Displacement structure approach to discrete-trigonometric-transform based preconditioners of G.Strang type and of T.Chan type

SIAM Journal on Matrix Analysis and Applications

Volumn 26, Issue 3, 2005, Pages 706-734

  Kailath, Thomas ^a   Olshevsky, Vadim ^b  

^a Stanford University   (United States)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

Conjugate gradient; DCT; Discrete convolution; Displacement structure; DST; Preconditioner

Indexed keywords

CONJUGATE GRADIENT; DCT; DISCRETE CONVOLUTION; DISPLACEMENT STRUCTURE; DST; PRECONDITIONER;

EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; LINEAR EQUATIONS; MATRIX ALGEBRA; REAL TIME SYSTEMS; SET THEORY;

DISCRETE TIME CONTROL SYSTEMS;

EID: 22944487678     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479896312560     Document Type: Article

References (45)

1. [BB90] D. BINI AND F. DI BENEDETTO, A new preconditioner for the parallel solution of positive definite Toeplitz systems, in Proceedings of the Second ACM Symposium on Parallel Algorithms and Architectures, Crete, Greece, 1990, pp. 220-223.
2. [BC83] D. BINI AND M. CAPOVANI, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52(1983), pp. 99-126.
3. [BF93] D. BINI AND P. FAVATI, On a matrix algebra related to the discrete Hartley transform, SIAM J. Matrix Anal. Appl., 14(1993), pp. 500-507.
4. [BK95] E. BOMAN AND I. KOLTRACHT, Fast transform based preconditioned for Toeplitz equations, SIAM J. Matrix Anal. Appl., 16(1995), pp. 628-645.
5. [BK094] T. BOROS, T. KAILATH, AND V. OLSHEVSKY, Pivoting and backward stability of fast algorithm for solving Cauchy linear equations, Linear Algebra Appl., 343/344(2002), pp. 63-99.
6. [C88] T. F. CHAN, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput., 9(1988), pp. 766-771.
7. [C89] R. H. CHAN, Circulant preconditioned for Hermitian Toeplitz systems, SIAM J. Matrix Anal. Appl., 10(1989), pp. 542-550.
8. [CJY91] R. CHAN, X. JIN, AND M. YEUNG, The circulant operator in the Banach algebra of matrices, Linear Algebra Appl., 149(1991), pp. 41-53.
9. [CNP94] R. CHAN, J. NAGY, AND R. PLEMMONS, Displacement preconditioner for Toeplitz least squares iteration, Electron. Trans. Numer. Anal., 2(1994), pp. 44-56.
10. [CNW96] R. CHAN, M. No, AND C. WONG, Sine transform based preconditioned for symmetric Toeplitz systems, Linear Algebra Appl., 232(1996), pp. 237-259.
11. [CS89] R. H. CHAN AND G. STRANG, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Stat. Comput., 10(1989), pp. 104-119.
12. [CY92] R. CHAN AND M. YENG, Circulant preconditioned for Toeplitz matrices with positive continuous generating functions, Math. Comp., 58(1992), pp. 233-240.
13. [CN96] R. H. CHAN AND M. K. No, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38(1996), pp. 427-482.
14. [DFZ95] C. DI FIORE AND P. ZELLINI, Matrix decompositions using displacement rank and classes of commutative matrix algebras, Linear Algebra Appl., 229(1995), pp. 49-99.
15. [GK89] I. GOHBERG AND I. KOLTRACHT, Efficient algorithm for Toeplitz plus Hankel matrices, Integral Equations Operator Theory, 12(1989), pp. 136-142.
16. [GKO95] I. GOHBERG, T. KAILATH, AND V. OLSHEVSKY, Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comp., 64(1995), pp. 1557-1576.
17. [GL89] G. GOLUB AND C. VAN LOAN, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
18. [G094a] I. GOHBERG AND V. OLSHEVSKY, Complexity of multiplication with vectors for structured matrices, Linear Algebra Appl., 202(1994), pp. 163-192.
19. [G094b] I. GOHBERG AND V. OLSHEVSKY, Fast state space algorithms for matrix Nehari and Nehari-Takagi interpolation problems, Integral Equations Operator Theory, 20(1994), pp. 44-83.
20. [GS84] U. GRENADER AND G. SZEGÖ, Toeplitz Forms and Their Applications, 2nd ed., Chelsea, New York, 1984.
21. [Gu98] M. Gu, Stable and efficient algorithms for structured systems of linear equations, SIAM J. Matrix Anal. Appl., 19(1998), pp. 279-306.
22. [He95a] G. REINIG, Inversion of generalized Cauchy matrices and other classes of structured matrices, in Linear Algebra Signal Processing, IMA Vol. Math. Appl. 69, Springer-Verlag, New York, 1995, pp. 63-81.
23. [He95b] G. HEINIG AND A. BOJANCZYK, Transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices. I. Transformations, Linear Algebra Appl., 254(1997), pp. 193-226.
24. [H94] T. HUCKLE, Iterative methods for Toeplitz matrices, Report SCCM-94-05, Computer Science Department, Stanford University, Stanford, CA, 1994.
25. [H95] T. HUCKLE, Cauchy matrices and iterative methods for Toeplitz matrices, Proc. SPIE, 2563(1995), pp. 281-292.
26. [HJR88] G. HEINIG, P. JANKOWSKI, AND K. ROST, Fast inversion of Toeplitz-plus-Hankel matrices, Numer. Math., 52(1988), pp. 665-682.
27. [K87] T. KAILATH, Signal processing applications of some moment problems, in Moments in Mathematics(San Antonio, TX, 1987), Proc. Sympos. Appl. Math. 37, H. Landau, ed., 1987, pp. 71-109.
28. [KKM79] T. KAILATH, S. KUNG, AND M. MORF, Displacement ranks of matrices and linear equations, 3. Math. Anal. Appl., 68(1979), pp. 395-407.
29. [KO94] T. KAILATH AND V. OLSHEVSKY, Displacement structure approach to polynomial Vandermonde and related matrices, Linear Algebra Appl., 261(1997), pp. 49-90.
30. [KO95a] T. KAILATH AND V. OLSHEVSKY, Diagonal pivoting for partially reconstructible Cauchy-like matrices, with applications to Toeplitz-like linear equations and to boundary rational matrix interpolation problems, Linear Algebra Appl., 254(1997), pp. 251-302.
31. [KO95b] T. KAILATH AND V. OLSHBVSKY, Displacement structure approach to Chebyshev-Vandermonde and related matrices, Integral Equations Operator Theory, 22(1995), pp. 65-92.
32. [KO96] T. KAILATH AND V. OLSHEVSKY, Displacement structure approach to discretetrigonometric-transform based preconditioners of G. Strang type and of T.Chan type, Calcolo, 33(1996), pp. 191-208.
33. [MB79] J. MAROULAS AND S. BARNETT, Polynomials with respect to a general basis. I. Theory, J. Math. Anal. Appl., 72(1979), pp. 177-194.
34. [O93a] V. OLSHEVSKY, A fast real-arithmetic algorithm for multiplication of a Toeplitz matrix by a vector, preprint, Stanford University, Stanford, CA, 1993.
35. [O93b] M. OHSMANN, Fast cosine transform of Toeplitz matrices, algorithm and applications, IEEE Trans. Signal Process., 41(1993), pp. 3057-3061.
36. [095] M. OHSMANN, Fast transforms of Toeplitz matrices, Linear Algebra Appl., 231(1995), pp. 181-192.
37. [O03] V. OLSHEVSKY, Pivoting for structured matrices and rational tangential interpolation, in Fast Algorithms for Structured Matrices: Theory and Applications(South Hadley, MA, 2001), Contemp. Math. 323, AMS, Providence, RI, 2003, pp. 1-75.
38. [O04] V. OLSHEVSKY, A displacement structure approach to the derivation of the eight versions of fast cosine and sine transforms, in preparation.
39. [OOW03] A. OLSHEVSKY, V. OLSHEVSKY, AND J. WANG, A comrade-matrix-based derivation of the eight versions of fast cosine and sine transforms, in Fast Algorithms for Structured Matrices: Theory and Applications(South Hadley, MA, 2001), Contemp. Math. 323, AMS, Providence, RI, 2003, pp. 119-149.
40. [P90] V. PAN, On computations with dense structured matrices, Math. Comp., 55(1990), pp. 179-190.
41. [S86] G. STRANG, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74(1986), pp. 171-176.
42. [SB95] D. SWEET AND R. BRENT, Error analysis of a partial pivoting method for structured matrices, Advanced Signal Processing Algorithms, Proc. SPIE, 2563, 1995, pp. 266-280.
43. [T92] E. E. TYRTYSHNIKOV, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl., 13(1992), pp. 459-473.
44. [T95] E. TYRTYSHNIKOV, Circulant preconditioners with unbounded inverses, Linear Algebra Appl., 216(1995), pp. 1-24.
45. [TS96] E. TYRTYSHNIKOV AND V. STRELA, Which preconditioner is better?, Math. Comp., 65 (1996), pp. 137-150.