Continuation methods for the computation of zeros of Szego polynomials

Linear Algebra and Its Applications

Volumn 249, Issue 1-3, 1996, Pages 125-155

  Ammar, G S ^a   Calvetti, D ^b   Reichel, L ^c  

^a NORTHERN ILLINOIS UNIVERSITY   (United States)

^b Stevens Institute of Technology   (United States)

^c KENT STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21444440987     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/0024-3795(95)00324-X     Document Type: Article

References (41)

1. E. L. Allgower and K. Georg, Numerical Continuation Methods, Springer-Verlag, Berlin, 1990.
2. E. L. Allgower and K. Georg, Continuation and path following, in Acta Numer. 1993, Cambridge U.P., Cambridge, 1993, pp. 1-64.
3. G. S. Ammar and W. B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in Rational Approximation and Its Applications in Mathematics and Physics (J. Gilewicz, M. Pindor, and W. Siemaszko, Eds.), Lecture Notes in Math. 1237, Springer-Verlag, Berlin, 1987, pp. 313-330.
4. G. S. Ammar and W. B. Gragg, Numerical experience with a superfast real Toeplitz solver, Linear Algebra Appl. 121:185-206 (1989).
5. G. S. Ammar, W. B. Gragg, and L. Reichel, On the eigenproblem for orthogonal matrices, in Proceedings 25th IEEE Conference on Decision and Control, Athens, 1986, pp. 1963-1966.
6. G. S. Ammar, W. B. Gragg, and L. Reichel, Downdating Szego polynomials and data fitting applications, Linear Algebra Appl. 172:315-336 (1992).
7. G. S. Ammar, W. B. Gragg, and L. Reichel, An analogue for Szego polynomials of the Clenshaw algorithm, J. Comput. Appl. Math. 40:211-216 (1993).
8. G. S. Ammar, L. Reichel, and D. C. Sorensen, An implementation of a divide and conquer algorithm for the unitary eigenproblem, ACM Trans. Math. Software 18:292-307 (1992).
9. G. S. Ammar, L. Reichel, and D. C. Sorensen, Algorithm 730: An implementation of a divide and conquer algorithm for the unitary eigenproblem, ACM Trans. Math. Software 20:161 (1994).
10. K. Brumme, Formantenbestimmung in der digitalen Sprachverarbeitung durch LPC - und verwandte Verfahren, Diplomarbeit, Dept. of Applied Mathematics, Univ. of Hamburg, Hamburg, Germany, 1987.
11. A. Bunse-Gerstner and L. Elsner, Schur parameter pencils for the solution of the unitary eigenproblem, Linear Algebra Appl. 154-156:741-778 (1991).
12. J. A. Cadzow, Foundations of Digital Signal Processing and Data Analysis, Macmillan, New York, 1987.
13. J. W. Demmel, Trading off parallelism and numerical stability, in Linear Algebra for Large Scale and Real-Time Applications (M. S. Moonen, G. H. Golub, and B. L. R. De Moor, Eds.), Kluwer, Dordrecht, 1993, pp. 49-68.
14. P. Deuflhard and A. Hohmann, Numerische Mathematik, de Gruyter, Berlin, 1991.
15. J. J. Dongarra and M. Sidani, A parallel algorithm for the non-symmetric eigenvalue problem, SIAM J. Sci. Comput. 14:542-569 (1993).
16. K. Fan and A. J. Hoffman, Some metric inequalities in the space of matrices, Proc. Amer. Math. Soc. 6:111-116 (1955).
17. S. Goedecker, Remark on algorithms to find roots of polynomials, SIAM J. Sci. Comput. 15:1059-1063 (1994).
18. G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins U.P., Baltimore, 1989.
19. W. B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle (in Russian), in Numerical Methods in Linear Algebra (E. S. Nikolaev, Ed.), Moscow U.P., Moscow, 1982, pp. 16-32; English transl., rev., J. Comput. Appl. Math. 46:183-198 (1993).
20. W. B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle (in Russian), in Numerical Methods in Linear Algebra (E. S. Nikolaev, Ed.), Moscow U.P., Moscow, 1982, pp. 16-32; English transl., rev., J. Comput. Appl. Math. 46:183-198 (1993).
21. W. B. Gragg, The QR algorithm for unitary Hessenberg matrices, J. Comput. Appl. Math. 16:1-8 (1986).
22. W. B. Gragg and L. Reichel, a divide and conquer method for the unitary eigenproblem, in Hypercube Multiprocessors 1987 (M. T. Heath, Ed.), SIAM, Philadelphia, 1987, pp. 639-647.
23. W. B. Gragg and L. Reichel, A divide and conquer method for unitary and orthogonal eigenproblems, Numer. Math. 57:695-718 (1990).
24. U. Grenander and G. Szego, Toeplitz Forms and Their Applications, 2nd ed., Chelsea, New York, 1984.
25. C. Jagels and L. Reichel, On the construction of Szego polynomials, J. Comput. Appl. Math. 46:241-254 (1993).
26. W. B. Jones, O. Njåstad, and E. B. Saff, Szego polynomials associated with Wiener-Levinson filters, J. Comput. Appl. Math. 32:387-406 (1990).
27. W. B. Jones, W. J. Thron, O. Njåstad, and H. Waadeland, Szego polynomials applied to frequency analysis, J. Comput. Appl. Math. 40:217-228 (1993).
28. T. Kailath, Linear estimation for stationary and near-stationary processes, in Modern Signal Processing (T. Kailath, Ed.), Hemisphere, New York, 1985, pp. 59-128.
29. T. Kailath, Signal processing applications of some moment problems, in Moments in Mathematics (H. J. Landau, Ed.), Amer. Math. Soc., Providence, 1987, pp. 71-109.
30. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1976.
31. T.-Y. Li and N. H. Rhee, Homotopy algorithm for symmetric eigenvalue problems, Numer. Math. 55:265-280 (1989).
32. T.-Y. Li and Z. Zeng, Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems, Math. Comp. 59:483-502 (1992).
33. T.-Y. Li, H. Zhang, and X.-H. Sun, Parallel homotopy algorithm for the symmetric tridiagonal eigenvalue problem, SIAM J. Sci. Statist. Comput. 12:469-487 (1991).
34. J. S. Lim, Fundamentals of digital signal processing, in Modern Signal Processing (T. Kailath, Ed.), Hemisphere, New York, 1985, pp. 1-57.
35. K. Pan and E. B. Saff, Asymptotics for zeros of Szego polynomials associated with trigonometric polynomial signals, J. Approx. Theory 71:239-251 (1992).
36. B. N. Parlett and J. Le, Forward instability of tridiagonal QR, SIAM J. Matrix Anal. Appl. 14:279-316 (1993).
37. L. Reichel and G. S. Ammar, Fast approximation of dominant harmonics by solving an orthogonal eigenvalue problem, in Mathematics in Signal Processing II, (J. G. McWhirter, Ed.), Clarendon, Oxford, 1990, pp. 575-591.
38. B. Smith, J. Boyle, Y. Ikebe, V. Klema, and C. Moler, Matrix Eigensystem Routines: EISPACK Guide, 2nd ed., Springer-Verlag, Berlin, 1976.
39. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
40. G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc., Providence, 1975.
41. K.-C. Toh and L. H. Trefethen, Pseudozeros of polynomials and pseudospectra of companion matrices, Numer. Math., 68:403-425 (1994).