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

accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

fuzzy wavelet neural network models for prediction and identification of dynamical systems this paper presents fuzzy wavelet neural network fwnn models for prediction and identification of nonlinear dynamical systems the proposed fwnn models are obtained from the traditional takagi sugeno kang fuzzy system by replacing the then part of fuzzy rules with wavelet basis functions that have the ability to localize both in time and frequency domains the first and last model use summation and multiplication of dilated and translated versions of single dimensional wavelet basis functions respectively and in the second model then parts of the rules consist of radial function of wavelets gaussian type of activation functions are used in if part of the fuzzy rules a fast gradient based training algorithm i e the broyden fletcher goldfarb shanno method is used to find the optimal values for unknown parameters of the fwnn models simulation examples are also given to compare the effectiveness of the models with the other known methods in the literature according to simulation results we see that the proposed fwnn models have impressive generalization ability

a priori error bounds on invariant subspace approximations by block krylov subspaces abstract we derive a priori error bounds for the block krylov subspace methods in terms of the sine between the desired invariant subspace and the block krylov subspace the obtained results can be seen as the block analogue of the classical a priori estimates for standard projection methods

vertex reinforced random walk on z has finite range a stochastic process called vertex reinforced random walk vrrw is defined in pemantle ann probab 16 1229 1241 we consider this process in the case where the underlying graph is an infinite chain i e process in the case where the underlying graph is an infinite chain i e the one dimensional integer lattice we show that the range is almost surely finite that at least five points are visited infinitely often almost surely and that with positive probability the range contains exactly five points there are always points visited infinitely often but at a set of times of zero density and we show that the number of visits to such a point to time n may be asymptotically n for a dense set of values 0 1 the power law analysis relies on analysis of a related urn model

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

sequentially interacting markov chain monte carlo methods we introduce a novel methodology for sampling from a sequence of probability distributions of increasing dimension and estimating their normalizing constants these problems are usually addressed using sequential monte carlo smc methods the alternative sequentially interacting markov chain monte carlo simcmc scheme proposed here works by generating interacting non markovian sequences which behave asymptotically like independent metropolis hastings mh markov chains with the desired limiting distributions contrary to smc methods this scheme allows us to iteratively improve our estimates in an mcmc like fashion we establish convergence of the algorithm under realistic verifiable assumptions and demonstrate its performance on several examples arising in bayesian time series analysis

an algorithmic approach for fuzzy inference to apply fuzzy logic two major tasks need to be performed the derivation of production rules and the determination of membership functions these tasks are often difficult and time consuming this paper presents an algorithmic method for generating membership functions and fuzzy production rules the method includes an entropy minimization for screening analog values membership functions are derived by partitioning the variables into the desired number of fuzzy terms and production rules are obtained from minimum entropy clustering decisions in the rule derivation process rule weights are also calculated this algorithmic approach alleviates many problems in the application of fuzzy logic to binary classification

vertex reinforced random walks and a conjecture of pemantle we discuss and disprove a conjecture of pemantle concerning vertex reinforced random walks the setting is a general theory of non markovian discrete time random processes on a finite space e 1 d for which the transition probabilities at each step are influenced by the proportion of times each state has been visited it is shown that under mild conditions the asymptotic behavior of the empirical occupation measure of the process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the d 1 unit simplex in particular any minimal attractor of this vector field has a positive probability to be the limit set of the sequence of empirical occupation measures these properties are used to disprove a conjecture and to extend some results due to pemantle some applications to edge reinforced random walks are also considered

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

free vibration analysis of three dimensional solids by bem in practical applications it is often necessary to determine the natural frequencies and mode shapes of a solid in addition to the static response under a variety of loading conditions this paper discusses the calculation of natural frequencies and mode shapes of three dimensional solids using the boundary element method the method developed is based on the use of particular integrals of the elastic equilibrium equations and employs only real arithmetic it represents an extension to three dimensions of work previously carried out for two dimensional problems in references 1 and 15 the method has been incorporated for multiregion analysis in the best3d boundary element solution technology three dimensional computer program 11 results of the boundary element calculations are compared with analytical finite element and experimental results

parallel computation of polynomials with minimal uniform norm and its application to large eigenproblems we investigate the parallelization of an algorithm that computes the polynomial with minimal uniform norm on polygonal domains the obtained polynomial is used to accelerate large non hermitian eigenvalue problems we report the performance results obtained on the machine paragon and compare this method with the one based on chebyshev acceleration techniques

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

krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices abstract in this paper we propose a general approach by which eigenvalues with a special property of a given matrix a can be obtained in this approach we first determine a scalar function c c whose modulus is maximized by the eigenvalues that have the special property next we compute the generalized power iterations uinj 1 a u j j 0 1 where u 0 is an arbitrary initial vector finally we apply known krylov subspace methods such as the arnoldi and lanczos methods to the vector u n for some sufficiently large n we can also apply the simultaneous iteration method to the subspace span x n 1 x n k with some sufficiently large n where x j 1 m a x j m j 0 1 m 1 k in all cases the resulting ritz pairs are approximations to the eigenpairs of a with the special property we provide a rather thorough convergence analysis of the approach involving all three methods as n for the case in which a is a normal matrix we also discuss the connections and similarities of our approach with the existing methods and approaches in the literature

computing eigenpairs of hermitian matrices in augmented krylov subspace produced by rayleigh quotient iterations abstract in this paper we first introduce a class of augmented krylov subspace that is a union of the standard krylov subspace and another low dimension subspace and then construct the corresponding augmented krylov subspace method for computing some extreme eigenvalues and their associated eigenvectors of hermitian matrices in order to improve the overall performance of this method we present some practical implementations with preconditioning which then results in the preconditioned augmented krylov subspace method in addition through convergence analysis it indicates that the preconditioned augmented krylov subspace method attains cubic convergence rate locally in the last we use some numerical examples to demonstrate the convergence property and the superiority of this method

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

lanczos versus subspace iteration for solution of eigenvalue problems in this paper we consider solution of the eigen problem in structural analysis using a recent version of the lanczos method and the subspace method the two methods are applied to examples and we conclude that the lanczos method has advantages which are too good to overlook

a handwritten numeral character classification using tolerant rough set proposes a data classification method based on the tolerant rough set that extends the existing equivalent rough set a similarity measure between two data is described by a distance function of all constituent attributes and they are defined to be tolerant when their similarity measure exceeds a similarity threshold value the determination of optimal similarity threshold value is very important for accurate classification so we determine it optimally by using the genetic algorithm ga where the goal of evolution is to balance two requirements such that 1 some tolerant objects are required to be included in the same class as many as possible and 2 some objects in the same class are required to be tolerant as much as possible after finding the optimal similarity threshold value a tolerant set of each object is obtained and the data set is grouped into the lower and upper approximation set depending on the coincidence of their classes we propose a two stage classification method such that all data are classified by using the lower approximation at the first stage and then the nonclassified data at the first stage are classified again by using the rough membership functions obtained from the upper approximation set we apply the proposed classification method to the handwritten numeral character classification problem and compare its classification performance and learning time with those of the feedforward neural network s backpropagation algorithm

evolutionary fuzzy modeling this study is concerned with a general methodology of identification of fuzzy models unlike numeric models fuzzy models operate at a level of information granules fuzzy sets and this aspect brings up an important design requirement of transparency of the model we propose a three phase development framework by distinguishing between structural and parametric optimization processes the underlying topology of the model dwells on fuzzy neural networks architectures governed by fuzzy logic and equipped with parametric flexibility two general optimization mechanisms are explored the structural optimization is realized via genetic programming whereas for the ensuing detailed parametric optimization we proceed with gradient based learning the main advantages of this approach are discussed in detail the study is illustrated with the aid of a numeric example that provides a detailed insight into the performance of the fuzzy models and quantifies crucial design issues

projection and deflation method for partial pole assignment in linear state feedback two projection methods are proposed for partial pole placement in linear control systems these methods are of interest when the system is very large and only a few of its poles must be assigned the first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace the second method can be regarded as a variant of the weilandt deflation technique used in eigenvalue methods

iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values abstract many problems in science require the computation of only one singular vector or more generally a singular subspace of a matrix instead of computing the complete singular value decomposition iterative methods can be used to improve the computational speed in particular when a priori information about the solution is available this paper deals with the computation of a basis of a singular subspace of a matrix associated with its smallest singular values three iterative methods inverse chebyshev and inverse chebyshev iteration are compared by analyzing their convergence properties based on the convergence rate and the operation counts per iteration step it is shown in which problems a particular iterative algorithm is most efficient if the gap between the singular values associated with the desired and undesired singular subspace is large inverse iteration is clearly the best choice if not convergence can be accelerated by applying inverse chebyshev iteration provided sufficiently tight bounds delineating the undesired singular value spectrum are known the smaller the gap the larger the gain in speed unless the matrix is structured or sparse this method is shown to be always more efficient than ordinary chebyshev iteration

an adaptive chebyshev iterative method for nonsymmetric linear systems based on modified moments large sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on chebyshev polynomials for an interval in the complex plane knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends adaptive chebyshev algorithms in which these parameters are determined by using eigenvalue estimates computed by the power method or modifications thereof have been described by manteuffel 1978 this paper presents adaptive chebyshev iterative methods in which eigenvalue estimates are computed from modified moments determined during the iterations the computation of eigenvalue estimates from modified moments requires less computer storage than when eigenvalue estimates are computed by a power method and yields faster convergence for many problems

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

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

developing a local least squares support vector machines based neuro fuzzy model for nonlinear and chaotic time series prediction local modeling approaches owing to their ability to model different operating regimes of nonlinear systems and processes by independent local models seem appealing for modeling identification and prediction applications in this paper we propose a local neuro fuzzy lnf approach based on the least squares support vector machines lssvms the proposed lnf approach employs lssvms which are powerful in modeling and predicting time series as local models and uses hierarchical binary tree hbt learning algorithm for fast and efficient estimation of its parameters the hbt algorithm heuristically partitions the input space into smaller subdomains by axis orthogonal splits in each partitioning the validity functions automatically form a unity partition and therefore normalization side effects e g reactivation are prevented integration of lssvms into the lnf network as local models along with the hbt learning algorithm yield a high performance approach for modeling and prediction of complex nonlinear time series the proposed approach is applied to modeling and predictions of different nonlinear and chaotic real world and hand designed systems and time series analysis of the prediction results and comparisons with recent and old studies demonstrate the promising performance of the proposed lnf approach with the hbt learning algorithm for modeling and prediction of nonlinear and chaotic systems and time series

arnoldi chebyshev approach for convection diffusion computations abstract in a convection diffusion equation if the convection term is very dominant the linear system of equations which result from either finite differencing or finite elementing will not have a strictly diagonally dominant coefficient matrix so if one tries a conventional iteration method jacobi or gauss seidel to solve the linear system of equations the iteration matrix may not satisfy the spectral radius condition for convergence and hence no converging solution may be obtained the problem can be overcome under certain conditions if one uses a two step iteration procedure involving the spectral enveloping ellipse for the iteration matrix in this paper we present such a two step method that combines an arnoldi chebyshev approach for convection diffusion computations

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

numerical solutions for large sparse quadratic eigenvalue problems abstract we study the quadratic eigenvalue problem a b 2c x 0 where a b and c are symmetric real n n matrices and a c are positive definite we propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors the new algorithm includes two parts the first part gives iterative methods which can be used to compute the smallest positive eigenvalue we develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations the second part uses the nonequivalence deflation technique this technique allows us to transform the original problem to a new problem with different a b and c the new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero therefore the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue proceeding in this way we can find out the 3rd 4th smallest positive eigenvalues our algorithm utilizes the symmetry and positivity of the given matrices and avoids computing the undesired complex conjugate eigenvalues under some mild conditions is efficient and reliable the above process can also be used to find a few of the largest negative eigenvalues

logic minimization as an efficient means of fuzzy structure discovery established methods of boolean minimization have previously unseen potential as an efficient and unrestricted means of fuzzy structure discovery becoming particularly useful within a design methodology for the automatic development of fuzzy models traditionally used in digital systems design logic minimization tools allow us to exploit the fundamental links between binary two valued and fuzzy multivalued logic in this paper we show how logic optimization plays an integral role in a two phase fuzzy model design process adaptive logic processing is realized as the discovered boolean structures are augmented with fuzzy granules and then refined by adjusting connections of fuzzy neurons helping to further capture the numeric details of the target systems behavior accurate and highly interpretable fuzzy models are the result of the entire development process

vertex reinforced random walk on ℤ eventually gets stuck on five points vertex reinforced random walk vrrw defined by pemantle in 1988 is a random process that takes values in the vertex set of a graph g which is more likely to visit vertices it has visited before pemantle and volkov considered the case when the underlying graph is the one dimensional integer lattice they proved that the range is almost surely finite and that with positive probability the range contains exactly five points they conjectured that this second event holds with probability 1 the proof of this conjecture is the main purpose of this paper

powerful and flexible fuzzy algorithm for nonlinear dynamic system identification a new powerful and flexible fuzzy algorithm for nonlinear dynamic system identification is presented it is based on the identification of the derivative of the system state instead of the future system state the membership functions of the underlying static fuzzy model are two sided gaussian functions and the learning algorithm is a hybrid nested routine based on least squares quasi newton and simplex optimization methods moreover a simple clustering algorithm based on an additional higher level fuzzy model is proposed the application to the identification of the mackey glass chaotic time series is presented and compared with previous results in terms of maximum error and nondimensional error index finally the application to a test nonlinear dynamic system is presented to show the capabilities of the clustering algorithm the obtained results show that the proposed algorithm can find wide application in practical problems such as in nonlinear electronic circuit design

on convergence of chains with occupational self interactions we consider stochastic chains on abstract measurable spaces whose evolution at any given time depends on the present position and on the occupation measure created by the path up to this instant this generalization of reinforced random walks enables us to impose conditions ensuring l p p 1 or a s convergence of the empirical measures towards some fixed point of a probability valued dynamical system we present two sets of hypotheses based on weak contraction properties leading to two different proofs but in both situations the rates of convergence are optimal in the examined level of generality

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