Download Solving for time-varying and static cube roots in real and complex

Neural Comput & Applic (2013) 23:255–268
DOI 10.1007/s00521-012-0842-4
ORIGINAL ARTICLE
Solving for time-varying and static cube roots in real and complex
domains via discrete-time ZD models
Yunong Zhang • Zhende Ke • Dongsheng Guo
Fen Li
•
Received: 18 October 2011 / Accepted: 11 January 2012 / Published online: 28 January 2012
Ó Springer-Verlag London Limited 2012
Abstract Different from conventional gradient-based
neural dynamics, a special type of neural dynamics has
been proposed by Zhang et al. for online solution of timevarying and/or static (or termed, time-invariant) problems.
The design of Zhang dynamics (ZD) is based on the
elimination of an indefinite error function, instead of the
elimination of a square-based positive (or at least lowerbounded) energy function usually associated with gradient
dynamics (GD). In this paper, we generalize, propose and
investigate the continuous-time ZD model and its discretetime models in two situations (i.e., the time-derivative of
the coefficient being known or unknown) for time-varying
cube root finding, including the complex-valued continuous-time ZD model for finding cube roots in complex
domain. In addition, to find the static scalar-valued cube
root, a simplified continuous-time ZD model and its discrete-time model are generated. By focusing on such a
static problem solving, Newton-Raphson iteration is found
to be a special case of the discrete-time ZD model by
utilizing the linear activation function and fixing the stepsize value to be 1. Computer-simulation and testing results
demonstrate the efficacy of the proposed ZD models
(including real-valued ZD models and complex-valued ZD
models) for time-varying and static cube root finding, as
well as the link and new explanation to Newton-Raphson
iteration.
Keywords Cube root finding Time-varying Zhang
dynamics Discrete-time models Real domain Complex
domain Newton-Raphson iteration
Y. Zhang (&) Z. Ke D. Guo F. Li
School of Information Science and Technology,
Sun Yat-sen University, Guangzhou 510006, China
e-mail: [email protected]
1 Introduction
Online solution of time-varying cube root in the form of
x3(t) - a(t) = 0 and/or static (or termed, constant, timeinvariant) cube root in the form of x3 - a = 0 is considered
to be a basic problem arising in science and engineering
fields, for example, computer graphics [1–3], scientific
computing [2, 4] and FPGA implementations [5]. It is
usually a fundamental part of many solutions. Thus, many
numerical algorithms are presented for such a problem
solving [1–8]. Generally speaking, it may not be efficient
enough for most numerical algorithms due to their serialprocessing nature performed on digital computers [9].
Suitable for analogue VLSI implementation [10, 11] and in
view of high-speed processing as well as parallel-distributed properties, the neural-dynamic approach is now
regarded as a powerful alternative to online and/or real-time
problems solving [12–25]. Besides, it is worth mentioning
that most reported computational-schemes were theoretically/intrinsically designed for time-invariant problems
solving and/or related to gradient methods [17, 19, 21].
Since March 2001, a special type of neural dynamics has
been formally proposed by Zhang et al. [18–21, 23–25] for
time-varying and/or static problems solving, such as timevarying matrix inversion [18, 24] and time-varying
Sylvester equation solving [20]. In addition, the proposed
Zhang dynamics (ZD) is designed based on an indefinite
error function, instead of a square-based positive (or at
least lower-bounded) energy function usually associated
with gradient-based models and/or Hopfield-type neural
networks [14–17, 19, 21].
In this paper, a novel real-valued continuous-time ZD
(CTZD) model and a complex-valued CTZD model are
generalized, developed and investigated for online solution
of time-varying real-valued and complex-valued cube
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