Iteration functions for pth roots of complex numbers

Cardoso, João R.; Loureiro, Ana F.

doi:10.1007/s11075-010-9431-8

Cited by 15 publications

(12 citation statements)

References 19 publications

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“…Note that the complex ZF is the basis of the complex ZNN model, as it is Comparison on the residual errors R(t) synthesized by the complex ZNN model (6) with γ = 5, 50, 500 for the inversion of (13) indefinite and quite different from the usual error functions in the study of conventional algorithms. Based on the complex ZF and the ZNN design method, the complex ZNN model utilizes the complex first-order time-derivative information of the time-varying complex matrix involved in the timevarying complex matrix-inversion problem and achieves the global convergence performance.…”

Section: Discussionmentioning

confidence: 99%

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Different complex ZFs leading to different complex ZNN models for time-varying complex matrix inversion

Zhang

Guo

2013

2013 10th IEEE International Conference on Control and Automation (ICCA)

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The Zhang neural network (ZNN), as a special class of recurrent neural network (RNN), has been proposed by Zhang et al. for the online solution of various time-varying problems. More importantly, such a ZNN is based on the Zhang function (ZF) as the error-monitoring function, which is indefinite and quite different from the usual error functions in the study of conventional algorithms, such as a scalar-valued norm-based energy function involved in the gradient-based neural network (GNN). Meanwhile, the resultant ZNN model can guarantee the global/exponential convergence performance for online time-varying problems solving by following Zhang et al.'s design method. In this paper, focusing on solving the time-varying complex matrix-inversion problem, the complex ZNN models are proposed, developed and investigated for timevarying complex matrix inversion. In addition, by introducing different complex ZFs, different corresponding complex ZNN models can be proposed and developed for time-varying complex matrix inversion. Finally, through some simulations and verifications, the illustrative results substantiate the efficacy of the complex ZNN models based on different complex ZFs for time-varying complex matrix inversion.

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Section: Discussionmentioning

confidence: 99%

“…5 and thus omitted due to similarity and space limitation. Therefore, we can draw a conclusion that, we can promote the convergence (6) and (7) with γ = 10 for the inversion of (13) performance of the proposed complex ZNN models (6) and (7) by choosing a larger value of design parameter γ.…”

Section: Simulations and Verificationsmentioning

confidence: 95%

Different complex ZFs leading to different complex ZNN models for time-varying complex matrix inversion

Zhang

Guo

2013

2013 10th IEEE International Conference on Control and Automation (ICCA)

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“…Because the convergence of the matrix iterations is determined by the convergence of the corresponding scalar iterations, there are papers that consider only the convergence of iterative methods applied on the scalar case of Equation . For the Padé family of iterations, the convergence for computing the sector function was considered by Laszkiewicz et al and Gomilko et al For a dual Padé family of iterations and the Schröder family of iterations, conditions on the starting point that guarantee the convergence to the principal p th root of a complex number were given by Zieȩtak and Cardoso et al, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we will consider the convergence and the computation of the famous Euler's method, a special case of the Schröder iterations, and a special one of the dual Padé family of iterations when k =0, m =2, for determining the principal p th root of the matrix A . Euler's method for Equation in the classical form

X_{k + 1} = X_{k} - [I + L_{f} (X_{k})] {f^{'} (X_{k})}^{- 1} f (X_{k}), k = 0, 1, 2, ⋯,

where I denotes the identity operator and

L_{f} (X) = \frac{1}{2} {f^{'} (X)}^{- 1} f^{''} (X) {f^{'} (X)}^{- 1} f (X)

can be rewritten as follows:

X_{k + 1} = E false(X_{k} false), k = 0, 1, 2, ⋯,

provided that X 0 commutes with A and X k is nonsingular for any k ≥0, where

E false(X false) = \frac{1}{2 p^{2}} X ([], false(2 p^{2} - 3 p + 1 false) I + 2 false(2 p - 1 false) A X^{- p} - false(p - 1 false) {((), A X^{- p})}^{2})

for any nonsingular matrix

X \in C^{n \times n}

and I the identity ma...…”

Section: Introductionmentioning

confidence: 99%

An analysis on the efficiency of Euler's method for computing the matrix pth root

Ling

Huang

2017

Numerical Linear Algebra App

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It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton's method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler's method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior.

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“…However, it is noted that, in some situations, complex matrices may also occur, when the problem contains online frequency domain identification processes, or when the input signals incorporate both the magnitude and phase information [13], [14]. Thus, the problems in the complex domain have attracted extensive attention of many researchers [13]- [21]. Furthermore, the time-varying matrix generalized inverse problem usually exists.…”

mentioning

confidence: 99%

Different Complex ZFs Leading to Different Complex ZNN Models for Time-Varying Complex Generalized Inverse Matrices

Liao

Zhang

2014

IEEE Trans. Neural Netw. Learning Syst.

145

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As a special class of recurrent neural network, Zhang neural network (ZNN) has been recently proposed since 2001 for solving various time-varying problems, and has shown high efficiency and excellent performance for solving the problems in the real domain. In this paper, to solve online the time-varying complex generalized inverse (in most cases, the pseudoinverse) problem in the complex domain, a new type of complex-valued ZNN is further proposed and investigated. The design of such a complex ZNN is based on a complex Zhang function (ZF) which is indefinite and quite different from the usual error function (specially, the scalar-valued energy function) in the studies of conventional algorithms. By introducing five different complex ZFs, five different complex ZNN models (termed complex ZNN-I, ZNN-II, ZNN-III, ZNN-IV, and ZNN-V models) are proposed, developed, and investigated for the online solution of the time-varying complex generalized inverse matrices. Theoretical results of convergence analysis are presented to show the desirable properties of complex ZNN models. In addition, we discover the link between the proposed complex ZNN models and the Getz-Marsden dynamic system in the complex domain. Computer-simulation results further demonstrate the effectiveness of complex ZNN models based on different complex ZFs for the time-varying complex generalized inverse matrices. Index Terms-Complex-valued Zhang neural network (ZNN), complex Zhang function (ZF), convergence analysis, generalized inverse, pseudoinverse, time-varying complex matrix.

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Iteration functions for pth roots of complex numbers

Cited by 15 publications

References 19 publications

Different complex ZFs leading to different complex ZNN models for time-varying complex matrix inversion

Different complex ZFs leading to different complex ZNN models for time-varying complex matrix inversion

An analysis on the efficiency of Euler's method for computing the matrix pth root

Different Complex ZFs Leading to Different Complex ZNN Models for Time-Varying Complex Generalized Inverse Matrices

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