Structured matrix methods for polynomial root-finding

Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Volumn , Issue , 2007, Pages 175-180

  Gemignani, Luca ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

Complexity; Eigenvalue computation; Polynomial root finding; Rank structured matrices

Indexed keywords

COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING;

EIGENVALUE COMPUTATION; POLYNOMIAL ROOT FINDING; RANK STRUCTURED MATRICES;

MATRIX ALGEBRA;

EID: 36349016037     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1277548.1277573     Document Type: Conference Paper

References (24)

1. D. A. Bini, F. Daddi, and L. Gemignani. On the shifted QR iteration applied to companion matrices. Electron. Trans. Numer. Anal., 18:137-152 (electronic), 2004.
2. D. A. Bini, Y. Eidelman, L. Gemignani, and I. Gohberg. Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices. Technical Report 1587, Dipartimento di Matematica, Università di Pisa, 2005. In press in SIAM J. Matrix Anal. Appl..
3. D. A. Bini, Y. Eidelman, L. Gemignani, and I. Gohberg. The unitary completion and QR iterations for a class of structured matrices. In press in Mathematics of Computation.
4. D. A. Bini and G. Fiorentino. Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms, 23(2-3):127-173, 2000.
5. D. A. Bini, L. Gemignani, and V. Y. Pan. Improved initialization of the accelerated and robust QR-like polynomial root-finding. Electron. Trans. Numer. Anal., 17:195-205 (electronic), 2004.
6. D. A. Bini, L. Gemignani, and V. Y. Pan. Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations. Numer. Math., 100(3):373-408, 2005.
7. C. Carstensen. Linear construction of companion matrices. Linear Algebra and Appl., 149:191-214, 1991.
8. S. Chandrasekaran, M. Gu, J. Xia, and J. Zhu. A fast QR algorithm for companion matrices. 2006.
9. J. Della-Dora. Numerical linear algorithms and group theory. Linear Algebra and Appl., 10:267-283, 1975.
10. S. Delvaux and M. Van Barel. Unitary rank structured matrices. Technical Report TW464, Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3000 Leuven (Heverlee), Belgium, July 2006.
11. P. Dewilde and A. J. van der Veen. Time-varying systems and computations. Kluwer Academic Publishers, Boston, MA, 1998.
12. Y. Eidelman, L. Gemignani, and I. Gohberg. On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms. Linear Algebra Appl., 420, 2007.
13. Y. Eidelman and I. Gohberg. On a new class of structured matrices. Integral Equations Operator Theory, 34:293-324, 1999.
14. Y. Eidelman and I. Gohberg. Fast inversion algorithms for a class of structured operator matrices. Linear Algebra Appl., 371:153-190, 2003.
15. Y. Eidelman, I. Gohberg, and V. Olshevsky. The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order. Linear Algebra Appl., 404:305-324, 2005.
16. L. Eisner. On some algebraic problems in connection with general eigenvalue algorithms. Linear Algebra Appl., 26:123-138, 1979.
17. D. Fasino. Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices. Numer. Linear Algebra Appl., 12(8):743-754, 2005.
18. S. Fortune. Polynomial root finding using iterated eigenvalue computation. In Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, pages 121-128 (electronic), New York, 2001. ACM.
19. S. Fortune. An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symbolic Comput., 33(5):627-646, 2002. Computer algebra (London, ON, 2001).
20. L. Gemignani. Quasiseparable structures of companion pencils under the QZ-algorithm. Calcolo, 42(3-4):215-226, 2005.
21. G. I. Hargreaves. Topics in matrix computations: stability and efficiency of algorithms. PhD thesis, School of Mathematics, University of Manchester, 2005.
22. B. T. Smith. Error bounds for zeros of a polynomial based upon Gerschgorin's theorems. J. Assoc. Comput. Mach., 17:661-674, 1970.
23. F. Uhlig. The DQR algorithm, basic theory, convergence, and conditional stability. Numer. Math., 76(4):515-553, 1997.
24. R. Vandebril, M. Van Barel, and N. Mastronardi. An implicit QR. algorithm for symmetric semiseparable matrices. Numer. Linear Algebra Appl., 12(7):625-658, 2005.