135 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
Sliding windows and lattice algorithms for computing QR factors in the least squares theory of linear prediction
Includes bibliographical references.In this correspondence we pose a sequence of linear prediction problems that differ a little from those previously posed. The solutions to these problems introduce a family of "sliding" window techniques into the least squares theory of linear prediction. By using these techniques we are able to QR factor the Toeplitz data matrices that arise in linear prediction. The matrix Q is an orthogonal version of the data matrix and the matrix R is a Cholesky factor of the experimental correlation matrix. Our QR and Cholesky algorithms generate generalized reflection coefficients that may be used in the usual ways for analysis, synthesis, or classification.This work was supported by the Office of Naval Research, Arlington, VA, under Contract N00014-85-K-0256
Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations
We describe fast algorithms for approximating the connection coefficients
between a family of orthogonal polynomials and another family with a
polynomially or rationally modified measure. The connection coefficients are
computed via infinite-dimensional banded matrix factorizations and may be used
to compute the modified Jacobi matrices all in linear complexity with respect
to the truncation degree. A family of orthogonal polynomials with modified
classical weights is constructed that support banded differentiation matrices,
enabling sparse spectral methods with modified classical orthogonal
polynomials
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