Applied Mathematics and Computation 217 (2011) 10066–10073

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Complex-valued Zhang neural network for online complex-valuedtime-varying matrix inversion

Yunong Zhang ⇑, Zhan Li, Kene LiSchool of Information Science and Technology, Sun Yat-sen University, Guangzhou 510006, China

a r t i c l e i n f o

Keywords:Complex-valued recurrent neural networkComplex-valued Zhang neural networkComplex-valued time-varying matrixinversionMatrix-valued error functionSuperior convergence

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.04.085

⇑ Corresponding author.E-mail addresses: zhynong@mail.sysu.edu.cn, yn

a b s t r a c t

In this paper, a new complex-valued recurrent neural network (CVRNN) called complex-valued Zhang neural network (CVZNN) is proposed and simulated to solve the complex-valued time-varying matrix-inversion problems. Such a CVZNN model is designed basedon a matrix-valued error function in the complex domain, and utilizes the complex-valuedfirst-order time-derivative information of the complex-valued time-varying matrix foronline inversion. Superior to the conventional complex-valued gradient-based neural net-work (CVGNN) and its related methods, the state matrix of the resultant CVZNN model canglobally exponentially converge to the theoretical inverse of the complex-valued time-varying matrix in an error-free manner. Moreover, by exploiting the design parameterc > 1, superior convergence can be achieved for the CVZNN model to solve such com-plex-valued time-varying matrix inversion problems, as compared with the situation with-out design parameter c involved (i.e., the situation with c ¼ 1). Computer-simulationresults substantiate the theoretical analysis and further demonstrate the efficacy of sucha CVZNN model for online complex-valued time-varying matrix inversion.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The problem of matrix inversion is a fundamental issue in science and engineering fields, e.g., in nonlinear optimization [1],GPS GDOP-approximation [2], robot-arm repetitive motion planning [3,4], signal-processing [5], and Gaussian regression [6].Generally speaking, there are two main types of solution to the problem of matrix inversion. The traditional one can be gen-eralized as the serial-processing numerical algorithms with the minimal arithmetic operations proportional to the cube of thecoefficient matrix dimension [6]. However, such serial-processing numerical algorithms may not be efficient enough for large-scale online (or real-time) applications due to high computational complexity [7].

Being the other important type of solution to online matrix inversion and relevant problems solving, many parallel-pro-cessing computational methods, based on artificial analog or biomimetic models have been proposed, analyzed, and imple-mented on specific architectures, e.g., the analog and neural-dynamic solvers [8–12]. Such a neural-dynamic approach is nowregarded as a powerful alternative to online computation and optimization, owing to its potential parallel-processing distrib-uted nature and convenience of hardware implementation.

However, a number of numerical methods and neural-dynamic approaches in the literature have been designed intrinsi-cally for online computation problems with static/constant coefficients in the real domain [7]. Such computational schemesmay be efficient for certain static applications, but may be less favorable for most of time-varying applications which may be

. All rights reserved.

zhang@ieee.org (Y. Zhang).

Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073 10067

more frequently encountered and requiring a faster convergence to exact time-varying solutions. In addition, dynamic neuralsolvers based on gradient methods would generate rather large steady-state (residual) error when applied to such time-vary-ing matrix/vector computational problems, e.g., online time-varying matrix inversion [13].

Differing from the conventional complex-valued gradient neural network (CVGNN) [9], in this paper a new type of complex-valued recurrent neural network (CVRNN) model is proposed, analyzed and simulated for online solution to complex-valuedtime-varying matrix inversion by following Zhang et al.’s design method [12,14–16]. For presentation and comparison conve-nience only, the complex-valued Zhang neural network (CVZNN) is named in this paper, which may help readers understandbetter its difference from the well-known gradient-based approach and related gradient-neural-network models. Differingfrom the conventional CVGNN model, such a CVZNN model makes full use of the first-order time-derivative information ofthe time-varying complex-valued matrix, and forces the unbounded/indefinite time-varying matrix-valued error function todecrease to zero during its solving process.

To the best of the authors’ knowledge, there is no literature dealing with such specific complex-valued time-varying matrixinversion at present stage, and the main contributions of the paper lie in the following facts emphatically.

(1) The online solution of complex-valued time-varying matrix inversion problems is investigated in this paper, ratherthan conventionally static complex-valued matrix inversion (in the real-domain) [9]. It is the main motivation for thiswork.

(2) In this paper, to solve complex-valued time-varying matrix inversion problems, an implicit-dynamics CVZNN model isdesigned based on an unbounded matrix-valued indefinite error function, rather than the usually-exploited scalar-val-ued nonnegative energy functions associated with CVGNN which belongs to the well-known Hopfield-type networks.

(3) A framework of the CVZNN model is proposed for solving complex-valued time-varying matrix inversion problems,which is guaranteed to have global exponential convergence to its theoretical time-varying solution. By increasingthe design parameter c, the aforementioned CVZNN model could then have superior convergence to the theoreticaltime-varying solution.

(4) Illustrative verification and comparison examples are presented, where the CVZNN and CVGNN models are bothexploited to solve the complex-valued time-varying matrix inversion problems. The efficacy, difference andadvantages of the proposed CVZNN model are thus shown evidently.

2. Preliminaries and neural-network models

In this section, we present some necessary preliminaries on the complex-valued time-varying matrix inversion prob-lem, and then introduce and analyze the proposed CVZNN model for solving such a problem.

2.1. Preliminaries

Consider a smoothly time-varying complex-valued nonsingular matrix AðtÞ 2 Cn�n. We aim at finding the unknownXðtÞ 2 Cn�n such that the following complex-valued matrix equation holds true:

AðtÞXðtÞ ¼ I; t 2 ½0;þ1Þ; ð1Þ

where I 2 Cn�n denotes the identity matrix. In practice, matrix AðtÞ 2 Cn�n is usually assumed bounded. Our objective in thiswork is to solve the complex-valued time-varying matrix-inversion problem (1) by the ensuing CVZNN model in an error-free manner (the residual error is small enough to be omitted by increasing the design parameter c to a suitable value). Toensure problem (1) being solvable and having the unique solution, all eigenvalues of complex-valued matrix AðtÞ 2 Cn�n mustbe nonzero. More specifically, problem (1) can be rewritten as

½AreðtÞ þ iAimðtÞ�½XreðtÞ þ iXimðtÞ� ¼ I; ð2Þ

where AreðtÞ 2 Rn�n and AimðtÞ 2 Rn�n denote the real and imaginary parts of AðtÞ 2 Cn�n, respectively; XreðtÞ 2 Rn�n andXimðtÞ 2 Rn�n denote the real and imaginary parts of neural state matrix XðtÞ 2 Cn�n, respectively; and i :¼

ffiffiffiffiffiffiffi�1p

denotesthe imaginary unit. Expanding Eq. (2) yields

AreðtÞXreðtÞ � AimðtÞXimðtÞ ¼ I;AreðtÞXimðtÞ þ AimðtÞXreðtÞ ¼ O:

�ð3Þ

An alternative to solving problem (1) via solving linear-coupled matrix-Eq. (3) in the real domain is the conventional (real-valued) GNN model [10,13]. However, this will increase the unnecessary complexities in theoretical analysis, simulation,computation, and applications, in addition to generating rather large errors which may not be eliminated by possible remedyschemes [10,13]. In the next subsection, we will introduce a superior CVZNN model for solving complex-valued time-varyingmatrix-inversion problem (1).

10068 Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073

2.2. CVZNN model

To monitor the process of solving complex-valued time-varying matrix-inversion problem (1), we can firstly define the fol-lowing matrix-valued and complex-valued error function (rather than the scalar-valued nonnegative energy functions used ingradient-based neural network approaches):

EðtÞ ¼ AðtÞXðtÞ � I 2 Cn�n; ð4Þ

of which each element is indefinite (i.e., positive, negative, zero, bounded, or even unbounded including lower-unbounded).Secondly, to make each element EpqðtÞ 2 C ð8p; q 2 f1;2; . . . ;ngÞ of error matrix EðtÞ 2 Cn�n converge to zero, the followingsimple ZNN design formula [12,14] in the complex-domain can be exploited:

_EðtÞ ¼ �cEðtÞ; ð5Þ

where design parameter c > 0, being a set of reciprocals of capacitance-parameters, should be set as large as the hardwarewould permit, or be set appropriately for simulative/experimental purposes [17].

Expanding ZNN design formula (5) leads to the following CVZNN model depicted in an implicit dynamic equation:

AðtÞ _XðtÞ ¼ � _AðtÞXðtÞ � cAðtÞXðtÞ þ cI; ð6Þ

where state matrix XðtÞ 2 Cn�n, starting from an initial condition Xð0Þ 2 Cn�n, corresponds to the theoretical complex-valuedtime-varying solution X�ðtÞ ¼ A�1ðtÞ 2 Cn�n of (1).

Remarks. It is worth pointing out that there is no need for CVZNN (6) to convert the complex-valued linear-matrix-equationform (1) to real-valued linear-matrix-equation form (3) and then invert complex-valued time-varying matrices in the realdomain. In this sense, the unnecessary computational complexity both in theoretical analysis and computer simulation canbe eliminated. In addition, CVZNN (6) and its related approach are handling simultaneously both of the real and imaginaryparts of the complex-valued time-varying matrix to be inverted. More importantly, CVZNN (6) makes use of the time-derivative information methodologically, which can thus guarantee the global exponential convergence to the complex-valued time-varying theoretical solution. Furthermore, similar to the handling manner of [11], by discretizing CVZNN (6) andfixing the step-size value to be 1, the well-known Newton iteration in the complex domain can be obtained, which can thusbe seen as a special case of the discrete-time form of CVZNN (6).

2.3. Comparison with CVGNN model

For comparison, it is worth pointing out here that a complex-valued gradient neural network (CVGNN) model has beenpresented to solve online a complex-valued matrix inversion problem [9]. However, similar to almost all numerical algo-rithms and neural-dynamic schemes mentioned earlier, the gradient neural networks are designed intrinsically for problemswith constant (real-valued) coefficient matrices and/or vectors. Now we show the CVGNN design procedure as follows.

(1) Firstly, a scalar-valued norm-based energy function, such as kAX � Ik2F=2 ¼ traceððAX � IÞHðAX � IÞÞ=2, is constructed

such that its minimum point is the solution of linear matrix equation AX � I ¼ 0. Here k � kF and ð�ÞH denote the Frobe-nius norm and complex conjugate transpose of a square matrix, respectively.

(2) Secondly, an algorithm is designed to evolve along a descent direction of this energy function until the minimum pointis reached. The typical descent direction is the negative gradient of energy function kAX � Ik2

F=2:

� @kAX � Ik2F=2

@X¼ �AHðAX � IÞ:

(3) Thirdly, by using the above negative gradient to construct and apply a neural network to the time-varying situation, wehave a linear CVGNN model [9] solving (1):

_XðtÞ ¼ �cAHðtÞAðtÞXðtÞ þ cAHðtÞ: ð7Þ

The main novelties of CVZNN (6) differing from CVGNN (7) lie in the following facts.

(1) CVZNN model (6) is designed based on the elimination of every element of the matrix-valued and complex-valued errorfunction EðtÞ ¼ AðtÞXðtÞ � I. In contrast, CVGNN model (7) is designed based on the elimination of the scalar-valuednorm-based energy function kAX � Ik2

F=2.(2) CVZNN model (6) is depicted in an implicit dynamics, i.e., AðtÞ _XðtÞ ¼ � � �, which coincides well with physical systems in

nature and in practice (e.g., in analogue electronic circuits and mechanical systems owing to well-known Kirchhoff’sand Newton’s laws, respectively). In contrast, CVGNN model (7) is depicted in an explicit dynamics, i.e., _XðtÞ ¼ � � �.

(3) CVZNN model (6) systematically and methodologically exploits the time-derivative information of coefficient matricesduring its real-time solving process. In contrast, CVGNN model (7) does not exploit such information, thus less effectiveon time-varying problems solving.

Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073 10069

(4) CVZNN model (6) can be viewed as a result of the predictive approach, which considers the variation tendency of thecomplex-valued time-varying matrix to be inverted [i.e., _AðtÞ]. However, CVGNN model (7) shows itself as a result ofthe passive tracking approach, which neglects the variation tendency of the complex-valued time-varying matrix tobe inverted.

(5) CVZNN model (6) considers (1) as a complex-valued matrix-equation problem to solve, which is more concise. Never-theless, CVGNN model (7) converts (1) to a (time-varying) minimization problem to solve, and may not be efficient foronline solution to (1).

(6) The derivation of CVZNN model (6) might only need less difficult knowledge of bachelors’ mathematical course, whilethat of CVGNN model (7) requires more complicated mathematical knowledge of postgraduates’ or even Ph.D.’s level.

2.4. Theoretical results

Proposition. Given a complex-valued time-varying matrix AðtÞ satisfying the invertibility condition, the state matrix XðtÞ 2 Cn�n

of CVZNN model (6) , starting from any initial state Xð0Þ 2 Cn�n, globally exponentially converges to the complex-valued time-varying theoretical inverse X�ðtÞ ¼ A�1ðtÞ 2 Cn�n.

Proof. The proof of this proposition is omitted due to space limitation, which can be generalized from [12,14] by taking intoaccount Lyapunov function candidate VðtÞ ¼ kEðtÞk2

F=2 and using Lyapunov theory [3]. h

3. Computer simulation and verification

In this section, four illustrative examples are presented to further verify the theoretical results and substantiate the effi-cacy of the CVZNN model (6). For illustration and simulation purposes, in the first and second examples, both CVZNN model(6) and CVGNN model (7) are exploited to solve online the complex-valued time-varying matrix inversion problem.

Example 1. Let us consider the following complex-valued time-varying matrix AðtÞ 2 C2�2:

Fig. 1.denotecolor in

AðtÞ ¼expð10itÞ �i expð�10itÞ�i expð10itÞ expð�10itÞ

� �: ð8Þ

It can be obtained that the theoretical inverse of (8) is

Trajectories of CVZNN model (6) with parameter c ¼ 10 for inverting a complex-valued time-varying matrix (8), where red dash-dotted curvestheoretical solution A�1ðtÞ of (8) and blue solid curves denote the solution computed by CVZNN model (6). (For interpretation of the references tothis figure legend, the reader is referred to the web version of this article.)

Fig. 2. Trajectories of CVGNN model (7) with parameter c ¼ 10 for inverting a complex-valued time-varying matrix, where red dash-dotted curves denotetheoretical solution A�1ðtÞ of (8) and blue solid curves denote the solution by CVGNN model (7). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

10070 Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073

A�1ðtÞ ¼0:5 expð10itÞ 0:5i expð10itÞ

0:5i expð�10itÞ 0:5 expð�10itÞ

� �;

which is used as an analytic theoretical solution to verify and compare the correctness of the solutions of CVZNN model (6)and CVGNN model (7). To solve the inverse of (8) in real time, we can take the following specific and simple form of CVZNNmodel (6):

expð10itÞ �i expð�10itÞ�i expð10itÞ expð�10itÞ

� �_x11 _x12

_x21 _x22

� �¼

10i expð10itÞ �10 expð�10itÞ10 expð10itÞ �10i expð�10itÞ

� �;

x11 x12

x21 x22

� �� c

expð10itÞ �i expð�10itÞ�i expð10itÞ expð�10itÞ

� �x11 x12

x21 x22

� �þ c

1 00 1

� �:

As shown in Fig. 1, starting from the randomly-generated initial state Xð0Þ 2 C2�2, the state matrix XðtÞ 2 C2�2 of CVZNNmodel (6) with parameter c ¼ 10 converges to the theoretical inverse A�1ðtÞ 2 C2�2 of (8) exactly within a rather short time(about 1s). In contrast, from Fig. 2 we observe that the state matrix XðtÞ 2 C2�2 of CVGNN model (7) does not fit well with thetheoretical inverse A�1ðtÞ 2 C2�2 of (8) with rather large computational errors throughout all the time t. All of these implythat the exact inverse A�1ðtÞ 2 C2�2 of (8) can be achieved by CVZNN model (6) with much higher accuracy. Fig. 3(a) furtherillustrates that the residual error kAðtÞXðtÞ � IkF synthesized by CVZNN model (6) with c ¼ 1 diminishes to zero within about5 s. However, from Fig. 3(b), we see that the residual error kAðtÞXðtÞ � IkF synthesized by CVGNN model (7) c ¼ 1 does notdescend to zero, but converge to about 1.4, which is a rather large error.

Example 2. Let us consider the following complex-valued time-varying matrix AðtÞ 2 C2�2:

AðtÞ ¼ 10sinðitÞ cosðitÞ� cosðitÞ sinðitÞ

� �: ð9Þ

To solve for the inverse of (9), we can take the following specific and simple form of CVZNN model (6):

10sinðitÞ 10cosðitÞ�10cosðitÞ 10sinðitÞ

� �_x11 _x12

_x21 _x22

� �¼

10icosðitÞ 10isinðitÞ�10isinðitÞ 10icosðitÞ

� �x11 x12

x21 x22

� ��10c

sinðitÞ cosðitÞ�cosðitÞ sinðitÞ

� �x11 x12

x21 x22

� �þc

1 00 1

� �:

To achieve superior (or to say, faster) convergence of CVZNN model (6) for inverting matrix (9), we can increase the designparameter c correspondingly. As seen from Fig. 4, the residual errors kAðtÞXðtÞ � IkF synthesized by CVZNN model (6) dimin-ish faster with increase of the design parameter c (in this example, we show the residual-error comparison with c = 1, 10 and

Fig. 3. Comparison on residual errors kAðtÞXðtÞ � IkF synthesized by complex-valued neural-network models with design parameter c ¼ 1 and starting fromtwelve complex-valued randomly-generated initial states for the inversion of (8).

Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073 10071

100). When c is set 1, 10 and 100, the residual error diminishes to almost zero within around 5, 0.5 and 0.05 s, respectively.Evidently, if faster convergence is required, the design parameter c could be set larger, e.g., 10 or 100. This further substan-tiates that the performance and the global-exponential-convergence property of CVZNN model (6) can be promoted bychoosing larger design parameter c .

Example 3. Let us consider the online inversion of the following complex-valued time-varying Toeplitz matrix:

Fig. 4

AðtÞ ¼

a1ðtÞ a2ðtÞ a3ðtÞ � � � anðtÞa2ðtÞ a1ðtÞ a2ðtÞ � � � an�1ðtÞa3ðtÞ a2ðtÞ a1ðtÞ � � � an�2ðtÞ

..

. ... ..

. . .. ..

.

anðtÞ an�1ðtÞ an�2ðtÞ � � � a1ðtÞ

266666664

3777777752 Cn�n: ð10Þ

. Comparison on residual errors kAðtÞXðtÞ � IkF synthesized by CVZNN model (6) with design parameter c from 1 to 100 for the inversion of (9).

Fig. 5. Comparison on residual errors kAðtÞXðtÞ � IkF synthesized by CVZNN model (6) with design parameter c ¼ 1 for the inversion of complex-valuedtime-varying Toeplitz matrix (10).

10072 Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073

Let a1ðtÞ ¼ nþ sinðitÞ and akðtÞ ¼ cosðitÞ=ðk� 1Þ ðk ¼ 2;3; . . . ;nÞ. Evidently, matrix AðtÞ is strictly diagonally dominant forany time instant t P 0 and is therefore invertible [16]. For illustrative purposes, we exploit CVZNN model (6) with c ¼ 1 tosolve the inverse of the above Toeplitz matrix AðtÞ under the situations of n ¼ 6 and n ¼ 8. Evidently, as seen from Fig. 5, theresidual errors kAðtÞXðtÞ � IkF diminish to zero within a short time (about 6s) for both n ¼ 6 and n ¼ 8, which implies that thestate matrix XðtÞ always exponentially converges to A�1ðtÞ.

Example 4. In the above three examples, the complex-valued time-varying matrices are inverted by using the proposedCVZNN model (6). In this example, the CVZNN model (6) is exploited for online inversion of a complex-valued constantmatrix, which could be viewed as a special case of the present complex-valued time-varying problem formulation andsolution. The following constant matrix A 2 C4�4 is presented [9]:

A ¼

�2� i �2 0:5i �10:7i 2� i 2 1

1 �2þ 0:5i �3 �2i 1 2 1þ 0:4i

26664

37775; ð11Þ

Fig. 6. Computational errors kXðtÞ � A�1kF synthesized by CVZNN model (6) with design parameter c ¼ 1 for the inversion of (11).

Y. Zhang et al. / Applied Mathematics and Computation 217 (2011) 10066–10073 10073

which has the theoretical inverse A�1 2 C4�4 [9] as below [used to check the correctness of the solutions generated by CVZNN(6)]:

A�1 ¼

0:036þ 0:416i �0:193þ 0:391i 0:489� 0:027i 1:030� 0:44i

0:141� 0:287i 1:000þ 0:295i 0:108� 0:224i �0:909� 0:666i

0:696þ 0:318i 0:298þ 0:788i 0:593� 0:047i 1:171� 1:033i�1:095þ 0:053i �1:617� 1:032i �1:196þ 0:309i �0:166þ 1:768i

26664

37775:

By using design parameter c ¼ 1, the state matrix XðtÞ computed by the proposed CVZNN model (6), starting form 50 ran-domly-generated initial states, always converges to the theoretical inverse A�1. This is shown in Fig. 6, where the computa-tional errors kXðtÞ � A�1kF synthesized by CVZNN model (6), starting with 50 randomly-generated initial matrices, alwaysconverge to zero in about 6 s.

In summary, all of the above four examples substantiate the efficacy of the proposed CVZNN model (6) on solving thecomplex-valued time-varying and constant matrices inversion problems, and substantiate as well the superiorness of CVZNNmodel (6) to the conventional CVGNN model (7).

4. Conclusions

In this paper, a new type of complex-valued recurrent neural network (CVRNN) called complex-valued Zhang neural net-work (CVZNN) is proposed, analyzed and simulated to solve complex-valued time-varying matrix-inversion problems. Sucha CVZNN model is designed based on eliminating a matrix-valued error function in the complex domain, and utilizes thecomplex-valued first-order time-derivative information of the complex-valued time-varying matrix for online inversion. Dif-ferent from the conventional complex-valued gradient-based neural network (CVGNN) and its related methods, the statematrix of the resultant CVZNN model can globally exponentially converge to the theoretical inverse of the complex-valuedtime-varying matrix in an error-free manner. By exploiting suitably the design parameter c, superior convergence can beachieved for the CVZNN model to solve such complex-valued time-varying matrix inversion problems. Computer simulationresults further demonstrate the efficacy of such a CVZNN model on online complex-valued time-varying matrix inversion.

Acknowledgements

This work is supported by the Natural Science Foundation of China under Grants 61075121 and 60935001, and also by theFundamental Research Funds for the Central Universities of China. In addition, the authors would like to thank the editor andanonymous reviewer(s) sincerely for their constructive comments and suggestions which have really improved the presen-tation and quality of this paper very much.

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