A generalized Schur-type algorithm for the joint factorization of a structured matrix and its inverse

A Schur-type algorithm is presented for the simultaneous triangular factorization of a given (non-degenerate) structured matrix and its inverse. The algorithm takes the displacement generator of a Hermitian, strongly regular matrixR as an input, and computes the displacement generator of the inverse matrixR−1 as an output. From these generators we can directly deduce theLD−1L* (lower-diagonal-upper) decomposition ofR, and theUD−1U* (upper-diagonallower) decomposition ofR−1. The computational complexity of the algorithm isO(rn2) operations, wheren andr denote the size and the displacement rank ofR, respectively. Moreover, this method is especially suited for parallel (systolic array) implementations: usingn processors the algorithm can be carried out inO(n) steps.

[1]  Thomas Kailath,et al.  Square-root algorithms for structured matrices, interpolation, and completion problems , 1995 .

[2]  Sun-Yuan Kung,et al.  A highly concurrent algorithm and pipeleined architecture for solving Toeplitz systems , 1983 .

[3]  T. Kailath,et al.  Generalized Displacement Structure for Block-Toeplitz,Toeplitz-Block, and Toeplitz-Derived Matrices , 1994 .

[4]  Philippe Delsarte,et al.  A polynomial approach to the generalized Levinson algorithm based on the Toeplitz distance , 1983, IEEE Trans. Inf. Theory.

[5]  J. Chun,et al.  Fast array algorithms for structured matrices , 1990 .

[6]  Ali Hussein Sayed.  Displacement structure in signal processing and mathematics , 1992 .

[7]  T. Boros.  Studies in displacement structure theory , 1996 .

[8]  T. Kailath,et al.  Triangular Factorization of Structured Hermitian Matrices , 1986 .

[9]  Thomas Kailath,et al.  Displacement ranks of a matrix , 1979 .

[10]  Ali H. Sayed,et al.  Time-variant displacement structure and triangular arrays , 1994, IEEE Trans. Signal Process..

[11]  Thomas Kailath,et al.  Correction to 'Lattice Filter Parametrization and Modeling of Nonstationary Processes' , 1984, IEEE Transactions on Information Theory.

[12]  Ali H. Sayed,et al.  Displacement Structure: Theory and Applications , 1995, SIAM Rev..

[13]  Thomas Kailath,et al.  State-space approach to factorization of lossless transfer functions and structured matrices☆ , 1992 .

[14]  Thomas Kailath,et al.  Displacement structure for Hankel, Vandermonde, and related (derived) matrices , 1991 .

[15]  L. Ljung,et al.  Extended Levinson and Chandrasekhar equations for general discrete-time linear estimation problems , 1978 .

[16]  Paul Van Dooren,et al.  On Sigma-lossless transfer functions and related questions , 1983 .

[17]  Thomas Kailath,et al.  Lattice filter parameterization and modeling of nonstationary processes , 1984, IEEE Trans. Inf. Theory.