of convergence for the method*, doctoral dissertation, Rijksuniversiteit Te Groningen, Groningen, The Netherlands, 1970. MR**52:4597****3.**T. J. Dekker and J. F. Traub,*The shifted algorithm for Hermitian matrices*, Linear Algebra Appl.**4**(1971), 137-154. MR**44:1207****4.**P. J. Eberlein and C. P. Huang,*Global convergence of the algorithm for unitary matrices with some results for normal matrices*, SIAM J. Numer. Anal.**12**(1975), 97-104. MR**50:8948****5.**J. G. F. Francis,*The transformation*. I, II, Comput. J.**4**(1961-1962), 265-271, 332-345. MR**25:744****6.**W. B. Gragg,*The algorithm for unitary Hessenberg matrices*, J. Comput. Appl. Math.**16**(1986), 1-8.**7.**W. B. Gragg,*Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle*, J. Comput. Appl. Math.**46**(1993), 183-198. MR**94e:65046****8.**W. Hoffmann and B. N. Parlett,*A new proof of global convergence for the tridiagonal algorithm*, SIAM J. Numer. Anal.**15**(1978), 929-937. MR**80a:65075****9.**C. P. Huang,*On the convergence of the QR algorithm with origin shifts for normal matrices*, IMA J. Numer. Anal.**1**(1981), 127-133. MR**87d:65130****10.**E. Jiang and Z. Zhang,*A new shift of the algorithm for irreducible symmetric tridiagonal matrices*, Linear Algebra Appl.**65**(1985), 261-272. MR**86g:65082****11.**B. N. Parlett,*Singular and invariant matrices under the transformation*, Math. Comp.**20**(1966), 611-615. MR**35:3870****12.**B. N. Parlett,*The Rayleigh quotient iteration and some generalizations for nonnormal matrices*, Math. Comp.**28**(1974), 679-693. MR**53:9615****13.**B. N. Parlett,*The Symmetric Eigenvalue Problem*, Prentice-Hall, Englewood Cliffs, N.J., 1980. MR**81j:65063****14.**T.-L. Wang,*Convergence of the algorithm with origin shifts for real symmetric tridiagonal and unitary Hessenberg matrices*, Ph.D. thesis, University of Kentucky, Lexington, Kentucky, 1988.**15.**T.-L. Wang and W. B. Gragg,*Convergence of the shifted algorithm for unitary Hessenberg matrices*, Technical Report NPS-53-90-007, Naval Postgraduate School, Monterey, California, 1990.**16.**D.S. Watkins and L. Elsner,*Convergence of algorithms of decomposition type for the eigenvalue problem*, Linear Algebra Appl.**143**(1991), 19-47. MR**91m:65114****17.**J. H. Wilkinson,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965.**18.**J. H. Wilkinson,*Global convergence of tridiagonal algorithm with origin shifts*, Linear Algebra Appl.**1**(1968), 409-420. MR**38:2938**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65F15,
15A18

Retrieve articles in all journals with MSC (2000): 65F15, 15A18

Additional Information

**Tai-Lin Wang**

Affiliation:
Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan, Republic of China

Email:
wang@math.nccu.edu.tw

**William B. Gragg**

Affiliation:
Department of Mathematics, Naval Postgraduate School, Monterey, California 93943

Email:
gragg@nps.navy.mil

DOI:
https://doi.org/10.1090/S0025-5718-01-01387-4

Keywords:
$QR$ algorithm,
shift strategy,
unitary Hessenberg matrices

Received by editor(s):
March 9, 1999

Received by editor(s) in revised form:
January 6, 2000, and November 7, 2000

Published electronically:
November 30, 2001

Additional Notes:
The research of the first author was supported by the Center for Computational Sciences at the University of Kentucky

The research of the second author was supported in part by the National Science Foundation under grant DMS-8704196

Dedicated:
Dedicated to the memory of James H. Wilkinson

Article copyright:
© Copyright 2001
American Mathematical Society