The stability of formulae of the Gohberg–Semencul–Trench type for Moore–Penrose and group inverses of Toeplitz matrices

Xie, Peng; Wei, Yimin

doi:10.1016/j.laa.2015.01.029

Cited by 15 publications

(4 citation statements)

References 23 publications

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“…The stability of the algorithms emerging from Toeplitz matrix inversion formulas is considered in [10,30]. Xie and Wei [31] presented a stability analysis of Gohberg-Semencul-Trench type for Moore-Penrose and group inverses of Toeplitz matrices. Toeplitz inversion formula involving circulant matrices have also been presented in [1,23,24].…”

Section: Let T = [T J−k ]mentioning

confidence: 99%

Gohberg-Semencul type formula and application for the inverse of a conjugate-Toeplitz matrix involving imaginary circulant matrices

Jiang¹,

Hong²

2017

J. Nonlinear Sci. Appl.

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Gohberg-Semencul type inverse formula of conjugate-Toeplitz (CT) is obtained by constructing a kind of imaginary cyclic displacement transform. The stability of decomposition formula of inverse is investigated, and its algorithm is also given. Numerical example is provided to verify the feasibility of the inverse formula. How the analogue of our formula leads to a more efficient way to solve the conjugate-Toeplitz linear system of equations is proposed. The corresponding inverse, stability, and algorithm of conjugate-Hankel (CH) matrix are also considered.

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Section: Let T = [T J−k ]mentioning

confidence: 99%

Gohberg-Semencul type formula and application for the inverse of a conjugate-Toeplitz matrix involving imaginary circulant matrices

Jiang¹,

Hong²

2017

J. Nonlinear Sci. Appl.

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“…Several methods have been developed for efficient inversion of Toeplitz matrices [15]- [21]. An important class of these methods are based on the Gohberg-Semencul formula [15], [17]. This formula, derived from Trench's method, involves the construction of the inverse from a low number of its columns and the entries of the Toeplitz matrix itself.…”

Section: Introductionmentioning

confidence: 99%

A Fast Quasi-Newton Adaptive Algorithm Based on Approximate Inversion of the Autocorrelation Matrix

2020

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The Newton adaptive filtering algorithm in its original form is computationally very complex as it requires inversion of the input-signal autocorrelation matrix at every time step. Also, it may suffer from stability problems due to the inversion of the input-signal autocorrelation matrix. In this paper, we propose to replace the inverse of the input-signal autocorrelation matrix by an approximate one, assuming that the input-signal autocorrelation matrix is Toeplitz. This assumption would help us in replacing the update of the inverse of the autocorrelation matrix by the update of the autocorrelation matrix itself, and performing the multiplication of R −1 x in the update equation by using the Fourier transform. This would increase the stability of the algorithm, in one hand, and decrease its computational complexity, on the other hand. Since the objective of the paper is to enhance the stability of the Newton algorithm, the performance of the proposed algorithm is compared to those of the Newton and the improved quasi-Newton (QN) algorithms in noise cancellation and system identification settings.

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“…In particular, the fact that some important space-time covariance matrices have the Toeplitz-block Toeplitz form stimulated the study of the inverted multi-dimensional Toeplitz matrices. Among various other interesting recent works on the Toeplitz matrices, convolution operators and their applications, we mention [1,4,5,9,12,15,21,31,38,39,60].…”

Section: Introductionmentioning

confidence: 99%

On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients

Roitberg¹,

Sakhnovich²

2020

Preprint

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The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the 1-D (one-dimensional) case are classical and have numerous applications. Last year, we considered the 2-D case of Toeplitz-block Toeplitz (TBT) matrices, described a minimal information, which is necessary to recover the inverse matrices, and gave a complete characterisation of the inverse matrices. Now, we develop our approach for the more complicated cases of block TBT-matrices and 3-D Toeplitz matrices.

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The stability of formulae of the Gohberg–Semencul–Trench type for Moore–Penrose and group inverses of Toeplitz matrices

Cited by 15 publications

References 23 publications

Gohberg-Semencul type formula and application for the inverse of a conjugate-Toeplitz matrix involving imaginary circulant matrices

Gohberg-Semencul type formula and application for the inverse of a conjugate-Toeplitz matrix involving imaginary circulant matrices

A Fast Quasi-Newton Adaptive Algorithm Based on Approximate Inversion of the Autocorrelation Matrix

On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients

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