Solving linear systems with a Levinson-like solver

Electronic Transactions on Numerical Analysis

Volumn 26, Issue , 2007, Pages 243-269

  Vandebril, Raf ^a   Mastronardi, Nicola ^b   Van Barel, Marc ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b National Council of Italy   (Italy)

Author keywords

Levinson; Levinson conform matrices; Look ahead; System solving; Yule Walker

Indexed keywords

ALGORITHMS; FACTORIZATION; LINEAR EQUATIONS; MATRIX ALGEBRA;

LEVINSON CONFORM MATRICES; SYSTEM SOLVER; SYSTEM SOLVING; TOEPLITZ SYSTEMS;

LINEAR SYSTEMS;

EID: 37649024046     PISSN: None     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. G. S. AMMAR, D. CALVETTI, W. B. GRAGG, AND L. REICHEL, Polynomial zerofinders based on Szegö polynomials, J. Comput. Appl. Math., 127 (2001), pp. 1-16.
2. G. S. AMMAR, D. CALVETTI, AND L. REICHEL, Continuation methods for the computation of zeros of Szegö polynomials, Linear Algebra Appl., 249 (1996), pp. 125-155.
3. S. BARNETT, Polynomials and Linear Control Systems, Marcel Dekker Inc. 1983.
4. D. A. BINI AND V. Y. PAN, Polynomial and Matrix Computations, vol. 1: Fundamental Algorithms, Birkhäuser, Boston, 1994.
5. C. F. BORGES AND W. B. GRAGG, A parallel divide and conquer algorithm for the generalized real symmetric definite tridiagonal eigenproblem, in Numerical Linear Algebra and Scientific Computing, L. Reichel, A. Ruttan, and R. S. Varga, eds., de Gruyter, Berlin, 1993, pp. 11-29.
6. D. CALVETTI, S. KIM, AND L. REICHEL, The restarted QR-algorithm for eigenvalue computation of structured matrices, J. Comput. Appl. Math., 149 (2002), pp. 415-422.
7. T. F. CHAN AND P. C. HANSEN, A look-ahead Levinson algorithm for indefinite Toeplitz systems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 490-506.
8. S. CHANDRASEKARAN AND M. GU, Fast and stable algorithms for banded plus semiseparable systems of linear equations, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 373-384.
9. J. J. M. CUPPEN, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math., 36 (1981), pp. 177-195.
10. A. M. CUYT AND B. VERDONK, Different techniques for the construction of multivariate rational interpolants and padé approximants, Universitaire instelling Antwerpen, 1988.
11. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
12. P. DEWILDE AND A.-J. VAN DER VEEN, Time-varying systems and computations, Kluwer Academic Publishers, Boston, June 1998.
13. Y. EIDELMAN, Fast recursive algorithm for a class of structured matrices, Appl. Math. Lett., 13 (2000), pp. 57-62.
14. Y. EIDELMAN AND I. C. GOHBERG, Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices, Comput. Math. Appl., 33 (1996), pp. 69-79.
15. _, A look-ahead block Schur algorithm for diagonal plus semiseparable matrices, Comput. Math. Appl., 35 (1997), pp. 25-34.
16. _, On a new class of structured matrices, Integral Equations Operator Theory, 34 (1999), pp. 293-324.
17. _, Fast inversion algorithms for a class of block structured matrices, Contemp. Math., 281 (2001), pp. 17-38.
18. _, A modification of the Dewilde-van der Veen method for inversion of finite structured matrices, Linear Algebra Appl., 343-344 (2002), pp. 419-450.
19. F. R. GANTMACHER AND M. G. KREǏN, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, AMS Chelsea Publishing, 2002.
20. I. C. GOHBERG, T. KAILATH, AND I. KOLTRACHT, Efficient solution of linear systems of equations with recursive structure, Linear Algebra Appl., 80 (1986), pp. 81-113.
21. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Third ed., The Johns Hopkins University Press, Baltimore, 1996.
22. F. A. GRAYBILL, Matrices with Applications in Statistics, Wadsworth International Group, Belmont, California, 1983.
23. L. GREENGARD AND V. ROKHLIN, On the numerical solution of two-point boundary value problems, Comm. Pure Appl. Math., 44 (1991), pp. 419-452.
24. T. KAILATH AND A. H. SAYED, eds., Fast Reliable Algorithms for Matrices with Structure, SIAM, Philadelphia, 1999.
25. N. LEVINSON, The Wiener RMS error criterion in filter desing and prediction, J. Math. Phys., 25 (1947), pp. 261-278.
26. N. MASTRONARDI, S. CHANDRASEKARAN, AND S. VAN HUFFEL, Fast and stable two-way algorithm for diagonal plus semi-separable systems of linear equations, Numer. Linear Algebra Appl., 8 (2001), pp. 7-12.
27. N. MASTRONARDI.M. VAN BAREL, AND R. VANDEBRIL,ALevinson-like algorithm for symmetric positive definite semiseparable plus diagonal matrices, Tech. Rep. TW423, Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3000 Leuven (Heverlee), Belgium, Mar. 2005.
28. H. P. J. STARR, On the numerical solution of one-dimensional integral and differential equations, PhD thesis, Yale University, 1992. Research Report YALEU/DCS/RR-888.
29. L.N. TREFETHEN AND D. BAU, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
30. E. VAN CAMP, N. MASTRONARDI, AND M. VAN BAREL, Two fast algorithms for solving diagonal-plus-semiseparable linear systems, J. Comput. Appl. Math., 164-165 (2004), pp. 731-747.
31. R. VANDEBRIL, N. MASTRONARDI, AND M. VAN BAREL, A Levinson-like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices, J. Comput. Appl. Math., 198 (2007), pp. 75-97.
32. R. VANDEBRIL, M. VAN BAREL, AND N. MASTRONARDI, A note on the representation and definition of semiseparable matrices, Numer. Linear Algebra Appl., 12 (2005), pp. 839-858.