Computations with quasiseparable polynomials and matrices

Theoretical Computer Science

Volumn 409, Issue 2, 2008, Pages 158-179

  Bella, T ^a   Eidelman, Y ^c   Gohberg, I ^c   Olshevsky, V ^b  

^a University of Rhode Island   (United States)

^b UNIVERSITY OF CONNECTICUT   (United States)

^c TEL AVIV UNIVERSITY   (Israel)

Author keywords

Bj rck Pereyra algorithm; Digital filters; Divide and conquer eigenvalue algorithms; Orthogonal polynomials; QR iteration; Quasiseparable matrices; Recurrence relations; Semiseparable matrices; Signal flow graphs; Szeg polynomials; Traub algorithm

Indexed keywords

ALGORITHMS; DIGITAL ARITHMETIC; DIGITAL FILTERS; EIGENVALUES AND EIGENFUNCTIONS; FOURIER ANALYSIS; NUMERICAL ANALYSIS; POLYNOMIAL APPROXIMATION; SURVEYS; WAVE FILTERS;

DIVIDE AND CONQUER EIGENVALUE ALGORITHMS; ORTHOGONAL POLYNOMIALS; QR ITERATION; QUASISEPARABLE MATRICES; RECURRENCE RELATIONS; SEMISEPARABLE MATRICES; SIGNAL FLOW GRAPHS; TRAUB ALGORITHM;

POLYNOMIALS;

EID: 55749098472     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2008.09.008     Document Type: Article

References (56)

1. T. Bella, Topics in numerical linear algebra related to quasiseparable and other structured matrices, Ph.D. Thesis, University of Connecticut, advisor: Vadim Olshevsky. Available online: http://www.tombella.com, 2008
2. Bella T., Eidelman Y., Gohberg I., Koltracht I., and Olshevsky V. A Björck-Pereyra-type algorithm for Szegö-Vandermonde matrices based on properties of unitary Hessenberg matrices. Linear Algebra and Its Applications 420 (2007) 634-647
3. T. Bella, Y. Eidelman, I. Gohberg, I. Koltracht, V. Olshevsky, A fast Bjorck-Pereyra like algorithm for solving Hessenberg-quasiseparable-Vandermonde systems, SIAM Journal of Matrix Analysis (SIMAX) (2007) (in press)
4. T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, Classifications of three-term and two-term recurrence relations and digital filter structures via subclasses of quasiseparable matrices, SIAM Journal of Matrix Analysis (SIMAX) (2007) (in press)
5. T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov, P. Zhlobich, A Traub-like algorithm for Hessenberg-quasiseparable-Vandermonde matrices of arbitrary order, in: V. Olshevsky (Ed.), Georg Heinig Memorial Volume, Birkhauser Verlag
6. T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, The QR iteration method for Hermitian out-of-band-quasiseparable matrices of an arbitrary order, preprint
7. T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov, Fast Traub-like inversion algorithm for Hessenberg order one quasiseparable Vandermonde matrices, Journal of Complexity (2007) (in press)
8. T. Bella, V. Olshevsky, P. Zhlobich, Classifications of recurrence relations via subclasses of (H, m)-quasiseparable matrices, in: S. Bhattacharyya, R. Chan, P. Van Dooren, V. Olshevsky, A. Routray (Eds.), Numerical Linear Algebra in Signals, Systems and Control, Springer-Verlag (in press)
9. Bini D.A., Gemignani L., and Pan V.Y. Fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equation. Numerische Math. 3 (2005) 373-408 Also Technical Report 1470, Department of Math., University of Pisa, Pisa, Italy, July 2003
10. Björck A., and Pereyra V. Solution of vandermonde systems of equations. Mathematics of Computation 24 (1970) 893-903
11. Bevilacqua R., and Del Corso G.M. Structural properties of matrix unitary reduction to semiseparable form. Calcolo 41 4 (2004) 177-202
12. Bini D., and Pan V.Y. Computing matrix eigenvalues and polynomial zeros where the output is real. SIAM Journal on Computing 27 4 (1998) 1099-1115 Proceedings version: Parallel complexity of tridiagonal symmetric eigenvalue problem, in: Proceedings of 2nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'91, ACM Press, New York, and SIAM Publications, Philadelphia, January 1991, pp. 384-393
13. Bini D.A., and Pan V.Y. Practical improvement of the divide-and-conquer eigenvalue algorithms. Computing 48 (1992) 109-123
14. Bini D.A., and Pan V.Y. Matrix and Polynomial Computations, Vol. I: Fundamental Algorithms (1994), Birkhauser, Boston
15. Cuppen J. A divide and conquer method for the symmetric tridiagonal eigenproblem. Numerische Mathematik 36 (1981) 177-195
16. Calvetti D., and Reichel L. Fast inversion of Vandermonde-like matrices involving orthogonal polynomials. BIT 33 (1993) 473-485
17. Chandrasekaran S., Gu M., Xia J., and Zhu J. A fast QR algorithm for companion matrices. In: Olshevsky V. (Ed). Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications vol. 179 (2007), Birkhauser, Basel 111-143
18. Datta K.B., and Mohan B.M. Orthogonal Functions in Systems and Control (1995), World Scientific
19. Dewilde P., and Van der Veen A.J. Time-Varying Systems and Computations (1998), Kluwer Academic Publishers, Boston
20. Eidelman Y., and Gohberg I. On a new class of structured matrices. Integral Equations and Operator Theory 3 4 (1999) 293-324
21. Eidelman Y., Gohberg I., and Olshevsky V. The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order. Linear Algebra and Its Applications 404 (2005) 305-324
22. Eidelman Y., Gohberg I., and Olshevsky V. Eigenstructure of order-one-quasiseparable matrices. Three-term and two-term recurrence relations. Linear Algebra and its Applications 405 (2005) 1-40
23. Fasino D. Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices. Numerical Linear Algebra and its Applications 12 (2005) 743-754
24. Fasino D., and Gemignani L. Direct and inverse eigenvalue problems for diagonal plussemiseparable matrices. Numerical Algorithms 34 (2003) 313-324
25. Gragg W.B. Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle. In: Nikolaev E.S. (Ed). Numerical Methods in Linear Algebra (1982), Moskow University Press 16-32 (in Russian). English translation in Journal of Computational and Applied Mathematics 46 (1993) 183-198
26. Gu M., and Eisenstat S. A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. SIAM Journal on Matrix Analysis and Applications 16 (1995) 172-191
27. Gohberg I., and Olshevsky V. Fast inversion of Chebyshev-Vandermonde matrices. Numerische Mathematik 67 1 (1994) 71-92
28. Gohberg I., and Olshevsky V. The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices. Journal of Complexity 13 2 (1997) 208-234
29. Gragg W., and Reichel L. A divide and conquer method for unitary and orthogonal eigenproblems. Numerische Mathematik 57 (1990) 695-718
30. Grenander U., and Szegö G. Toeplitz Forms and Applications (1958), University of California Press
31. Higham N.J. Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM Journal on Matrix Analysis and Applications 11 1 (1990) 23-41
32. Heinig G., and Rost K. Algebraic Methods for Toeplitz-like Matrices and Operators (1984), Birkhauser
33. Jackson D. Fourier Series and Orthogonal Polynomials (2004), Dover, New York
34. Kailath T., and Olshevsky V. Displacement structure approach to discrete trigonometric transform based preconditioners of G. Strang and T. Chan types. SIAM Journal on Matrix Analysis and Applications 26 3 (2005) 706-734
35. In: Kailath T., and Sayed A. (Eds). Fast Reliable Algorithms for Matrices with Structure (1999), SIAM, PA
36. Kailath T. A theorem of I. Schur and its Impact on modern signal processing. In: Gohberg (Ed). Operator Theory: Advances and Applications vol. 18 (1986), Birkhauser, Boston 9-30
37. Kailath T. Signal processing applications of some moment problems. In: Landau (Ed). Moments in Mathematics vol. 37 (1987), American Mathematical Society 71-109
38. Kailath T., Kung S.-Y., and Morf M. Displacement ranks of matrices and linear equations. Journal of Mathematical Analysis and Applications 68 2 (1979) 395-407
39. Mastronardi N., Van Camp E., and Van Barel M. Divide and conquer algorithms for computing the eigendecomposition of symmetric diagonal-plus-semiseparable matrices. Numerical Algorithms 9 (2005) 379-398
40. Olshevsky V. Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szego-Vandermonde matrices via Discrete Transmission lines. Linear Algebra and its Applications 285 (1998) 37-67
41. Olshevsky V. Unitary Hessenberg matrices and the generalized Parker-Forney-Traub algorithm for inversion of Szegö-Vandermonde matrices. In: Bini D., Tyrtyshnikov E., and Yalamov. P. (Eds). Structured Matrices: Recent Developments in Theory and Computation (2001), NOVA Science Publ. 67-78
42. In: Olshevsky V. (Ed). Structured Matrices in Mathematics, Computer Science, and Engineering. CONM280 vol. 1, 2 (2001), AMS Publications
43. Olshevsky V. Pivoting for structured matrices and rational tangential interpolation. In: Olshevsky V. (Ed). Fast Algorithms for Structured Matrices: Theory and Applications. CONM/323 (2003), AMS Publications 1-75
44. Olshevsky V., and Pan V.Y. A unified superfast algorithm for boundary rational tangential interpolation problems and for inversion and factorization of dense structured matrices. Proc. of 39th Annual Symposium on Foundations of Computer Science. FOCS'98 (1998), IEEE Computer Society, Los Alamitos, CA 192-201
45. Olshevsky V., and Shokrollahi A. A displacement structure approach to efficient decoding of Reed-Solomon and algebraic geometric codes. A Conference Paper Appeared in Proc of of the Thirty First ACM Symposium on Theory of Computing. STOC'99 (1999), ACM, New York 235-244
46. Olshevsky V., and Shokrollahi A. Fast matrix-vector multiplication algorithms for confluent Cauchy-like matrices with applications. Proc of the Thirty Second ACM Symposium on Theory of Computing. STOC'00 (2000), ACM, New York 573-581
47. Porat B. A Course in Digitial Signal Processing (1996), Wiley
48. Pan V.Y. Structured Matrices and Polynomials: Unified Superfast Algorithms (2001), Birkhäuser/Springer, Boston/New York
49. Reichel L., and Opfer G. Chebyshev-Vandermonde systems. Mathematics of Computation 57 (1991) 703-721
50. Szegö G. Orthogonal Polynomials (1991), AMS Publications
51. Stoer J., and Bulirsch R. Introduction to Numerical Analysis. 3rd ed. (2002), Springer
52. Traub J. Associated polynomials and uniform methods for the solution of linear problems. SIAM Review 8 3 (1966) 277-301
53. Tyrtyshnikov E.E. How bad are Hankel matrices?. Numerische Mathematik 67 (1994) 261-269
54. Van Barel M., Vandebril R., and Mastronardi N. An orthogonal similarity reduction of a matrix into semiseparable form. SIAM Journal on Matrix Analysis and Applications 27 1 (2005) 176-197
55. Vandebril R., Van Barel M., and Mastronardi N. An implicit QR algorithm for symmetric semiseparable matrices. Numerical Linear Algebra with Applications 12 7 (2005) 659-682
56. Vandebril R., Van Barel M., and Mastronardi N. Matrix Computations and Semiseparable Matrices: Linear Systems (December 18, 2007), The Johns Hopkins University Press