on a higher order secant method in this paper we carry out a modification of the classical secant method our goal is to accelerate the convergence until order two but without using any derivative two convergence theorems are presented finally some numerical experiments are analyzed

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

a convergence theorem for newton like methods under generalized chen yamamoto type assumptions in this study we provide sufficient conditions for the convergence of a newton like method to a locally unique solution of a nonlinear equation with a nondifferentiable term in a banach space setting we introduce new conditions by assuming the existence of real functions of two variables that serve as upper bounds on the operators involved special choices of the real functions mentioned above lead to chen yamamoto type assumptions or the usual lipschitz conditions associated with the frechet derivatives of the nonlinear operators involved using the majorant method we show that if a certain scalar inequality has a minimum positive solution then the abstract equation has a locally unique solution also we also show that our upper bounds on the distances x n 1 x n and x n x improve on earlier ones found by chen yamamoto dennis rheinboldt and potra et al finally our results can apply to solve nonlinear integral equations of uryson type

a special matrix equation and its application in microelectronics abstract a special matrix equation is examined which arises in designing electrical interconnections between microelectronic circuits and systems the equation is first rewritten as a fixed point problem a globally convergent iteration method is then proposed and monotone convergence is proved a sufficient and necessary condition for the existence of the solution is presented the solution can be used to minimize the reflection coefficients of the active signals

a recurrent neural network computing the largest imaginary or real part of eigenvalues of real matrices as the efficient calculation of eigenpairs of a matrix especially a general real matrix is significant in engineering and neural networks run asynchronously and can achieve high performance in calculation this paper introduces a recurrent neural network rnn to extract some eigenpair the rnn whose connection weights are dependent upon the matrix can be transformed into a complex differential system whose variable z t is a complex vector by the analytic expression of z t 2 the convergence properties of the rnn are analyzed in detail with general nonzero initial complex vector the rnn obtains the largest imaginary part of all eigenvalues by a rearrangement of connection matrix the largest real part is obtained a practice of a 7x7 matrix indicates the validity of this method two matrices whose dimensionalities are 50 and 100 respectively are employed to test the efficiency of this approach when dimension number becomes large the results imply that the iteration number at which the network enters into equilibrium state is not sensitive with dimensionality this rnn can be used to estimate the largest modulus of eigenvalues etc compared with other neural networks designed for the similar aims this rnn is applicable to general real matrices

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

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

results on newton methods part i a unified approach for constructing perturbed newton like methods in banach spaces and their applications many applied problems can be formulated to fit the model of a nonlinear equation undoubtedly newton methods are the most popular methods for solving such equations here we report the results of our recent investigations in a series of two papers we unified newton methods and provided convergence results to solve nonlinear equations we used newton kantorovich type hypotheses and the majorant method our results depend on the existence of a lipschitz function with which we replace the lipschitz constant usually appearing in the hypotheses of these investigations we find our methods useful for many reasons from the computational point of view we have allowed our iterations to be corrected at every step and still converge to a solution of the equation at hand in most earlier results the iterates are assumed to be computed exactly to achieve convergence moreover we showed that under weaker hypotheses our iterations converge faster than others already in the literature finally we used our results to solve uryson and fredholm type integral equations appearing in radiative transfer

a recurrent neural network for solving sylvester equation with time varying coefficients presents a recurrent neural network for solving the sylvester equation with time varying coefficient matrices the recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time varying solution of a given sylvester equation theoretical results of convergence and sensitivity analysis are presented to show the desirable properties of the recurrent neural network simulation results of time varying matrix inversion and online nonlinear output regulation via pole assignment for the ball and beam system and the inverted pendulum on a cart system are also included to demonstrate the effectiveness and performance of the proposed neural network

common nature of learning between back propagation and hopfield type neural networks for generalized matrix inversion with simplified models in this paper two simple structure neural networks based on the error back propagation bp algorithm i e bp type neural networks bpnns are proposed developed and investigated for online generalized matrix inversion specifically the bpnn l and bpnn r models are proposed and investigated for the left and right generalized matrix inversion respectively in addition for the same problem solving task two discrete time hopfield type neural networks hnns are developed and investigated in this paper similar to the classification of the presented bpnn l and bpnn r models the presented hnn l and hnn r models correspond to the left and right generalized matrix inversion respectively comparing the bpnn weight updating formula with the hnn state transition equation for the specific i e left or right generalized matrix inversion we show that such two derived learning expressions turn out to be the same in mathematics although the bp and hopfield type neural networks are evidently different from each other a great deal in terms of network architecture physical meaning and training patterns numerical results with different illustrative examples further demonstrate the efficacy of the presented bpnns and hnns for online generalized matrix inversion and more importantly their common natures of learning

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

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

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

on the convergence of a certain class of iterative procedures under relaxed conditions with applications we provide sufficient convergence conditions for a certain class of inexact newton like methods to a locally unique solution of a nonlinear equation in a banach space the equation contains a nondifferentiable term and at each step we use the inverse of the same linear operator we use ptak like conditions that are weaker than earlier ones our results apply whenever earlier ones do but not vice versa a semilocal convergence result is also given based on the contraction mapping principle finally our results apply to solve a nonlinear integral equation of uryson type appearing in elasticity theory that cannot be solved with existing methods

product cosines of angles between subspaces abstract let cos l m i 1 cos i denote the product of the cosines of the principal angles i between the subspaces l and m the direction cosines of an r dimensional subspace l are the n r numbers cos l r n j j q r n where q r n the set of increasing sequences of r elements from 1 n and r n j x x k r n x k 0 for k j the basic decomposition of a linear operator a r n r m with rank a r 0 is a i i a j j a cos 2 r a r m l cos 2 r a t r n j b ij a convex combination of nonsingular linear operators b ij r n j r n i here j a i q r m rank a i r and j a j q r n rank a j r the product cosines are related to the matrix volume defined as the product of its nonzero singular values the moore penrose inverse a is characterized as having the minimal volume among all 1 2 inverses of a indeed if g is a 1 2 inverse of a with range r g t and null space n g s then vol g vol a cos t r a t cos s n a t

on the convergence of two step methods generated by point to point operators in this study we examine conditions for the convergence of two step methods generated by point to point operators iterations of this type have a great importance in optimization theory stability analysis of dynamic systems and many other fields of applied mathematics the speed of convergence is also examined using the theory of majorants in a banach space setting the monotone convergence of these methods is also examined in a partially ordered topological space special cases of our results reduce to corresponding results already in the literature in particular relevant work can be found in i k argyros appl math comput 61 1 1994 51 69 and f szidarovasky and o palusinski appl math comput 64 1994 115 119 and references therein some applications are also given to the solution of systems of nonlinear integral equations of uryson type

on the stability of the cut locus abstract in r d we consider a riemannian metric g and an open bounded subset we study the stability of the cut locus associated with and g w r t perturbations both of the set and of the metric g in order to have the stability of the cut locus we assume c 2 regularity of the data the metrics and the sets in the case of sets with c 1 1 boundaries the cut locus may be unstable we prove that to c 2 perturbations both of the set and of the metric correspond small changes of the cut locus w r t the hausdorff distance i e the cut locus is stable in the c 2 category we give some examples showing that c 1 perturbations may lead to large variations of the cut locus

a dual neural network for redundancy resolution of kinematically redundant manipulators subject to joint limits and joint velocity limits in this paper a recurrent neural network called the dual neural network is proposed for online redundancy resolution of kinematically redundant manipulators physical constraints such as joint limits and joint velocity limits together with the drift free criterion as a secondary task are incorporated into the problem formulation of redundancy resolution compared to other recurrent neural networks the dual neural network is piecewise linear and has much simpler architecture with only one layer of neurons the dual neural network is shown to be globally exponentially convergent to optimal solutions the dual neural network is simulated to control the pa10 robot manipulator with effectiveness demonstrated

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

on the extension of the solutions of hamilton jacobi equations abstract we consider the viscosity solution of a homogeneous dirichlet problem for the eikonal equation in a bounded set we suppose that the hamiltonian h x p a x p p 1 is strictly convex w r t the variables p and of class c 1 1 w r t the variables x then the solution of the dirichlet problem admits an extension to a neighbourhood of u such that u is still a viscosity solution of the eikonal equation if and only if satisfies an exterior sphere condition the above result in particular provides a characterization of the boundary singularities and a regularity theorem up to the boundary for the solution of the eikonal equation

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

frequency estimation by principal component ar spectral estimation method without eigendecomposition an eigenvalue filtering method is proposed that applies a transformation to an autocorrelation matrix which has the effect of truncating the undesired eigenvalues so that the corresponding matrix function closely approximates the pseudoinverse it is shown using a computer simulation that compared to the forward backward method the proposed method enhances the threshold in snr by about 6 8 db further improvement is obtained by a simple subset selection method and a second eigenvalue filtering iteration

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

sufficient conditions for constructing methods faster than newton s in this study we use inexact newton like methods to approximate a locally unique solution of a nonlinear operator equation in a banach space using the majorant method and newton kantorovich type hypotheses we provide sufficient convergence conditions and an error analysis for our method our method is also shown under very natural assumptions to be faster than newton s under the same computational cost finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant flux emerging from a plane radiation field

on the monotone convergence of algorithmic models in this paper we provide first monotone convergence conditions for general algorithmic models practical conditions for linear spaces are then derived from the general theory in particular our results can apply to show monotone convergence of newton like methods

real time computation of the eigenvectors of a class of positive definite matrices this paper proposes an analog circuit approach to computing in real time the eigenvectors corresponding to all the eigenvalues of a class of positive definite matrices we show analytically and by simulations that the proposed circuit can provide the desired eigenvectors with arbitrarily small error during an elapsed time of only a few characteristic time constants of the circuit

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

a modification of newton s method for nondifferentiable equations in this paper we study the problem of approximating a solution of a nonlinear equation where the operator involved is nondifferentiable having the newton method as origin we construct a uniparametric family of iterative processes to approximate a solution of the equation to finish we consider several test problems

trace bounds on the solution of the algebraic matrix riccati and lyapunov equation lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix riccati and lyapunov equation are derived the upper trace bound obtained in this note in many cases results in a tighter bound as compared to the upper bound for the maximal eigenvalue proposed in 1 and 2

brief augmented gradient flows for on line robust pole assignment via state and output feedback this paper is concerned with robust pole assignment in synthesis of linear control systems via state and output feedbacks first both the pole assignment and robustness requirements are appropriately formulated as two optimization problems then gradient flow models are developed for the on line computation of feedback gain matrices that result in robust pole assignment by solving these two optimization problems a technique is introduced to facilitate the real time matrix inverse involved for realizing the gradient flow models the resulting augmented gradient flows have desired convergence properties simulation results are included to show the effectiveness of the proposed approach

convergence rates for inexact newton like methods at singular points and applications newton s method converges linearly rather than quadratically if the initial guess is close to the solution and the frechet derivative is singular at the solution several authors have already tried to improve the rate of convergence for this method in this case the limitations of their approach is twofold first the residuals at each step are assumed to be zero which is not the case in general second the previous works are limited only to the ordinary newton method here we extend and improve on these results by providing convergence rates for inexact newton like methods on a banach space setting moreover our results reduce to the ones already in the literature for special choices of the operators involved we complete our study with some numerical examples appearing in connection with some predator prey problems appearing in population modeling that show how our results improve on the earlier ones

a lagrangian network for kinematic control of redundant robot manipulators a recurrent neural network called the lagrangian network is presented for the kinematic control of redundant robot manipulators the optimal redundancy resolution is determined by the lagrangian network through real time solution to the inverse kinematics problem formulated as a quadratic optimization problem while the signal for a desired velocity of the end effector is fed into the inputs of the lagrangian network it generates the joint velocity vector of the manipulator in its outputs along with the associated lagrange multipliers the proposed lagrangian network is shown to be capable of asymptotic tracking for the motion control of kinematically redundant manipulators

rectifiability results for singular and conjugate points of optimal exit time problems we consider the value function v of optimal control problems with exit time under suitable assumptions through the study of the conjugate points we prove that the closure of the singular set of v is rectifiable moreover a sharper hausdorff estimate is given on the set of the conjugate nonsingular points

upper bounds for the eigenvalues of the solution of the lyapunov matrix equation eigenvalue upper bounds for the solution of the lyapunov matrix equation are presented for single eigenvalues summations of eigenvalues including the trace and products of eigenvalues including the determinant these bounds are compared to those for the trace and the extreme eigenvalues

a newton kantorvich theorem for equations involving m frechet differentiable operators and applications in radiative transfer abstract in this study we approximate a locally unique solution of a nonlinear operator equation in banach space using newton s method a complete error analysis showing the quadratic convergence of our method is also given our new theorem uses lipschitz or holder continuity assumptions on m frechet differentiable operators where m 2 is a positive integer a numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results a second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer

equilibrium characterization of dynamical neural networks and a systematic synthesis procedure for associative memories several novel results concerning the characterization of the equilibrium conditions of a continuous time dynamical neural network model and a systematic procedure for synthesizing associative memory networks with nonsymmetrical interconnection matrices are presented the equilibrium characterization focuses on the exponential stability and instability properties of the network equilibria and on equilibrium confinement viz ensuring the uniqueness of an equilibrium in a specific region of the state space while the equilibrium confinement result involves a simple test the stability results given obtain explicit estimates of the degree of exponential stability and the regions of attraction of the stable equilibrium points using these results as valuable guidelines a systematic synthesis procedure for constructing a dynamical neural network that stores a given set of vectors as the stable equilibrium points is developed

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

on some bounds in the algebraic riccati and lyanupov equations some bounds for the trace and determinant of the solution to the algebraic riccati and lyapunov matrix equations are presented these bounds give information on mean size of the solution to the equations in a simple way the obtained determinant bound for the lyapunov equation shows the improvement over the previously reported one

the regularity of the distance function propagates along minimizing geodesics abstract we consider the distance function from the boundary of an open bounded set r n associated to a riemannian metric with c 1 1 coefficients we show that the c 1 1 regularity propagates towards the boundary along the distance minimizing geodesics hence we show that the cut locus is invariant with respect to the generalized gradient flow associated to the distance function and that it has the same homotopy type as

on newton s method under mild differentiability conditions and applications in this study we are concerned with the problem of approximating a locally unique solution of a nonlinear operator equation in a banach space we examine the convergence of the newton kantorovich method by assuming that the frechet derivative of the nonlinear operator involved is only holder continuous many results exist already in the literature to cover the stronger case when the operator is lipschitz continuous moreover we use this study to counter some claims made by galperin and waksman on related work of ours furthermore we use our results to solve a two point boundary value problem appearing in kinetic theory of gasses elasticity and other applied areas

a generalized hopfield network for nonsmooth constrained convex optimization lie derivative approach this paper proposes a generalized hopfield network for solving general constrained convex optimization problems first the existence and the uniqueness of solutions to the generalized hopfield network in the filippov sense are proved then the lie derivative is introduced to analyze the stability of the network using a differential inclusion the optimality of the solution to the nonsmooth constrained optimization problems is shown to be guaranteed by the enhanced fritz john conditions the convergence rate of the generalized hopfield network can be estimated by the second order derivative of the energy function the effectiveness of the proposed network is evaluated on several typical nonsmooth optimization problems and used to solve the hierarchical and distributed model predictive control four tank benchmark

analog approach for the eigen decomposition of positive definite matrices abstract this paper proposes an analog approach for performing the eigen decomposition of positive definite matrices we show analytically and by simulations that the proposed circuit is guaranteed to converge to the desired eigenvectors and eigenvalues of positive definite matrices

minimal properties of moore penrose inverses abstract let ax y be consistent let x0 gy be any minimum norm solution satisfying ag ag and let a be the moore penrose inverse of a it is shown that g a for any in a class containing the unitarily invariant matrix norms the conditioning of the system ax y is studied via condition numbers c a g it is shown that c a g c a a for every moreover bounds on c a g are given in terms of singular values parallel results are found when a and g are symmetric with applications to linear models of less than full rank

a novel iterative method for computing generalized inverse in this letter we propose a novel iterative method for computing generalized inverse based on a novel kkt formulation the proposed iterative algorithm requires making four matrix and vector multiplications at each iteration and thus has low computational complexity the proposed method is proved to be globally convergent without any condition furthermore for fast computing generalized inverse we present an acceleration scheme based on the proposed iterative method the global convergence of the proposed acceleration algorithm is also proved finally the effectiveness of the proposed iterative algorithm is evaluated numerically

on the local semiconcavity of the solutions of the eikonal equation abstract we show that if the hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave furthermore in the 1d case we show that every viscosity solution of the eikonal equation is semiconcave if and only if the hamiltonian is lipschitz continuous with respect to the state variable

neural network for computing pseudoinverses and outer inverses of complex valued matrices we propose two continuous time neural networks for computing generalized inverses of complex valued matrices with rank deficient cases the first of them is applicable in the pseudoinverse computation and the second one is applicable in construction of outer inverses the proposed continuous time neural networks have a low complexity of implementation and they are proved to be globally convergent without any condition compared with the existing algorithms for computing the pseudoinverse and outer inverses of matrices the global convergence of the proposed continuous time neural networks is analyzed in the complex domain effectiveness of the proposed continuous time neural networks is evaluated numerically via examples

on the convergence of generalized newton methods and implicit functions abstract in this article we provide sufficient conditions for the convergence of generalized newton methods the linear operators involved are obtained recursively from the previous ones

on the cut locus of closed sets abstract let c be a closed subset of a smooth manifold of dimension n m and let m be endowed with a riemannian metric of class c 2 we study the cut locus of c cut c first we show that cut c is a set of measure zero then we assume that c is the boundary of an open bounded set m in particular this assumption implies that cut c 0 we deduce that cut c is invariant w r t the generalized gradient flow associated with the distance function from the set c as a consequence of the invariance we have that cut c has the same homotopy type as the set furthermore if m is a compact manifold then cut c has the same homotopy type as m c finally we show that the closure of the cut locus stays away from c if and only if c is a manifold of class c 1 1

stability analysis of a family of matrices the authors extend the use of the lyapunov equation method to check the stability of a family of matrices the family of interval matrices is defined as the convex hull of a finite number of matrices and is a special case where each element has independent perturbation while the stability of interval matrices cannot be inferred from the stability of its vertex matrices the authors show that the family of matrices will be stable if the symmetric matrices associated with the vertex matrices have matrix measures less than two

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

a convergence theorem for newton like methods in banach spaces on the basis of the results obtained in a series of papers 25 28 a convergence theorem for newton s method in banach spaces is given which improves the theorems of kantorovich 4 lancaster 8 and ostrowski 10 the error bounds obtained improve the recent results of potra 17

a uniparametric family of iterative processes for solving nondifferentiable equations in this work we study a class of secant like iterations for solving nonlinear equations in banach spaces we consider a condition for divided differences which generalizes the usual ones i e lipschitz and holder continuous conditions a semilocal convergence result is obtained for nondifferentiable operators for that we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution finally we apply our results to the numerical solution of several examples