Solution of Linear Equations with Rational Toeplitz Matrices

Dickinson, B.

doi:10.2307/2006230

Cited by 3 publications

(6 citation statements)

References 8 publications

(13 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Levinson's algorithm was in fact for solving the more general Toeplitz system (1.4) T,x=y, but this involved the explicit computation of the solutions to the Yule-Walker equations of all orders strictly less than p. Durbin [12] subsequently streamlined Levinson's algorithm to solve only (1.3), and his name has been associated with the algorithm in much of the statistical and engineering literature where the Yule-Walker equations are most often encountered. The basic algorithm has been independently discovered, generalized, and modified more recently by a number of authors [1], [10], [16], [26], [27], [29], [35]. In 2, this basic algorithm will be presented and will henceforth be called the Levinson-Durbin algorithm.…”

mentioning

confidence: 99%

The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations

Cybenko¹

1980

SIAM J. Sci. and Stat. Comput.

192

View full text Add to dashboard Cite

The numerical stability of the Levinson-Durbin algorithm for solving the Yule-Walker equations with a positive-definite symmetric Toeplitz matrix is studied. Arguments based on the analytic results of an error analysis for fixed-point and floating-point arithmetics show that the algorithm is stable and in fact comparable to the Cholesky algorithm. Conflicting evidence on the accuracy performance of the algorithm is explained by demonstrating that the underlying Toeplitz matrix is typically ill-conditioned in most applications.

show abstract

mentioning

confidence: 99%

The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations

Cybenko¹

1980

SIAM J. Sci. and Stat. Comput.

192

View full text Add to dashboard Cite

show abstract

“…An important property for We note that a fast recursive direct solver for (3.6) was proposed in [14] under the assumption that all the principal minor determinants of the coefficient matrix (T n [l 2 ]T n [u 1 ] + T n [l 1 ]T n [u 2 ]) are not equal to zero.…”

Section: Product Preconditioners For Matrices Generated By Rational Fmentioning

confidence: 99%

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

Lin

2009

Numer Algor

View full text Add to dashboard Cite

In this paper, we first propose product Toeplitz preconditioners (in an inverse form) for non-Hermitian Toeplitz matrices generated by functions with zeros. Our inverse product-type preconditioner is of the formwhere T F , T L , and T U are full, band lower triangular, and band upper triangular Toeplitz matrices, respectively. Our basic idea is to decompose the generating function properly such that all factors T F , T L , and T U of the preconditioner are as well-conditioned as possible. We prove that under certain conditions, the preconditioned matrix has eigenvalues and singular values clustered around 1. Then we use a similar idea to modify the preconditioner proposed in Ku and Kuo (SIAM J Sci Stat Comput 13: [1470][1471][1472][1473][1474][1475][1476][1477][1478][1479][1480][1481][1482][1483][1484][1485][1486][1487] 1992) to handle the zeros in rational generating functions. Numerical results, including applications to the computation of the stationary probability distribution of Markovian queuing models with batch arrival, are given to illustrate the good performance of the proposed preconditioners.

show abstract

“…Toeplitz matrices with a rational generating function can be transformed to banded ones [13]. We describe the transformation briefly as follows.…”

Section: Rational Toeplitzmentioning

confidence: 99%

“…Since power series multiplication is commutative, we have TN LbTNUd LaUd + LbU. (13) where TN iS banded and nearly Toeplitz characterized by the following lemma.…”

Section: Rational Toeplitzmentioning

confidence: 99%

“…The MPLU preconditioner FN is a product of the shift matrix SN and triangular banded Toeplitz matrices LN and UN, the preconditioning step z = F1r can be achieved with a computational complexity proportional to 0(N) only. The total computational complexity for solving a rational Toeplitz system by MPLU-preconditioned iterative methods is 0(N), which is lower than the O(N log N) operations required by the circulant-preconditioned iterative methods and is in the same order as that required by several direct methods [12], [13], [32], [33]. However, there is a drawback of the MPLU preconditioner in the context of parallel processing.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Minimum-Phase LU Factorization Preconditioner for Toeplitz Matrices

Ku¹,

Kuo²

1992

SIAM J. Sci. and Stat. Comput.

View full text Add to dashboard Cite

A new preconditioner is proposed for the solution of an N x N Toeplitz system TNX = b, where TN can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner FN is obtained based on factorizing the generating function T(z) into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and therefore called the minimum-phase LU (MPLU) factorization preconditioner. Due to the minimum-phase property, IIFJ1II is bounded. For rational Toeplitz TN with generating function T(z) = A(z1)/B(z1) + C(z)/D(z), where A(z), B(z), C(z) and D(z) are polynomials of orders p , q , P2 and q , we show that the eigenvalues of FJ1 TN are repeated exactly at 1 except at most cF outliers, where cF depends on Pi ,qi , P2 , q2 and the number w of the roots of T(z) = A(z1)D(z) + B(z1)C(z) outside the unit circle. A preconditioner IN in circulant form generalized from the symmetric case is also presented for comparison.

show abstract

Solution of Linear Equations with Rational Toeplitz Matrices

Abstract: Abstract.We associate a sequence of Toeplitz matrices with the rational formal power series T(z). An algorithm for solving linear equations with a Toeplitz matrix from this sequence is given. The algorithm requires 0(n) operations to solve a set of « equations, for n sufficiently large.

Cited by 3 publications

References 8 publications

The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations

The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

A Minimum-Phase LU Factorization Preconditioner for Toeplitz Matrices

Contact Info

Product

Resources

About