Abstract
We propose a newO(p 3 n 2) algorithm for solving complexnp×np linear systems that have block Hankel structure, where the blocks are square matrices of sizep×p. Via FFTs the block Hankel system is transformed into a block Loewner system. An inversion formula enables us to calculate the inverse of the block Loewner matrix explicitely. The parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle. We have implemented our algorithm in Fortran 90. Numerical examples are included.
Similar content being viewed by others
References
E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen, LAPACK Users' Guide, SIAM, 1994.
D. Bini, V. Pan, Polynomial and Matrix Computations, volume 1: Fundamental Algorithms, Birkhäuser, 1994.
M. Fiedler,Hankel and Loewner matrices, Linear Algebra Appl.58, (1984) 75–95.
K. A. Gallivan, S. Thirumalai, P. Van Dooren, V. Vermaut,High performance algorithms for Toeplitz and block Toeplitz matrices Linear Algebra Appl.241–243, (1996).
I. Gohberg, T. Kailath, V. Olshevsky,Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comput.64 (212), (1995) 1557–1576.
G. Heinig,Transformation approaches for fast and stable solution of Toeplitz systems and polynomial equations. In Proceedings of the International Workshop “Recent Advances in Applied Mathematics”, pages 223–238, State of Kuwait, May 4–7 1996.
G. Heinig, A. Bojanczyk,Transformation techniques for Toeplitz and Toeplitzplus-Hankel matrices: I. Transformations, Submitted to Linear Algebra Appl.
G. Heinig, A. Bojanczyk,Transformation techniques for Toeplitz and Toeplitzplus-Hankel matrices: II. Algorithms, Submitted to Linear Algebra Appl.
T. Kailath, A. H. Sayed,Displacement structure: theory and applications, SIAM Review37, (1995) 297–386.
P. Kravanja, M. Van Barel,A fast Hankel solver based on an inversion formula for Loewner matrices. Submitted to Linear Algebra Appl.
C. Van Loan, Computational Frameworks for the Fast Fourier Transform, volume 10 of Frontiers in Applied Mathematics, SIAM, Philadelphia, 1992.
K. Loewner,Über monotone matrixfunktionen, Math. Z.38, (1934) 177–216.
M. Van Barel, A. Bultheel,A look-ahead algorithm for the solution of block Toeplitz systems, Accepted for publication in Linear Algebra Appl. (37 pages).
M. Van Barel, A. Bultheel A general module theoretic framework for vector M-Padé and matrix rational interpolation, Numer. Algorithms3 (1992) 451–462.
M. Van Barel, Z. Vavřín,Inversion of a block Löwner matrix, J. Comput. Appl. Math.69, (1996) 261–284.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kravanja, P., Van Barel, M. A fast block Hankel solver based on an inversion formula for block Loewner matrices. Calcolo 33, 147–164 (1996). https://doi.org/10.1007/BF02575714
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02575714