Convergence of the unitary algorithm with a unimodular Wilkinson shift

Authors:
Tai-Lin Wang and William B. Gragg

Journal:
Math. Comp. **72** (2003), 375-385

MSC (2000):
Primary 65F15, 15A18

DOI:
https://doi.org/10.1090/S0025-5718-02-01444-8

Published electronically:
June 25, 2002

MathSciNet review:
1933826

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In applying the algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a *projected* Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

**1.**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****2.**W. B. Gragg,*The algorithm for unitary Hessenberg matrices*, J. Comput. Appl. Math.**16**(1986), 1-8.**3.**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****4.**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****5.**R. S. Martin, G. Peters, and J. H. Wilkinson,*The algorithm for real Hessenberg matrices*, Numer. Math.**14**(1970), 219-231.**6.**B. N. Parlett,*The Symmetric Eigenvalue Problem*, Prentice-Hall, Englewood Cliffs, NJ, 1980. MR**81j:65063****7.**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, KY, 1988.**8.**T.-L. Wang,*Convergence of the tridiagonal algorithm*, Linear Algebra Appl.**322**(2001), 1-17.**9.**T.-L. Wang and W. B. Gragg,*Convergence of the shifted algorithm for unitary Hessenberg matrices*, to appear in Math. Comp.**10.**T.-L. Wang and W. B. Gragg,*Convergence of the unitary Hessenberg algorithm with unimodular shifts*, Report NPS-53-90-008, Naval Postgraduate School, Monterey, CA, 1990.**11.**J. H. Wilkinson,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965. MR**32:1894****12.**J. H. Wilkinson,*Global convergence of tridiagonal algorithm with origin shifts*, Linear Algebra Appl.**1**(1968), 409-420. MR**38:2938**

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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/S0025-5718-02-01444-8

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

Received by editor(s):
May 5, 1999

Received by editor(s) in revised form:
April 3, 2001

Published electronically:
June 25, 2002

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

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

Article copyright:
© Copyright 2002
American Mathematical Society