Block matrices with L-block-banded inverse: Inversion algorithms

IEEE Transactions on Signal Processing

Volumn 53, Issue 2 I, 2005, Pages 630-642

  Asif, Amir ^a   Moura, José M F ^b  

^a YORK UNIVERSITY   (Canada)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Block banded matrix; Cholesky decomposition; Covariance matrix; Gauss Markov random process; Kalman Bucy filter; Matrix inversion; Sparse matrix

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BOUNDARY CONDITIONS; COMPUTER SIMULATION; DIFFERENCE EQUATIONS; FINITE DIFFERENCE METHOD; INVERSE PROBLEMS; MARKOV PROCESSES; MATHEMATICAL MODELS; MATRIX ALGEBRA; PARTIAL DIFFERENTIAL EQUATIONS; VECTORS;

BLOCK BANDED MATRIX; CHOLESKY DECOMPOSITION; COVARIANCE MATRIX; GAUS-MARKOV RANDOM PROCESS; KALMAN-BUCY FILTER; MATRIX INVERSION; SPARSE MATRIX;

KALMAN FILTERING;

EID: 13244272097     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2004.840709     Document Type: Article

References (22)

1. J. W. Woods, "Two-dimensional discrete Markovian fields," IEEE Trans. Inf. Theory, vol. IT-18, no. 2, pp. 232-240, Mar. 1972.
2. J. M. F. Moura and N. Balram, "Recursive structure of noncausal Gauss Markov random fields," IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 334-354, Mar. 1992.
3. S. J. Reeves, "Fast algorithm for solving block-banded Toeplitz systems with banded Toeplitz blocks," in Proc. Int. Conf. Acoust., Speech, Sgnal Process., vol. 4, May 2002, pp. 3325-3329.
4. M. Chakraborty, "An efficient algorithm for solving general periodic Toeplitz system," IEEE Trans. Signal Process., vol. 46. no. 3, pp. 784-787, Mar. 1998.
5. G. S. Ammar and W. B. Gragg, "Superfast solution of real positive definite Toeplitz systems," SIAM J. Matrix Anal. Applicat., vol. 9, pp. 61-76, Jan. 1988.
6. A. Kavcic and J. M. F. Moura, "Matrix with banded inverses: algorithms and factorization of Gauss-Markov processes," IEEE Trans. Inf. Theory, vol. 46, no. 4, pp. 1495-1509, Jul. 2000.
7. S. Chandrasekaran and A. H. Syed, "A fast stable solver for nonsymmetric Toeplitz and quasi-Toeplitz systems of linear equations," SIAM J. Matrix Anal. Applicat., vol. 19, no. 1, pp. 107-139, Jan. 1998.
8. P. Dewilde and A. J. van der Veen, "Inner-outer factorization and the inversion of locally finite systems of equations," Linear Algebra Applicat., vol. 313, no. 1-3, pp. 53-100, Jul. 2000.
9. A. M. Erishman and W. Tinney, "On computing certain elements of the inverse of a sparse matrix," Commun. ACM, vol. 18, pp. 177-179, Mar. 1975.
10. I. S. Duff, A. M. Erishman, and W. Tinney, Direct Methods for Sparse Matrices. Oxford, U.K.: Oxford Univ. Press, 1992.
11. G. H. Golub and C. V. Loan, Matrix Computations, 3rd ed. Baltimore, MD: ohn Hopkins Univ. Press, 1996, pp. 133-192.
12. J. Chun and T. Kailath, Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms. New York: Springer-Verlag, 1991, pp. 215-236.
13. N. Kalouptsidis, G. Carayannis, and D. Manolakis, "Fast algorithms for block-Toeplitz matrices with Toeplitz entries," Signal Process., vol. 6, no. 3, pp. 77-81, 1984.
14. T. Kailath, S.-Y. Kung, and M. Morf, "Displacement ranks of matrices and linear equations," J. Math. Anal. Applicat., vol. 68, pp. 395-407, 1979.
15. D. A. Bini and B. Meini, "Toeplitz system," SIAM J. Matrix Anal. Applicat., vol. 20, no. 3, pp. 700-719, Mar. 1999.
16. A. E. Yagle, "A fast algorithm for Toeplitz-block-Toeplitz linear systems," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. vol. 3, Phoenix, AZ, May 2001, pp. 1929-1932.
17. C. Corral, "Inversion of matrices with prescribed structured inverses," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. vol. 2, May 2002, pp. 1501-1504.
18. A. Asif and J. M. F. Moura, "Data assimilation in large time-varying multidimensional fields," IEEE Trans. Image Process., vol. 8, no. 11, pp. 1593-1607, Nov. 1999.
19. A. Asif, and J. M. F. Moura, "Fast inversion of L-block-banded matrices and their inverses," in Proc. IEEE Int. Conf Acoust., Speech, Signal Process. vol. 2, Orlando, FL, May 2002, pp. 1369-1372.
20. T. M. Chin, W. C. Karl, and A. S. Willsky, "Sequential filtering for multi-frame visual reconstruction," Signal Process., vol. 28, no. 1, pp. 311-333, 1992.
21. J. C. Strikwerda, Finite Element Schemes and Partial Differential Equations. Pacific Grove, CA: Wadsworth, Brooks/Cole, 1989.
22. A. Asif and J. M. F. Moura, "Image codec by noncausal prediction, residual mean removal, and cascaded VQ," IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 1, pp. 42-55, Feb. 1996.