2,010 research outputs found
A fast algorithm for LR-2 factorization of Toeplitz matrices
In this paper a new order recursive algorithm for the efficient â1 factorization of Toeplitz matrices is described. The proposed algorithm can be seen as a fast modified Gram-Schmidt method which recursively computes the orthonormal columns i, i = 1,2, âŠ,p, of , as well as the elements of Râ1, of a Toeplitz matrix with dimensions L Ă p. The factor estimation requires 8Lp MADS (multiplications and divisions). Matrix â1 is subsequently estimated using 3p2 MADS. A faster algorithm, based on a mixed and â1 updating scheme, is also derived. It requires 7Lp + 3.5p2 MADS. The algorithm can be efficiently applied to batch least squares FIR filtering and system identification. When determination of the optimal filter is the desired task it can be utilized to compute the least squares filter in an order recursive way. The algorithm operates directly on the experimental data, overcoming the need for covariance estimates. An orthogonalized version of the proposed â1 algorithm is derived. Matlab code implementing the algorithm is also supplied
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
A fast solver for linear systems with displacement structure
We describe a fast solver for linear systems with reconstructable Cauchy-like
structure, which requires O(rn^2) floating point operations and O(rn) memory
locations, where n is the size of the matrix and r its displacement rank. The
solver is based on the application of the generalized Schur algorithm to a
suitable augmented matrix, under some assumptions on the knots of the
Cauchy-like matrix. It includes various pivoting strategies, already discussed
in the literature, and a new algorithm, which only requires reconstructability.
We have developed a software package, written in Matlab and C-MEX, which
provides a robust implementation of the above method. Our package also includes
solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as
these structures can be reduced to Cauchy-like by fast and stable transforms.
Numerical experiments demonstrate the effectiveness of the software.Comment: 27 pages, 6 figure
Fast Algorithms for Displacement and Low-Rank Structured Matrices
This tutorial provides an introduction to the development of fast matrix
algorithms based on the notions of displacement and various low-rank
structures
Generalized companion matrix for approximate GCD
We study a variant of the univariate approximate GCD problem, where the
coefficients of one polynomial f(x)are known exactly, whereas the coefficients
of the second polynomial g(x)may be perturbed. Our approach relies on the
properties of the matrix which describes the operator of multiplication by gin
the quotient ring C[x]=(f). In particular, the structure of the null space of
the multiplication matrix contains all the essential information about GCD(f;
g). Moreover, the multiplication matrix exhibits a displacement structure that
allows us to design a fast algorithm for approximate GCD computation with
quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201
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