integration enhanced zhang neural network for real time varying matrix inversion in the presence of various kinds of noises matrix inversion often arises in the fields of science and engineering many models for matrix inversion usually assume that the solving process is free of noises or that the denoising has been conducted before the computation however time is precious for the real time varying matrix inversion in practice and any preprocessing for noise reduction may consume extra time possibly violating the requirement of real time computation therefore a new model for time varying matrix inversion that is able to handle simultaneously the noises is urgently needed in this paper an integration enhanced zhang neural network ieznn model is first proposed and investigated for real time varying matrix inversion then the conventional znn model and the gradient neural network model are presented and employed for comparison in addition theoretical analyses show that the proposed ieznn model has the global exponential convergence property moreover in the presence of various kinds of noises the proposed ieznn model is proven to have an improved performance that is the proposed ieznn model converges to the theoretical solution of the time varying matrix inversion problem no matter how large the matrix form constant noise is and the residual errors of the proposed ieznn model can be arbitrarily small for time varying noises and random noises finally three illustrative simulation examples including an application to the inverse kinematic motion planning of a robot manipulator are provided and analyzed to substantiate the efficacy and superiority of the proposed ieznn model for real time varying matrix inversion

a new method of constructing symmetry adapted linear combinations for a finite group ii correspondence theorem and basis operators abstract a correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry adapted linear combination belonging to the same irreducible representation of a point group by means of basis operators the general expressions are given for the symmetry adapted linear combinations the coupling coefficients the lattice harmonics and the hybrid atomic orbitals illustrative examples are discussed to demonstrate the effectiveness of the formalism

different complex zfs leading to different complex znn models for time varying complex generalized inverse matrices 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

recurrent neural network for computing the drazin inverse this paper presents a recurrent neural network rnn for computing the drazin inverse of a real matrix in real time this recurrent neural network rnn is composed of n independent parts subnetworks where n is the order of the input matrix these subnetworks can operate concurrently so parallel and distributed processing can be achieved in this way the computational advantages over the existing sequential algorithms can be attained in real time applications the rnn defined in this paper is convenient for an implementation in an electronic circuit the number of neurons in the neural network is the same as the number of elements in the output matrix which represents the drazin inverse the difference between the proposed rnn and the existing ones for the drazin inverse computation lies in their network architecture and dynamics the conditions that ensure the stability of the defined rnn as well as its convergence toward the drazin inverse are considered in addition illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

the representation and approximation for the weighted moore penrose inverse in hilbert space we present a unified representation theorem of the weighted moore penrose inverse in hilbert space specific expressions and computational procedures for the weighted moore penrose inverse in hilbert space can be uniformly derived

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

neural network approach to computing matrix inversion a neural network for matrix inversion is proposed we show both analytically and by simulations that this network is guaranteed to be stable and to provide results arbitrarily close to the accurate inversion of a matrix within an elapsed time of only a few characteristic time constants of the network the parameters such as the interconnection strengths of this network can be obtained from the known matrix without any computations

general square pattern discretization formulas via second order derivative elimination for zeroing neural network illustrated by future optimization previous works provide a few effective discretization formulas for zeroing neural network znn of which the precision is a square pattern however those formulas are separately developed via many relatively blind attempts in this paper general square pattern discretization spd formulas are proposed for znn via the idea of the second order derivative elimination all existing spd formulas in previous works are included in the framework of the general spd formulas the connections and differences of various general formulas are also discussed furthermore the general spd formulas are used to solve future optimization under linear equality constraints and the corresponding general discrete znn models are proposed general discrete znn models have at least one parameter to adjust thereby determining their zero stability thus the parameter domains are obtained by restricting zero stability finally numerous comparative numerical experiments including the motion control of a puma560 robot manipulator are provided to substantiate theoretical results and their superiority to conventional euler formula

symbolic computation of drazin inverses by specializations in this paper we show how to reduce the computation of drazin inverses over certain computable fields to the computation of drazin inverses of matrices with rational functions as entries as a consequence we derive a symbolic algorithm to compute the drazin inverse of matrices whose entries are elements of a finite transcendental field extension of a computable field the algorithm is applied to matrices over the field of meromorphic functions in several complex variables on a connected domain and to matrices over the field of laurent formal power series

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

solving simultaneous linear equations using recurrent neural networks abstract simultaneous linear algebraic equations can be found in many mathematical model formulations in monitoring and control of dynamic systems there is often a need for solving simultaneous linear algebraic equations in real time in this paper recurrent neural networks for solving simultaneous linear algebraic equations are proposed the asymptotic stability of the proposed neural networks and solvability of simultaneous linear equations by using the neural networks are substantiated a circuit schematic for realizing the neural networks is described the results of numerical simulations are discussed via illustrative examples an extension of the recurrent neural networks for solving quadratic programming problems subject to equality constraints is also discussed

bounds on the smallest eigenvalue of a pinned laplacian matrix in this note we study a networked system with single multiple pinning given a weighted and undirected network we derive lower and upper bounds on its algebraic connectivity with respect to the reference signal the bounds are derived by partitioning the network in terms of distance of each node from the pinning set upper and lower bounds for two networks with differing topologies are computed to demonstrate the tightness of the derived results it is shown using the derived bounds how requirements on the number of pinning nodes and pinning gain required for achieving stability or a specified convergence rate for the network can be easily obtained

recurrent neural network approach based on the integral representation of the drazin inverse in this letter we present the dynamical equation and corresponding artificial recurrent neural network for computing the drazin inverse for arbitrary square real matrix without any restriction on its eigenvalues conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the drazin inverse are considered several illustrative examples present the results of computer simulations

neural network approach to computing outer inverses based on the full rank representation abstract the first intention of this paper is to emphasize equivalence between two well known general representations of outer inverses with prescribed range and null space of a given matrix two dynamical state equations corresponding to particular expressions which are included in these representations are defined within the second objective in this way two gradient based neural networks initiated by introduced dynamical equations are exploited in generating the class of outer inverses the results corresponding to most common generalized inverses are obtained in particular cases numerical results are presented

the representation and approximation for drazin inverse we present a unified representation theorem for drazin inverse specific expression and computational procedures for drazin inverse can be uniformly derived numerical examples are tested and the application to the singular linear equations is also considered

complex neural network models for time varying drazin inverse two complex zhang neural network znn models for computing the drazin inverse of arbitrary time varying complex square matrix are presented the design of these neural networks is based on corresponding matrix valued error functions arising from the limit representations of the drazin inverse two types of activation functions appropriate for handling complex matrices are exploited to develop each of these networks theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex valued znn models numerical results further demonstrate the effectiveness of the proposed models

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

finite time convergent recurrent neural network with a hard limiting activation function for constrained optimization with piecewise linear objective functions this paper presents a one layer recurrent neural network for solving a class of constrained nonsmooth optimization problems with piecewise linear objective functions the proposed neural network is guaranteed to be globally convergent in finite time to the optimal solutions under a mild condition on a derived lower bound of a single gain parameter in the model the number of neurons in the neural network is the same as the number of decision variables of the optimization problem compared with existing neural networks for optimization the proposed neural network has a couple of salient features such as finite time convergence and a low model complexity specific models for two important special cases namely linear programming and nonsmooth optimization are also presented in addition applications to the shortest path problem and constrained least absolute deviation problem are discussed with simulation results to demonstrate the effectiveness and characteristics of the proposed neural network

on the smallest eigenvalue of grounded laplacian matrices we provide bounds on the smallest eigenvalue of grounded laplacian matrices which are obtained by removing certain rows and columns of the laplacian matrix of a given graph the gap between our upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded laplacian we provide a graph theoretic bound on this ratio and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs specifically for weighted erdos renyi random graphs we show that when a sufficiently small set s of rows and columns is removed from the laplacian and the probability p of adding an edge is sufficiently large the smallest eigenvalue of the grounded laplacian asymptotically almost surely approaches mu w vert s vert p where mu w is the mean edge weight we also show that for weighted random d regular graphs with a single row and column removed the smallest eigenvalue is theta 1 n where n is the number of nodes in the network our bounds have applications to the study of the convergence rate in consensus dynamics with stubborn or leader nodes

three dimensional structured networks for matrix equation solving two three dimensional structured networks are developed for solving linear equations and the lyapunov equation the basic idea of the structured network approaches is to first represent a given equation solving problem by a 3 d structured network so that if the network matches a desired pattern array the weights of the linear neurons give the solution to the problem then train the 3 d structured network to match the desired pattern array using some training algorithms and finally obtain the solution to the specific problem from the converged weights of the network the training algorithms for the two 3 d structured networks are proved to converge exponentially fast to the correct solutions simulations were performed to show the detailed convergence behaviors of the 3 d structured networks

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

gradient methods for computing the drazin inverse solution the problem of computing the vector a db in the cases when it provides a solution of a singular linear system is reduced to the problem of finding a least squares solution of an appropriate linear system later the presented ideas are extended to arbitrary linear systems as a consequence we present iterative methods for computing the vector a db in general the functionality of the exposed algorithms is based on a specific representation of the drazin inverse solution as well as the properties that we introduce convergence results of the proposed methods are also considered illustrative numerical examples are presented

structural analysis of laplacian spectral properties of large scale networks using methods from algebraic graph theory and convex optimization we study the relationship between local structural features of a network and the eigenvalues of its laplacian matrix in particular we propose a series of semidefinite programs to find new bounds on the spectral radius and the spectral gap of the laplacian matrix in terms of a collection of local structural features of the network our analysis shows that the laplacian spectral radius is strongly constrained by local structural features on the other hand we illustrate how local structural features are usually insufficient to accurately estimate the laplacian spectral gap as a consequence random graph models in which only local structural features are prescribed are in general inadequate to faithfully model laplacian spectral properties of a network

a new method of constructing the symmetry coordinates of molecular vibrations based on the correspondence theorem a general method of constructing the symmetry coordinates of molecular vibrations is presented via the correspondence theorem which describes the parallelism between a salc symmetry adapted linear combination and an elementary basis function belonging to the same irreducible representation of the symmetry group of the molecule

solving a kind of restricted matrix equations and cramer rule the problem of finding determinantal formula for solutions of some restricted linear systems ax b has been discussed in linear multilinear algebra 15 1984 319 linear algebra appl 116 1989 27 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 this paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsax d r x r a k 1 xb d n x n b k 2 axb d r x r a k 1 n x n b k 2 where a c n x n with ind a k 1 b c m x m with ind b k 2 and d c n x m the results in linear multilinear algebra 15 1984 319 linear algebra appl 74 1986 213 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 are partially the special cases in our paper the classic cramer rule is also a special case of our results

global exponential convergence and stability of gradient based neural network for online matrix inversion wang proposed a gradient based neural network gnn to solve online matrix inverses global asymptotical convergence was shown for such a neural network when applied to inverting nonsingular matrices as compared to the previously presented asymptotical convergence this paper investigates more desirable properties of the gradient based neural network e g global exponential convergence for nonsingular matrix inversion and global stability even for the singular matrix case illustrative simulation results further demonstrate the theoretical analysis of gradient based neural network for online matrix inversion

neural network approach for solving nonsingular multi linear tensor systems abstract the main propose of this paper is to develop two neural network models for solving nonsingular multi linear tensor system theoretical analysis shows that each of the neural network models ensures the convergence performance for possible hardware implementation of the proposed neural network models based on digital circuits we adopt the euler type difference rule to discretize the corresponding gradient neural network gnn models the computer simulation results further substantiate that the models can solve a multi linear system with nonsingular tensors

a unified theory of the point groups iii classification and basis functions of improper point groups this paper introduces a new system of classification of improper point groups which is most effective for describing their general irreducible representations the complete set of the general angular momentum eigenfunctions is classified according to the general irreducible representations of the seven sets of the improper point groups corresponding to cn c and dn d for an aribtrary n

design of recurrent neural networks for solving constrained least absolute deviation problems recurrent neural networks for solving constrained least absolute deviation lad problems or l1 norm optimization problems have attracted much interest in recent years but so far most neural networks can only deal with some special linear constraints efficiently in this paper two neural networks are proposed for solving lad problems with various linear constraints including equality two sided inequality and bound constraints when tailored to solve some special cases of lad problems in which not all types of constraints are present the two networks can yield simpler architectures than most existing ones in the literature in particular for solving problems with both equality and one sided inequality constraints another network is invented all of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems the different networks designed for solving the same types of problems possess the same structural complexity which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks by this means some flexibility for circuits realization is provided numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks

a new method of constructing symmetry adapted linear combinations for a finite group i salc s of equivalent invariant functions abstract a theory of representation for a finite group g has been presented based on the basis functions defined on a set of equivalent points s n with respect to the symmetry operations of g the general expressions are given for the symmetry adapted linear combinations of equivalent invariant functions and the bonding hybrid atomic orbitals

two finite time convergent zhang neural network models for time varying complex matrix drazin inverse abstract this paper concerns the computation of the drazin inverse of a complex time varying matrix based on two zhang functions constructed from two limit representations of the drazin inverse we present two complex zhang neural network znn models with the li activation function for computing the drazin inverse of a complex time varying square matrix we prove that our znn models globally converge in finite time in addition upper bounds of the convergence time are derived analytically via the lyapunov theory our simulation results verify the theoretical analysis and demonstrate the superiority of our znn models over the gradient based gnn models

recurrent neural network for computing the w weighted drazin inverse abstract two gradient based recurrent neural networks rnns for computing the w weighted drazin inverse of a real rectangular matrix are proposed and considered usage of the first rnn is limited by a specific constraint on the spectrum of a certain matrix the second rnn is usable without restrictions the stability of the recurrent neural networks as well as their convergence are considered numerical examples are given to show the efficiency of the proposed neural networks

recurrent neural networks for solving linear matrix equations abstract recurrent neural networks for solving linear matrix equations are proposed the proposed recurrent neural networks consist of two bidirectionally connected layers and each layer consists of an array of neurons the proposed recurrent neural networks are shown to be asymptotically stable in the large and capable of computing inverse matrices and solving lyapunov matrix equations the operating characteristics of the proposed recurrent neural networks are demonstrated via several illustrative examples

the drazin inverse in multibody system dynamics the numerical analysis of multibody system dynamics is based on the equations of motion as differential algebraic systems a thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches the main tool is the drazin inverse a generalized matrix inverse which preserves the eigenvalues the results are illustrated by a realistic truck model finally the approach is extended to the nonlinear index 2 formulation

laplacian matrices of general complex weighted directed graphs abstract we introduce the concept of general complex weighted directed graphs where each edge is assigned a complex number necessary and sufficient conditions for the laplacian matrix to be singular nonsingular are derived our results give the relationship between the laplacian matrix and the structure of its corresponding directed graph compared with the existing results our main contribution is that our results are established without the restriction that the adjacency matrix is hermitian