refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights recently a dynamic neural system was presented and analyzed due to its good performance in optimization computation and low complexity for implementation the global asymptotic stability of such a dynamic neural system with symmetric connection weights was well studied in this note based on a new lyapunov function we investigate the global asymptotic stability of the dynamic neural system with asymmetric connection weights since the dynamic neural system with asymmetric weights is more general than that with symmetric ones the new results are significant in both theory and applications specially the new result can cover the asymptotic stability results of linear systems as special cases

a dynamic thick restarted semi refined able algorithm for computing a few selected eigentriplets of large nonsymmetric matrices able method can be used to compute eigentriplets of large scale nonsymmetric matrices however there is no guarantee for the ritz vectors obtained by this method to converge even if the subspace is good enough furthermore for the sake of storage limitation and amount of computations restarting techniques are often needed in order to deal with these problems we propose a dynamic thick restarted semi refined able algorithm sable in which we use semi refined ritz vectors to approximate the desired eigenvectors the relationship between the novel method and the classical one is given theoretical results show that the new method can circumvent the possible danger that exists in the standard one in some degree numerical experiments are made on real world problems and comparisons are drown on the thick restarted able algorithm table and the dynamic thick restarted semi refined able algorithm sable they show that the latter is often more powerful and attractive than its standard counterpart

perfectly secure index coding in this paper we investigate the index coding problem in the presence of an eavesdropper messages are to be sent from one transmitter to a number of legitimate receivers who have side information about the messages and share a set of secret keys with the transmitter to do this the transmitter communicates to the legitimate receivers the public code c which is also heard by the eavesdropper we assume perfect secrecy meaning that the eavesdropper should not be able to retrieve any information about the message set from the public communication we study the minimum key lengths for zero error and perfectly secure index coding problem on one hand this problem is a generalization of the index coding problem and thus a difficult one on the other hand it is a generalization of the shannon s cipher system we show that a generalization of shannon s one time pad strategy is optimal up to a multiplicative constant meaning that it obtains the entire boundary of the cone formed by looking at the secure rate region from the origin this shows the optimality of the generalized one time pad for minimizing the consumption of shared secret keys per message bits when public communication is free the transmitter is not charged for the rate of the public communication finally we consider relaxation of the perfect secrecy and zero error constraints to weak secrecy and asymptotically vanishing probability of error and provide a secure version of the result obtained by langberg and effros on the equivalence of zero error and epsilon error regions in the conventional index coding problem

polynomial characterizations of the approximate eigenvectors by the refined arnoldi method and an implicitly restarted refined arnoldi algorithm abstract it is well known that a ritz vector obtained by the lanczos method and the arnoldi method can be characterized by an elegant explicit polynomial whose roots are just the other m 1 ritz values of the matrix a in question from the m dimensional krylov subspace involved analogous results hold for the recently developed harmonic lanczos and arnoldi methods and the roots of the corresponding polynomial are now the other m 1 harmonic ritz values of a from the same subspace in this paper we investigate a polynomial characterization of the refined arnoldi method proposed by the author in recent years we derive a polynomial characterization of the refined arnoldi method and apply the implicitly restarting scheme proposed by sorensen to the refined arnoldi method the roots of this polynomial are used as shifts called refined shifts within an implicitly restarted refined arnoldi algorithm the shifts are interpreted to have the same nature as the exact shifts used within the implicitly restarted arnoldi algorithm proposed by sorensen numerical experiments compare the use of refined shifts with exact shifts within an implicitly restarted arnoldi algorithm the results also show that implicitly restarting the refined arnoldi method is far superior to the saad s explicitly restarted scheme

a refined jacobi davidson method and its correction equation a central problem in the jacobi davidson method is to expand a projection subspace by solving a certain correction equation it has been commonly accepted that the correction equation always has a solution however it is proved in this paper that this is not true conditions are given to decide when it has a unique solution or many solutions or no solution a refined jacobi davidson method is proposed to overcome the possible nonconvergence of ritz vectors by computing certain refined approximation eigenvectors from the subspace a corresponding correction equation is derived for the refined method numerical experiments are conducted and efficiency of the refined method is confirmed

a novel neural network for variational inequalities with linear and nonlinear constraints variational inequality is a uniform approach for many important optimization and equilibrium problems based on the sufficient and necessary conditions of the solution this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of lyapunov and converges to an exact solution of the original problem meanwhile the proposed network with a gradient mapping is also proved to be stable in the sense of lyapunov and to have a finite time convergence under some mild conditions by using a new energy function compared with the existing neural networks the new model can be applied to solve some nonmonotone problems has no adjustable parameter and has lower complexity thus the structure of the proposed network is very simple since the proposed network can be used to solve a broad class of optimization problems it has great application potential the validity and transient behavior of the proposed neural network are demonstrated by several numerical examples

on a generalization of a normal map and equation the class of normal maps was recently investigated by robinson and ralph in connection with the study of a variational inequality defined on a polyhedral set in this paper a generalization of such a map is considered and the associated generalized normal equation is studied the latter provides a unified formulation of several generalized variational inequality and complementarity problems using degree theory some sufficient conditions for the existence of a zero of a generalized normal map are established and the stability of a generalized normal equation at a solution is analyzed specializations of the results to various applications are discussed

quasi kernel polynomials and their use in non hermitian matrix iterations abstract some of the most efficient iterative algorithms for large sparse hermitian matrix computations are based on orthogonal or kernel polynomials for the case of non hermitian matrices methods based on orthogonal or kernel polynomials are less satisfactory in that the resulting algorithms involve long recurrences consequently it is usually too expensive to run the full algorithms and restarts are necessary a typical example is the generalized minimal residual method gmres for solving non hermitian linear systems where work and storage per iteration grow linearly with the iteration number recently two quasi minimal residual methods qmr for solving non hermitian linear systems have been proposed which unlike gmres are based on short recurrences and hence can be used as true iterative schemes without restarts in this paper the concept of quasi kernel polynomials is introduced some general theory for quasi kernel polynomials is developed such as recurrence relations and a characterization of roots of quasi kernel polynomials as generalized eigenvalues it is pointed out that the qmr approaches are based on two particular instances of quasi kernel polynomials also the use of quasi kernel polynomials for approximating eigenvalues or pseudospectra of large sparse non hermitian matrices is briefly discussed

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

an analysis of the rayleigh ritz method for approximating eigenspaces this paper concerns the rayleigh ritz method for computing an approximation to an eigenspace x of a general matrix a from a subspace w that contains an approximation to x the method produces a pair n x that purports to approximate a pair l x where x is a basis for x and ax xl in this paper we consider the convergence of n x as the sine e of the angle between x and w approaches zero it is shown that under a natural hypothesis called the uniform separation condition the ritz pairs n x converge to the eigenpair l x when one is concerned with eigenvalues and eigenvectors one can compute certain refined ritz vectors whose convergence is guaranteed even when the uniform separation condition is not satisfied an attractive feature of the analysis is that it does not assume that a has distinct eigenvalues or is diagonalizable

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

on the correction equation of the jacobi davidson method abstract the jacobi davidson method is one of the most popular approaches for iteratively computing a few eigenvalues and their associated eigenvectors of a large matrix the key of this method is to expand the search subspace via solving the jacobi davidson correction equation whose coefficient matrix is singular it is believed by scholars that the jacobi davidson correction equation is consistent and has a unique solution in this paper however we point out that the correction equation either has a unique solution or has no solution and we derive a computable necessary and sufficient condition for cheaply judging the existence and uniqueness of the solution furthermore we consider the problem of stagnation and verify that if the jacobi davidson method stagnates then the corresponding ritz value is a defective eigenvalue of the projection matrix finally we provide a computable criterion for expanding the search subspace successfully the properties of some alternative jacobi davidson correction equations are also discussed

natural convection in steady solidification finite element analysis of a two phase rayleigh benard problem abstract galerkin finite element approximations and newton s method for solving free boundary problems are combined with computer implemented techniques from nonlinear perturbation analysis to study solidification problems with natural convection in the melt the newton method gives rapid convergence to steady state velocity temperature and pressure fields and melt solid interface shapes and forms the basis for algebraic methods for detecting multiple steady flows and assessing their stability the power of this combination is demonstrated for a two phase rayleigh benard problem composed of melt and and solid in a vertical cylinder with the thermal boundary conditions arranged so that a static melt with a flat melt solid interface is always a solution multiple cellular flows bifurcating from the static state are detected and followed as rayleigh number is varied changing the boundary conditions to approach those appropriate for the vertical bridgman solidification system causes imperfections that eliminate the static state the flow structure in the bridgman system is related to those for the rayleigh benard system by a continuous evolution of the boundary conditions

a block arnoldi chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices in this paper we describe a block version of arnoldi s method for computing a few eigenvalues with largest or smallest real parts the method is accelerated via chebyshev iteration and a procedure is developed to identify the optimal ellipse which encloses the spectrum a parallel implementation of this method is investigated on the eight processor alliant fx 80 numerical results and comparisons with simultaneous iteration on some harwell boeing matrices are reported

a new recurrent neural network for solving convex quadratic programming problems with an application to the k winners take all problem in this paper a new recurrent neural network is proposed for solving convex quadratic programming qp problems compared with existing neural networks the proposed one features global convergence property under weak conditions low structural complexity and no calculation of matrix inverse it serves as a competitive alternative in the neural network family for solving linear or quadratic programming problems in addition it is found that by some variable substitution the proposed network turns out to be an existing model for solving minimax problems in this sense it can be also viewed as a special case of the minimax neural network based on this scheme a k winners take all k wta network with o n complexity is designed which is characterized by simple structure global convergence and capability to deal with some ill cases numerical simulations are provided to validate the theoretical results obtained more importantly the network design method proposed in this paper has great potential to inspire other competitive inventions along the same line

information inequalities for joint distributions with interpretations and applications upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies these inequalities generalize shannon s chain rule for entropy as well as inequalities of han fujishige and shearer a duality between the upper and lower bounds for joint entropy is developed all of these results are shown to be special cases of general new results for submodular functions thus the inequalities presented constitute a richly structured class of shannon type inequalities the new inequalities are applied to obtain new results in combinatorics such as bounds on the number of independent sets in an arbitrary graph and the number of zero error source channel codes as well as determinantal inequalities in matrix theory a general inequality for relative entropies is also developed finally revealing connections of the results to literature in economics computer science and physics are explored

an improved dual neural network for solving a class of quadratic programming problems and its k winners take all application this paper presents a novel recurrent neural network for solving a class of convex quadratic programming qp problems in which the quadratic term in the objective function is the square of the euclidean norm of the variable this special structure leads to a set of simple optimality conditions for the problem based on which the neural network model is formulated compared with existing neural networks for general convex qp the new model is simpler in structure and easier to implement the new model can be regarded as an improved version of the dual neural network in the literature based on the new model a simple neural network capable of solving the k winners take all k wta problem is formulated the stability and global convergence of the proposed neural network is proved rigorously and substantiated by simulation results

improved algorithms for the lowest few eigenvalues and associated eigenvectors of large matrices abstract some modifications of davidson s eigenvalue algorithm are discussed and their performances on a number of test cases are assessed they are found to offer improvements over the original algorithm a method for solving the equations stemming from the quasi degenerate variational perturbation theory is presented solutions can be obtained as simply as with the eigenvalue algorithm for both ground and excited states

a refined shift and invert arnoldi algorithm for large unsymmetric generalized eigenproblems abstract the shift and invert arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and or the associated eigenvectors of a large unsymmetric matrix pair but there is no guarantee for the approximate eigenvectors ritz vectors obtained by this method to converge even though the subspace is good enough in order to correct this problem a refined shift and invert arnoldi method is proposed that uses certain refined ritz vectors to approximate the desired eigenvectors the refined ritz vectors can be computed cheaply and reliably by small sized singular value decompositions it is shown that the refined method converges a refined shift and invert arnoldi algorithm is developed and several numerical examples are reported comparisons are drawn on the refined algorithm and the shift and invert arnoldi algorithm indicating that the former is considerably more efficient than the latter

residuals of refined projection methods for large matrix eigenproblems abstract in terms of the deviation of the eigenvector from a projection subspace e a priori error bounds are established for the residual norm of an approximate eigenpair obtained by a refined projection method it is shown that the residual converges to zero as the deviation tends to zero finally how to efficiently compute refined ritz vectors is discussed in the refined symmetric lanczos method

reformulation of variational inequalities on a simplex and compactification of complementarity problems many variational inequality problems vips can be reduced by a compactification procedure to a vip on the canonical simplex reformulations of this problem are studied including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized fischer burmeister function it is proved that bounded level set results hold for these reformulations under quite general assumptions on the operator therefore it can be guaranteed that minimization algorithms generate bounded sequences and under monotonicity conditions these algorithms necessarily find solutions of the original problem some numerical experiments are presented

a nonfeasible gradient projection recurrent neural network for equality constrained optimization problems in this paper a recurrent neural network for both convex and nonconvex equality constrained optimization problems is proposed which makes use of a cost gradient projection onto the tangent space of the constraints the proposed neural network constructs a generically nonfeasible trajectory satisfying the constraints only as t rarr infin local convergence results are given that do not assume convexity of the optimization problem to be solved global convergence results are established for convex optimization problems an exponential convergence rate is shown to hold both for the convex case and the nonconvex case numerical results indicate that the proposed method is efficient and accurate

generalized block lanczos methods for large unsymmetric eigenproblems generalized block lanczos methods for large unsymmetric eigenproblems are presented which contain the block arnoldi method and the block arnoldi algorithms are developed the convergence of this class of methods is analyzed when the matrix a is diagonalizable upper bounds for the distances between normalized eigenvectors and a block krylov subspace are derived and a priori theoretical error bounds for ritz elements are established compared with generalized lanczos methods which contain arnoldi s method the convergence analysis shows that the block versions have two advantages first they may be efficient for computing clustered eigenvalues second they are able to solve multiple eigenproblems however a deep analysis exposes that the approximate eigenvectors or ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix a even if corresponding approximate eigenvalues or ritz values do since the convergence of ritz vectors needs more sufficient conditions which may be impossible to satisfy theoretically than that of ritz values does the issues of how to restart and to solve multiple eigenproblems are addressed and some numerical examples are reported to confirm the theoretical analysis

an implicitly restarted lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets e kokiopoulou et al developed an implicitly restarted lanczos bidiagonalization method named irlanb for computing the smallest singular triplets they use the wanted harmonic ritz values and the corresponding refined harmonic ritz vectors to approximate the desired smallest singular triplets however they use the unwanted harmonic ritz values as shifts named the harmonic shifts to implicitly restart their algorithm therefore although they replace the harmonic ritz vectors by the refined harmonic ritz vectors which are always better than the harmonic ritz vectors to improve the convergence of irlanb the subspace after implicitly restarting is the same as that without this replacement in this paper using the information of the refined harmonic ritz vectors we present a new shift strategy named the refined harmonic shifts to replace the harmonic shifts used in irlanb and obtain a new method called irrlanb the refined harmonic shifts are better than the harmonic shifts and can be computed cheaply and reliably the numerical experiments are reported to indicate that irrlanb is superior than irlanb

block arnoldi and davidson methods for unsymmetric large eigenvalue problems we present two methods for computing the leading eigenpairs of large sparse unsymmetric matrices namely the block arnoldi method and an adaptation of the davidson method to unsymmetric matrices we give some theoretical results concerning the convergence and discuss implementation aspects of the two methods finally some results of numerical tests on a variety of matrices in which we compare these two methods are reported

solving generally constrained generalized linear variational inequalities using the general projection neural networks generalized linear variational inequality glvi is an extension of the canonical linear variational inequality in recent years a recurrent neural network nn called general projection neural network gpnn was developed for solving glvis with simple bound often box type or sphere type constraints the aim of this paper is twofold first some further stability results of the gpnn are presented second the gpnn is extended for solving glvis with general linear equality and inequality constraints a new design methodology for the gpnn is then proposed furthermore in view of different types of constraints approaches for reducing the number of neurons of the gpnn are discussed which results in two specific gpnns moreover some distinct properties of the resulting gpnns are also explored based on their particular structures numerical simulation results are provided to validate the results

on the use of harmonic ritz pairs in approximating internal eigenpairs the goal of this paper is to increase our understanding of harmonic rayleigh ritz for real symmetric matrices we do this by discussing different though related topics a priori error analysis a posteriori error analysis a comparison with refined rayleigh ritz and the selection of a suitable harmonic ritz vector

a delayed projection neural network for solving linear variational inequalities in this paper a delayed projection neural network is proposed for solving a class of linear variational inequality problems the theoretical analysis shows that the proposed neural network is globally exponentially stable under different conditions by the proposed linear matrix inequality lmi method the monotonicity assumption on the linear variational inequality is no longer necessary by employing lagrange multipliers the proposed method can resolve the constrained quadratic programming problems finally simulation examples are given to demonstrate the satisfactory performance of the proposed neural network

on expansion of search subspaces for large non hermitian eigenproblems abstract how to expand a given subspace efficiently is an important task for large sparse eigenvalue problems in this paper we are interested in the following question which vector from a given subspace after multiplied by the matrix a will give a better expanded search subspace in 2008 33 ye investigated this problem and determined how to choose the optimal expansion vector theoretically unfortunately the result is not computable since it involves some unknown information on the desired eigenvector when a is symmetric it was suggested that the ritz vector may lead to a good candidate for subspace expansion however when a is non hermitian the ritz vector may not be a satisfactory choice the contribution of this paper is twofold first we suggest to use the refined ritz vector to expand the search subspace and propose a residual expansion arnoldi method for subspace expansion theoretical results justify the use of the refined ritz vector second we prove that the elements of the primitive refined ritz vector have a decreasing pattern going to zero we then show that the decreasing pattern exists in an arbitrary primitive approximate eigenvector and derive an inexact residual expansion arnoldi method for subspace expansion numerical examples show the effectiveness of our theoretical results and illustrate the numerical behavior of the new method for subspace expansion

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

deflated block krylov subspace methods for large scale eigenvalue problems we discuss a class of deflated block krylov subspace methods for solving large scale matrix eigenvalue problems the efficiency of an arnoldi type method is examined in computing partial or closely clustered eigenvalues of large matrices as an improvement we also propose a refined variant of the arnoldi type method comparisons show that the refined variant can further improve the arnoldi type method and both methods exhibit very regular convergence behavior

the convergence of harmonic ritz values harmonic ritz vectors and refined harmonic ritz vectors this paper concerns a harmonic projection method for computing an approximation to an eigenpair x of a large matrix a given a target point r and a subspace w that contains an approximation to x the harmonic projection method returns an approximation x to x three convergence results are established as the deviation e of x from w approaches zero first the harmonic ritz value r converges to a if a certain rayleigh quotient matrix is uniformly nonsingular second the harmonic ritz vector x converges to x if the rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic ritz values third better error bounds for the convergence of are derived when x converges however we show that the harmonic projection method can fail to find the desired eigenvalue in other words the method can miss if it is very close to r to this end we propose to compute the rayleigh quotient p of a with respect to x and take it as a new approximate eigenvalue p is shown to converge to once x tends to x no matter how r is close to finally we show that if the rayleigh quotient matrix is uniformly nonsingular then the refined harmonic ritz vector or more generally the refined eigenvector approximation introduced by the author converges we construct examples to illustrate our theory

a new shift strategy for the implicitly restarted refined harmonic lanczos method abstract the implicitly restarted refined harmonic lanczos method is widely used for computing the interior eigenpairs of large real symmetric matrices in each restart it takes the wanted refined harmonic ritz pairs to be the approximate eigenpairs and takes the refined harmonic shifts to implicitly restart the algorithm in this paper we modify this method by replacing the refined harmonic shifts by new shifts based on a special direct sum decomposition of the projected subspace we build a new subspace in each restart and then take the harmonic ritz values onto it to be the new shifts the new shifts can be obtained from a small generalized eigenvalue problem and the computational cost is negligible numerical experiments show that the new shifts are generally better than the refined harmonic shifts

robust preconditioning of large sparse symmetric eigenvalue problems abstract iterative methods for solving large sparse symmetric eigenvalue problems often encounter convergence difficulties because of ill conditioning the generalized davidson method is a well known technique which uses eigenvalue preconditioning to surmount these difficulties preconditioning the eigenvalue problem entails more subtleties than for linear systems in addition the use of an accurate conventional preconditioner i e as used in linear systems may cause deterioration of convergence or convergence to the wrong eigenvalue the purpose of this paper is to assess the quality of eigenvalue preconditioning and to propose strategies to improve robustness numerical experiments for some ill conditioned cases confirm the robustness of the approach

tchebychev acceleration technique for large scale nonsymmetric matrices the acceleration by tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed a simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue new criteria are established to identify the optimal ellipse of the eigenspectrum the algorithm is fast reliable and does not require a search for all possible ellipses which enclose the spectrum the procedure is applicable to nonsymmetric linear systems as well

global exponential stability of neural networks with globally lipschitz continuous activations and its application to linear variational inequality problem this paper investigates the existence uniqueness and global exponential stability ges of the equilibrium point for a large class of neural networks with globally lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations the provided sufficient condition for ges is mild and some conditions easily examined in practice are also presented the ges of neural networks in the case of locally lipschitz continuous activations is also obtained under an appropriate condition the analysis results given in the paper extend substantially the existing relevant stability results in the literature and therefore expand significantly the application range of neural networks in solving optimization problems as a demonstration we apply the obtained analysis results to the design of a recurrent neural network rnn for solving the linear variational inequality problem vip defined on any nonempty and closed box set which includes the box constrained quadratic programming and the linear complementarity problem as the special cases it can be inferred that the linear vip has a unique solution for the class of lyapunov diagonally stable matrices and that the synthesized rnn is globally exponentially convergent to the unique solution some illustrative simulation examples are also given

an iterative block arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems a modified m step arnoldi algorithm proposed by jia and elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version in the new method we use new modified approximate eigenvectors obtained from a linear combination of ritz vectors and the wasted m 1 th block basis vector whose residual norms are satisfied with some p 1 dimensional minimization problems the resulting block arnoldi algorithm is not only better than the standard m step one both in theory and in practice but also cheaper than the standard m 1 step one the relationships among the residual norms of ritz pairs and those of new approximate eigenpairs are analyzed theoretical results show that the new method can overcome the drawback of nonconvergence which exists in the conventional one in some extent we carry out some numerical experiments which exhibit that the new algorithm is more efficient and works better than its counterparts

a refined harmonic lanczos bidiagonalization method and an implicitly restarted algorithm for computing the smallest singular triplets of large matrices the harmonic lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix a we prove that for good enough projection subspaces harmonic ritz values converge if the columns of a are strongly linearly independent on the other hand harmonic ritz values may miss some desired singular values when the columns of a are almost linearly dependent furthermore harmonic ritz vectors may converge irregularly and may even fail to converge based on the refined projection principle for large matrix eigenproblems due to the first author we propose a refined harmonic lanczos bidiagonalization method that takes the rayleigh quotients of the harmonic ritz vectors as approximate singular values and extracts the best approximate singular vectors called the refined harmonic ritz approximations from the given subspaces in the sense of residual minimizations the refined approximations are shown to converge to the desired singular vectors once the subspaces are sufficiently good and the rayleigh quotients converge an implicitly restarted refined harmonic lanczos bidiagonalization algorithm irrhlb is developed we study how to select the best possible shifts and suggest refined harmonic shifts that are theoretically better than the harmonic shifts used within the implicitly restarted harmonic lanczos bidiagonalization algorithm irhlb we propose a novel procedure that can numerically compute the refined harmonic shifts efficiently and accurately numerical experiments are reported that compare irrhlb with five other algorithms based on the lanczos bidiagonalization process it appears that irrhlb is at least competitive with them and can be considerably more efficient when computing the smallest singular triplets

a thick restarted block arnoldi algorithm with modified ritz vectors for large eigenproblems the block arnoldi method is one of the most commonly used techniques for large eigenproblems in this paper we exploit certain modified ritz vectors to take the place of ritz vectors in the thick restarted block arnoldi algorithm and propose a modified thick restarted block arnoldi algorithm for large eigenproblems we then consider how to periodically combine the refined subspace iterative method with the modified thick restarting block arnoldi algorithm for computing a few dominant eigenpairs of a large matrix the resulting algorithm is called a subspace block arnoldi algorithm numerical experiments show the efficiency of our new algorithms

a numerical study on the wiretap network with a simple network topology in this paper we study a security problem on a simple wiretap network consisting of a source node s a destination node d and an intermediate node r the intermediate node connects the source and the destination nodes via a set of noiseless parallel channels with sizes n 1 and n 2 respectively a message m is to be sent from s to d the information in the network may be eavesdropped by a set of wiretappers the wiretappers cannot communicate with one another each wiretapper can access a subset of channels called a wiretap set all the chosen wiretap sets form a wiretap pattern a random key k is generated at s and a coding scheme on m k is employed to protect m we define two decoding classes at d in class i only m is required to be recovered and in class ii both m and k are required to be recovered the objective is to minimize h k h m for a given wiretap pattern under the perfect secrecy constraint the first question we address is whether routing is optimal on this simple network by enumerating all the wiretap patterns on the class i ii 3 3 networks and harnessing the power of shannon type inequalities we find that gaps exist between the bounds implied by routing and the bounds implied by shannon type inequalities for a small fraction min h k h m for the remaining wiretap patterns where gaps exist we study some simple wiretap patterns and find that their shannon bounds i e the lower bound induced by shannon type inequalities can be achieved by linear codes which means routing is not sufficient even for the 3 3 network for some complicated wiretap patterns we study the structures of linear coding schemes under the assumption that they can achieve the corresponding shannon bounds this paper indicates that the determination of the entropic region of six linear vector spaces cannot be sidestepped some subtle issues on the network models are discussed and interesting observations are stated

on secure network coding with nonuniform or restricted wiretap sets the secrecy capacity of a network for a given collection of permissible wiretap sets is the maximum rate of communication such that observing links in any permissible wiretap set reveal no information about the message this paper considers secure network coding with nonuniform or restricted wiretap sets for example networks with unequal link capacities where a wiretapper can wiretap any subset of k links or networks where only a subset of links can be wiretapped existing results show that for the case of uniform wiretap sets networks with equal capacity links packets where any k can be wiretapped the secrecy capacity is given by a cut set bound if random keys are injected at the source and decoded at the sink whether or not the communicating users have information about the choice of wiretap set in contrast we show that for the nonuniform case this secrecy rate is achievable for the case of known but not unknown wiretap set we give achievable linear optimization based strategies where random keys are canceled at intermediate nonsink nodes or injected at intermediate nonsource nodes finally we show that determining the secrecy capacity is an np hard problem

performance bounds on a wiretap network with arbitrary wiretap sets consider a communication network represented by a directed graph cal g cal v cal e where cal v is the set of nodes and cal e is the set of point to point channels in the network on the network a secure message m is transmitted and there may exist wiretappers who want to obtain information about the message in secure network coding we aim to find a network code which can protect the message against the wiretapper whose power is constrained cai and yeung studied the model in which the wiretapper can access any one but not more than one set of channels called a wiretap set out of a collection cal a of all possible wiretap sets in order to protect the message the message needs to be mixed with a random key k they proved tight fundamental performance bounds when cal a consists of all subsets of cal e of a fixed size r however beyond this special case obtaining such bounds is much more difficult in this paper we investigate the problem when cal a consists of arbitrary subsets of cal e and obtain the following results 1 an upper bound on h m and 2 a lower bound on h k in terms of h m the upper bound on h m is explicit while the lower bound on h k can be computed in polynomial time when vert cal a vert is fixed the tightness of the lower bound for the point to point communication system is also proved

a refined iterative algorithm based on the block arnoldi process for large unsymmetric eigenproblems when the matrix in question is unsymmetric the approximate eigenvectors or ritz vectors obtained by orthogonal projection methods including arnoldi s method and the block arnoldi method cannot be guaranteed to converge in theory even if the corresponding approximate eigenvalues or ritz values do in order to circumvent this danger a new strategy has been proposed by the author for arnoldi s method the strategy used is generalized to the block arnoldi method in this paper it discards ritz vectors and instead computes refined approximate eigenvectors by small sized singular value decompositions it is proved that the new strategy can guarantee the convergence of refined approximate eigenvectors if the corresponding ritz values do the resulting refined iterative algorithm is realized by the block arnoldi process numerical experiments show that the refined algorithm is much more efficient than the iterative block arnoldi algorithm