The QR algorithm revisited

SIAM Review

Volumn 50, Issue 1, 2008, Pages 133-145

  Watkins, David S ^a  

^a WASHINGTON STATE UNIVERSITY   (United States)

Author keywords

Eigenvalue; Eigenvector; Hessenberg form; QR algorithm

Indexed keywords

ALGORITHMS; EIGENVALUES AND EIGENFUNCTIONS;

HESSENBERG FORM; QR ALGORITHM;

MATRIX ALGEBRA;

EID: 40749131323     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/060659454     Document Type: Article

References (25)

1. E. ANDERSON, Z. BAI, C. BISCHOF, S. BLACKFORD, J. DEMMEL, J. DONGARRA, J. Du CROZ, A. GREENBAUM, S. HAMMARLING, A. McKENNEY, AND D. SORENSEN, LAPACK Users' Guide, 3rd ed., SIAM, Philadelphia, 1999.
2. K. BRAMAN, R. BYERS, AND R. MATHIAS, The multishift QR algorithm. Part I: Maintaining well-focused shifts and level 3 performance, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 929-947.
3. K. BRAMAN, R. BYERS, AND R. MATHIAS, The multishift QR algorithm. Part II: Aggressive early deflation, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 948-973.
4. J. J. M. CUPPEN, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math., 36 (1981), pp. 177-195.
5. 2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem, Ph.D. thesis, University of California, Berkeley, 1997.
6. J. J. DONGARRA AND D. C. SORENSEN, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sci. Statist. Comput., 8 (1987), pp. s139-s154.
7. J. G. F. FRANCIS, The QR transformation: A unitary analogue to the LR transformation, I., Comput. J., 4 (1961), pp. 265-271.
8. J. G. F. FRANCIS, The QR transformation. II, Comput. J., 4 (1961), pp. 332-345.
9. V. N. KUBLANOVSKAYA, On some algorithms for the solution of the complete eigenvalue problem, U.S.S.R. Comput. Math. and Math. Phys., 3 (1961), pp. 637-657.
10. R. B. LEHOUCQ, D. C. SORENSEN, AND C. YANG, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998. http://www.caam.rice.edu/software/ARPACK/index.html.
11. B. N. PARLETT, The new qd algorithms, Acta Numer., (1995), pp. 459-491.
12. B. N. PARLETT AND I. S. DHILLON, Relatively robust representations of symmetric tridiagonals, Linear Algebra Appl., 309 (2000), pp. 121-151.
13. D. C. SORENSEN, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357-385.
14. D.C. SORENSEN, Numerical methods for large eigenvalue problems, Acta Numer., (2002), pp. 519-584.
15. G. W. STEWART, A Krylov-Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 601-614.
16. D. S. WATKINS, Understanding the QR algorithm, SIAM Rev., 24 (1982), pp. 427-440.
17. D. S. WATKINS, Fundamentals of Matrix Computations, 1st ed., John Wiley and Sons, New York, 1991.
18. D. S. WATKINS, Some perspectives on the eigenvalue problem, SIAM Rev., 35 (1993), pp. 430-471.
19. D. S. WATKINS, QR-like algorithms - an overview of convergence theory and practice, in The Mathematics of Numerical Analysis, J. Renegar, M. Shub, and S. Smale, eds., Lectures in Appl. Math. 32, AMS, Providence, RI, 1996, pp. 879-893.
20. D.S. WATKINS, The transmission of shifts and shift blurring in the QR algorithm, Linear Algebra Appl., 241/243 (1996), pp. 877-896.
21. D. S. WATKINS, QR-like algorithms for eigenvalue problems, J. Comput. Appl. Math, 123 (2000), pp. 67-83.
22. D. S. WATKINS, Fundamentals of Matrix Computations, 2nd ed., Wiley Interscience, New York, 2002.
23. D. S. WATKINS, The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods, SIAM, Philadelphia, 2007.
24. D. S. WATKINS AND L. ELSNER, Convergence of algorithms of decomposition type for the eigenvalue problem, Linear Algebra Appl., 143 (1991), pp. 19-47.
25. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford University, Oxford, 1965.