an iterative method for nonsymmetric systems with multiple right hand sides we propose a method for the solution of linear systems ax b where a is a large possibly sparse nonsymmetric matrix of order n and b is an arbitrary rectangular matrix of order n times s with s of moderate size the method uses a single krylov subspace per step as a generator of approximations a projection process and a richardson acceleration technique it thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right hand sides numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right hand sides

on the global krylov subspace methods for solving general coupled matrix equations in the present paper we propose the global full orthogonalization method gl fom and global generalized minimum residual gl gmres method for solving large and sparse general coupled matrix equations j 1pa i jx jb i j c i i 1 p where a i j r m x m b i j r n x n c i r m x n i j 1 2 p are given matrices and x i r m x n i 1 2 p are the unknown matrices to do so first a new inner product and its corresponding matrix norm are defined then using a linear operator equation and new matrix product we demonstrate how to employ gl fom and gl gmres algorithms for solving general coupled matrix equations finally some numerical experiments are given to illustrate the validity and applicability of the results obtained in this work

minimal volume product near hanner polytopes abstract mahler s conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed dimension it is known that every hanner polytope has the same volume product as the cube or the cross polytope in this paper we prove that every hanner polytope is a strict local minimizer for the volume product in the class of symmetric convex bodies endowed with the banach mazur distance

clustering spatiotemporal data an augmented fuzzy c means in spatiotemporal data commonly encountered in geographical systems biomedical signals and the like each datum is composed of features comprising a spatial component and a temporal part clustering of data of this nature poses challenges especially in terms of a suitable treatment of the spatial and temporal components of the data in this study proceeding with the objective function based clustering such as e g fuzzy c means we revisit and augment the algorithm to make it applicable to spatiotemporal data an augmented distance function is discussed and the resulting clustering algorithm is provided two optimization criteria i e a reconstruction error and a prediction error are introduced and used as a vehicle to optimize the performance of the clustering method experimental results obtained for synthetic and real world data are reported

a remark on the mahler conjecture local minimality of the unit cube we prove that the unit cube b n is a strict local minimizer for the mahler volume product voln k voln k in the class of origin symmetric convex bodies endowed with the banach mazur distance

an efficient ensemble algorithm for numerical approximation of stochastic stokes darcy equations abstract we propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic stokes darcy model with a random hydraulic conductivity tensor the algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately moreover it decouples the stokes darcy system into two smaller sub physics problems which reduces the size of the linear systems and allows parallel computation of the two sub physics problems we prove the ensemble method is long time stable and first order in time convergent under a time step condition and two parameter conditions numerical examples are presented to support the theoretical results and illustrate the application of the algorithm

a modified gradient based algorithm for solving sylvester equations abstract in this paper a modified gradient based algorithm for solving sylvester equations is presented different from the gradient based method introduced by ding and chen 7 and the relaxed gradient based algorithm proposed by niu et al 18 the information generated in the first half iterative step is fully exploited and used to construct the approximate solution theoretical analysis shows that the new method converges under certain assumptions numerical results are given to verify the efficiency of the new method

rapid solution of parameter dependent linear systems for electromagnetic problems in the frequency domain the minimal residual interpolation method reduces the number of iterations in an iterative method for multiple right hand sides it uses computed solutions to initialize an iterative solver with an accurate guess this work demonstrates the efficiency of the method for frequency sweeps and solving scattering problems by plane waves incident from multiple angles a bound on the number of solutions required for plane wave scattering before the remaining solutions obtained by minimal residual interpolation only is given we discuss the performance of the method compared to iterative seed techniques in a numerical example a reduction factor of 60 is obtained on the number of matrix vector multiplications

new convergence results on the global gmres method for diagonalizable matrices in the present paper we give some new convergence results of the global gmres method for multiple linear systems in the case where the coefficient matrix a is diagonalizable we derive new upper bounds for the frobenius norm of the residual we also consider the case of normal matrices and we propose new expressions for the norm of the residual

global simpler gmres for nonsymmetric systems with multiple right hand sides a new version of the simpler gmres for solving nonsymmetric linear system of equations with multiple right hand sides is presented the main advantage over the simpler block gmres is that the new method requires less storage and computational work per cycle preconditioning techniques can also be used to further accelerate the convergence of the new method finally numerical examples are given to illustrate the effectiveness of the proposed method and its some preconditioned versions

a robust preconditioner with low memory requirements for large sparse least squares problems this paper describes a technique for constructing robust preconditioners for the cgls method applied to the solution of large and sparse least squares problems the algorithm computes an incomplete ldlt factorization of the normal equations matrix without the need to form the normal matrix itself the preconditioner is reliable pivot breakdowns cannot occur and has low intermediate storage requirements numerical experiments illustrating the performance of the preconditioner are presented a comparison with incomplete qr preconditioners is also included

flexible variants of block restarted gmres methods with application to geophysics in a wide number of applications in computational science and engineering the solution of large linear systems of equations with several right hand sides given at once is required direct methods based on gaussian elimination are known to be especially appealing in that setting nevertheless when the dimension of the problem is very large preconditioned block krylov space solvers are often considered as the method of choice the purpose of this paper is thus to present iterative methods based on block restarted gmres that allow variable preconditioning for the solution of linear systems with multiple right hand sides the use of flexible methods is especially of interest when approximate possibly iterative solvers are considered in the preconditioning phase first we introduce a new variant of block flexible restarted gmres that includes a strategy for detecting when a linear combination of the systems has approximately converged this explicit block size reduction is often called deflation we analyze the main properties of this flexible method based on deflation and notably prove that the frobenius norm of the block residual is always nonincreasing we also present a flexible variant based on both deflation and truncation to especially be used in case of limited memory finally we illustrate the numerical behavior of these flexible block methods for large industrial simulations arising in geophysics where indefinite linear systems of size up to 1 billion unknowns with multiple right hand sides have been successfully solved in a parallel distributed memory environment

a quasi minimal residual variant of idrstab using the residual smoothing technique abstract quasi minimal residual qmr type methods have been developed for improving the oscillations in the residual norms of the bi conjugate gradient bi cg method and the hybrid bi cg methods the idrstab method which combines the induced dimension reduction idr s method with higher order stabilizing polynomials is often more effective than idr s and the hybrid bi cg methods but the residual norms oscillate in this paper we design a qmr variant of idrstab by using a residual smoothing technique in order to obtain smoother convergence behavior numerical experiments show the effectiveness of our qmr variant of idrstab

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

a hybrid block gmres method for nonsymmetric systems with multiple right hand sides the block gmres bgmres method is a natural generalization of the gmres algorithm for solving large non symmetric systems with multiple right hand sides unfortunately its cost increases significantly per iteration frequently rendering the method impractical in this paper we propose a hybrid block gmres method which offers significant performance improvements over bgmres this method uses the matrix polynomial obtained in the course of a bgmres step and combines the advantages of the block approach with those of successful hybrid methods we discuss the properties and several implementation variants of the method and report results from numerical experiments we also describe how to use these techniques in order to solve multiply shifted systems with multiple right hand sides

a global harmonic arnoldi method for large non hermitian eigenproblems with an application to multiple eigenvalue problems the global arnoldi method can be used to compute exterior eigenpairs of a large non hermitian matrix a but it does not work well for interior eigenvalue problems based on the global arnoldi process that generates an f orthonormal basis of a matrix krylov subspace we propose a global harmonic arnoldi method for computing certain harmonic f ritz pairs that are used to approximate some interior eigenpairs we propose computing the f rayleigh quotients of the large non hermitian matrix with respect to harmonic f ritz vectors and taking them as new approximate eigenvalues they are better and more reliable than the harmonic f ritz values the global harmonic arnoldi method inherits convergence properties of the harmonic arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix some properties of the harmonic f ritz vectors are presented as an application assuming that a is diagonalizable we show that the global harmonic arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice to be practical we develop an implicitly restarted global harmonic arnoldi algorithm with certain harmonic f shifts suggested in particular this algorithm can be adaptively used to solve multiple eigenvalue problems numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill conditioned multiple eigenproblems

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

a parametric model for fusing heterogeneous fuzzy data presented is a model that integrates three data types numbers intervals and linguistic assessments data of these three types come from a variety of sensors one objective of sensor fusion models is to provide a common framework for data integration processing and interpretation that is what our model does we use a small set of artificial data to illustrate how problems as diverse as feature analysis clustering cluster validity and prototype classifier design can all be formulated and attacked with standard methods once the data are converted to the generalized coordinates of our model the effects of reparameterization on computational outputs are discussed numerical examples illustrate that the proposed model affords a natural way to approach problems which involve mixed data types

classification of type 2 fuzzy sets represented as sequences of vertical slices granulation of information by using type 2 fuzzy sets is receiving more attention nowadays this is due to the superior capability of type 2 fuzzy sets in handling the data uncertainty from a theoretical perspective a set containing type 2 fuzzy sets has no trivial geometric structure therefore a proper metric cannot be easily defined as a consequence common pattern recognition systems which in one way or another rely on some geo metric structure of the input space are not easily applicable to a space of type 2 fuzzy sets in this paper we study the problem of designing a classifier in the input space of type 2 fuzzy sets type 2 fuzzy sets are hence interpreted as granular patterns forming a given input dataset by decomposing a type 2 fuzzy set into a sequence of simpler lower type fuzzy sets we explore the possibility of defining and building dissimilarity and kernel based classification systems on input spaces of type 2 fuzzy sets such an interpretation provided in terms of sequences allows us to conceive an effective sequence matching strategy which can be suitably embedded into well established pattern recognition systems we support the methodological developments by performing experiments on synthetically generated classification problems for datasets composed of type 2 fuzzy sets with adjustable and controlled level of difficulty results are promising and suggest to further investigate on the possibility of interpreting type 2 fuzzy sets as input patterns of a given data driven inference system

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

shadow systems of asymmetric lp zonotopes abstract shadow systems are used to establish new asymmetric l p volume product and asymmetric l p volume ratio inequalities along with their equality conditions these inequalities have reisner s volume product inequality for l 1 zonotopes as a special case moreover uniqueness of the extremals in the symmetric setting is obtained

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

implicitly restarted global fom and gmres for nonsymmetric matrix equations and sylvester equations the global gmres and global fom algorithm are recently proposed by jbilou et al for solving the linear equations with multiple right hand sides k jbilou a messaoudi h sados global fom and gmres algorithms for matrix equations appl numer math 31 1999 49 63 like gmres for the linear equations they generally uses restarting which slows the convergence however some information can be retained at the time of the restart and used in the next cycle we present algorithms that use implicit restarting in order to retain this information as morgan have proposed recently r b morgan implicitly restarted gmres and arnoldi methods for nonsymmetric systems of equations siam j matrix anal appl 21 2000 1112 1135 at the same time we prove that global gmres and global fom methods for matrix equations are equivalent with the corresponding methods for linear equations and propose implicitly restarted global fom and gmres numerical examples show that our methods are efficient

taiex forecasting based on fuzzy time series and fuzzy variation groups in this paper we present a new method to forecast the daily taiwan stock exchange capitalization weighted stock index taiex based on fuzzy time series and fuzzy variation groups where the main input factor is the previous day s taiex and the secondary factor is either the dow jones the nasdaq the m 1b or their combination first the proposed method fuzzifies the historical training data of the taiex into fuzzy sets to form fuzzy logical relationships second it groups the fuzzy logical relationships into fuzzy logical relationship groups flrgs based on the fuzzy variations of the secondary factor third it evaluates the leverage of the fuzzy variations between the main factor and the secondary factor to construct fuzzy variation groups fourth it gets the statistics of the fuzzy variations appearing in each fuzzy variation group fifth it calculates the weights of the statistics of the fuzzy variations appearing in each fuzzy variation group respectively finally based on the weights of the statistics of the fuzzy variations appearing in the fuzzy variation groups and the flrgs it performs the forecasting of the daily taiex because the proposed method uses both fuzzy variation groups and flrgs to analyze in detail the historical training data it gets higher forecasting accuracy rates to forecast the taiex than the existing methods

a mutually recurrent interval type 2 neural fuzzy system mrit2nfs with self evolving structure and parameters in this paper a mutually recurrent interval type 2 neural fuzzy system mrit2nfs is proposed for the identification of nonlinear and time varying systems the mrit2nfs uses type 2 fuzzy sets in order to enhance noise tolerance of the system in the mrit2nfs the antecedent part of each recurrent fuzzy rule is defined using interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang type with interval weights the antecedent part of mrit2nfs forms a local internal feedback and interaction loop by feeding the rule firing strength of each rule to others including itself the consequent is a linear combination of exogenous input variables the learning of mrit2nfs starts with an empty rule base and all rules are learned online via structure and parameter learning the structure learning of mrit2nfs uses online type 2 fuzzy clustering for parameter learning the consequent part parameters are tuned by rule ordered kalman filter algorithm to reinforce parameter learning ability the type 2 fuzzy sets in the antecedent and weights representing the mutual feedback are learned by the gradient descent algorithm after the training a weight elimination scheme eliminates feedback connections that do not have much effect on the network behavior this method can efficiently remove redundant recurrence and interaction weights finally the mrit2nfs is used for system identification under both noise free and noisy environments for this we consider both time series prediction and nonlinear plant modeling compared with type 1 recurrent fuzzy neural networks simulation results show that our approach produces smaller root mean squared errors using the same number of iterations

clustering granular data and their characterization with information granules of higher type the study is devoted to the clustering of granular data and an evaluation of the results of such clustering a comprehensive and systematic approach is developed which is composed of three fundamental phases 1 representation of granular data 2 clustering carried out in the representation space of information granules and 3 evaluation of quality of clusters following the reconstruction criterion the reconstruction criterion formed originally for numeric data and leading to an idea of granular prototypes is revisited we show here an emergence of granular information of higher type which are used to implement granular interval prototypes we discuss a way of forming granular data in the context of representation of time series and present clustering of granular time series

the block cmrh method for solving nonsymmetric linear systems with multiple right hand sides abstract cmrh method changing minimal residual with hessenberg process is an iterative method for solving nonsymmetric linear systems this method is similar to qmr method but based on the hessenberg process instead of the lanczos process on dense matrices the cmrh method is less expensive and requires less storage than other krylov methods this paper presents a block version of the cmrh algorithm for solving linear systems with multiple right hand sides the new algorithm is based on the block hessenberg process and the iterates are characterized by a block version of the quasi minimization property we analyze its main properties and show that under the condition of full rank of block residual the block cmrh method cannot break down finally some numerical examples are presented to show the efficiency of the new method in comparison with the traditional cmrh method and a comparison with the block gmres method is also provided

fourier analysis of frequency filtering decomposition preconditioners in this paper frequency filtering decomposition ffd preconditioner is analyzed by the approach of fourier analysis the condition number estimation of a preconditioned 2 d model problem is presented analysis reveals that condition number of the preconditioned matrix grows like o h 1 with h be the mesh size by using the framework of ffd a stabilized frequency filtering decomposition sffd method is proposed and analyzed by fourier method results show that sffd preconditioner is superior to ffd preconditioner in the sense that k m s f f d 1a k m f f d 1a numerical tests are performed to illustrate the theoretical results and the superiority of sffd preconditioner

isotropic constants and mahler volumes abstract this paper contains a number of results related to volumes of projective perturbations of convex bodies and the laplace transform on convex cones first it is shown that a sharp version of bourgain s slicing conjecture implies the mahler conjecture for convex bodies that are not necessarily centrally symmetric second we find that by slightly translating the polar of a centered convex body we may obtain another body with a bounded isotropic constant third we provide a counter example to a conjecture by kuperberg on the distribution of volume in a body and in its polar

piecewise two dimensional normal cloud representation for time series data mining many high level dimensionality reduction approaches for mining time series have been proposed e g sax pwca and feature based due to the rapid performance degradation of time series data mining in much lower dimensionality and the continuously increasing amount of time series data with uncertainty there remains a burning need to develop new time series representations that can retain good performance in much lower reduced space and address uncertainty efficiently in this work we propose a novel time series representation namely two dimensional normal cloud representation 2d ncr based on cloud model theory the representation achieves dimensionality reduction by transforming the raw time series into a sequence of two dimensional normal cloud models moreover a new similarity measure between the transformed time series is presented the proposed method can reflect the characteristic data distribution of the time series and capture the variation with time we validate the performance of our representation on the various data mining tasks of classification clustering and query by content the experimental results demonstrate that 2d ncr is an effective and competitive representation for time series data mining

a block gcrot m k method for linear systems with multiple right hand sides this study is mainly focused on iterative solution to multiple linear systems with several right hand sides for solving such systems efficiently we explore a new block gcrot m k bgcrot m k method which is derived by extending gcrot m k method jason e hicken david w zingg a simplified and flexible variant of gcrot for solving nonsymmetric linear systems siam j sci comput 32 2010 1672 1694 we analyze its main properties it is shown that under the condition of full rank of block residual the frobenius norm of the block residual generated by the proposed method is always nonincreasing moreover we also present its block flexible version bfgcrot m k finally numerical examples demonstrate that the bgcrot m k method and its flexible variant can achieve a smoothed residual and can be more competitive than some other block solvers

block minimum perturbation algorithm based on block arnoldi process for nonsymmetric linear systems with multiple right hand sides abstract in many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right hand sides customarily one uses the residual error as a stopping condition but small residue does not imply accurate approximate solutions therefore minimizing quantities other than the residual norm may be more adequate in this context based on the above consideration a block minimum perturbation krylov subspace method bminpet for the solution xm satisfying a a x m b b at each step of nonsymmetric linear systems a x b is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm a b f of the matrix a b and a stopping criterion is given for consistent system with multiple right hand sides to get a more robust way to stop the iterations this process is a generalization of minimum perturbation algorithm minpert by kasenally and simoncini 1997 analysis of a minimum perturbation algorithm for nonsymmetric linear systems siam j numer anal 34 48 66 to multiple right hand sides for the benefit of the amount of calculation we give lower and upper bounds about min a b f that are easier to compute as a by product we come up with a conception of function f g which is a generalization of function that is a symmetric gauge sg function the algorithm is used to analyze the performance of bgmres method by evaluating the proximity of their solutions to block joint backward perturbation optimality the relationships between bminpet method and relative methods are investigated numerical examples show us that bminpert shows its superiority compared with bfgmres s m pf gsgmres bl bicg rq bgmres and barnoldi in solving the large sparse ill conditioned problems

overlapped multicolor milu preconditioning milu preconditioned iterative methods are useful for solving large sparse linear systems which arise from the finite difference approximations of three dimensional second order elliptic partial differential equations in these schemes the milu preconditioning which accelerates the convergence is the most difficult part to vectorize on vector supercomputers currently the reordering techniques like the multicolor ordering strategy are commonly used to obtain sufficiently long vector lengths however these reordering techniques deteriorate the convergence compared with the standard milu preconditioner and gustafsson s acceleration does not work well for them in this paper a new preconditioning is proposed and its advantages compared with the multicolor and hyperplane orderings are shown through experiments on nec vector supercomputer sx 3

zonoids with minimal volume product of course only the only if part needs a proof we give here some preliminaries and notations first we make the remark that our approach is based on the analysis of random polytopes made in 7 and we make extensive reference to 7 and 6 in order to avoid repeating calculations that are done there we use standard geometric terminology as may be found e g in 3 or 4 the dimension n will be fixed throughout the paper b is the euclidean unit ball in s the unit sphere of with the relative topology co is the rotation invariant measure on s normalized so that

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

global scd algorithm for real positive definite linear systems with multiple right hand sides abstract in the present paper we present the left conjugate direction lcd method for linear systems with multiple right hand sides we define the method by using left and right conjugate direction matrices and petrov galerkin condition the method has no breakdown for real positive definite systems the method reduces to the usual cg type method when a is symmetric positive definite we also show how to apply these methods for solving the lyapunov matrix equation finally a numerical experiment is given to illustrate the result

a new implementation of the lanczos method in linear problems the lanczos algorithm has proved to be a powerful solution method not only for finding the eigenvalues but for solving linear systems of equations in this work a new implementation of the algorithm is presented for solving linear systems of equations with a sequence of right hand sides the versions of the method proposed in the past treat the right hand side vectors successively by keeping the tridiagonal matrix and the orthonormal basis in fast or secondary storage the new technique handles all approximations to the solution vectors simultaneously without the necessity for keeping the tridiagonal matrix or the orthonormal basis in fast or secondary storage thus when the first solution vector has converged to a required accuracy good approximations to the remaining solution vectors have simultaneously been obtained it then takes fewer iterations to reach the final accuracy by working separately on each of the remaining vectors

incomplete hyperbolic gram schmidt based preconditioners for the solution of large indefinite least squares problems abstract we propose to precondition the cgils method based on the incomplete hyperbolic gram schmidt methods for the solution of the large and sparse indefinite least squares ils problem which requires minimization of an indefinite quadratic form numerical experiments show that the incomplete hyperbolic classical modified gram schmidt ihcgs ihmgs preconditioners can generally greatly reduce the number of iterations and ihmgs has a better quality for ill conditioned problems but may be expensive in computation cost

the volume product of convex bodies with many hyperplane symmetries mahler s conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume we confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense their hyperplanes of symmetries have a one point intersection moreover we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflection groups namely direct products of the isometry groups of regular polytopes

hierarchical clustering problems and analysis of fuzzy proximity relation on granular space in this paper based on granular space some hierarchical clustering problems and analysis for fuzzy proximity relation are developed by using rigorous mathematical descriptions and four results are obtained first the granular representation of a fuzzy proximity relation is studied an algorithm is obtained to compute the granular space or hierarchical clustering structure and the min transitive closure derived from the same fuzzy proximity relation on a finite universe and the consistent clustering properties are discussed second a consistency index that is based on granular space is developed and a global optimizing mathematical model is established to obtain the optimal clustering of a fuzzy proximity relation furthermore for given two fuzzy proximity relations on different granulations of the same universe a collaborative clustering method is studied by the intersection operation and this method can provide a better hierarchical clustering structure finally the theory for hierarchical clustering analysis of fuzzy proximity relations on granular space is established for given two fuzzy proximity relations on the same universe two sufficient conditions theorems 6 1 and 6 2 of isomorphism are given and the necessary condition theorem 6 3 and the sufficient condition theorem 6 4 for e similarity are obtained we also illustrate the effectiveness of these theories and methods with some practical examples these results may help form a comprehensive theory and methodology for related potential application and will help us to obtain a deeper understanding of the essence of hierarchical clustering procedures

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

modified gath geva clustering for fuzzy segmentation of multivariate time series partitioning a time series into internally homogeneous segments is an important data mining problem the changes of the variables of a multivariate time series are usually vague and do not focus on any particular time point therefore it is not practical to define crisp bounds of the segments although fuzzy clustering algorithms are widely used to group overlapping and vague objects they cannot be directly applied to time series segmentation because the clusters need to be contiguous in time this paper proposes a clustering algorithm for the simultaneous identification of local probabilistic principal component analysis ppca models used to measure the homogeneity of the segments and fuzzy sets used to represent the segments in time the algorithm favors contiguous clusters in time and is able to detect changes in the hidden structure of multivariate time series a fuzzy decision making algorithm based on a compatibility criteria of the clusters has been worked out to determine the required number of segments while the required number of principal components are determined by the screeplots of the eigenvalues of the fuzzy covariance matrices the application example shows that this new technique is a useful tool for the analysis of historical process data

global lsmr gl lsmr method for solving general linear systems with several right hand sides abstract the global solvers are an attractive class of iterative solvers for solving linear systems with multiple right hand sides in this paper first a new global method for solving general linear systems with several right hand sides is presented this method is the global version of the lsmr algorithm presented by fong and saunders 2011 then some theoretical properties of the new method are discussed finally numerical experiments from real applications are used to confirm the effectiveness of the proposed method

volume inequalities for orlicz zonotopes abstract in this paper we define the orlicz zonotopes z using the notion of shadow systems we give a sharp upper estimate for the volume of z and a sharp lower estimate for the volume of the polar of z in terms of the volume of z 1

on improving linear solver performance a block variant of gmres the increasing gap between processor performance and memory access time warrants the re examination of data movement in iterative linear solver algorithms for this reason we explore and establish the feasibility of modifying a standard iterative linear solver algorithm in a manner that reduces the movement of data through memory in particular we present an alternative to the restarted gmres algorithm for solving a single right hand side linear system ax b based on solving the block linear system ax b algorithm performance i e time to solution is improved by using the matrix a in operations on groups of vectors experimental results demonstrate the importance of implementation choices on data movement as well as the effectiveness of the new method on a variety of problems from different application areas

symmetric mahler s conjecture for the volume product in the 3 dimensional case we prove mahler s conjecture concerning the volume product of centrally symmetric convex bodies in rn in the case where n 3 more precisely we show that for every 3 dimensional centrally symmetric convex body k r3 the volume product k k is greater than or equal to 32 3 with equality if and only if k or k is a parallelepiped

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

a new family of global methods for linear systems with multiple right hand sides the global bi conjugate gradient gl bcg method is an attractive matrix krylov subspace method for solving nonsymmetric linear systems with multiple right hand sides but it often show irregular convergence behavior in many applications in this paper we present a new family of global a biorthogonal methods by using short two term recurrences and formal orthogonal polynomials which contain the global bi conjugate residual gl bcr algorithm and its improved version finally numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals

two nonparametric models for fusing heterogeneous fuzzy data two models are discussed that integrate heterogeneous fuzzy data of three types real numbers real intervals and real fuzzy sets the architecture comprises three modules 1 an encoder that converts the mixed data into a uniform internal representation 2 a numerical processing core that uses the internal representation to solve a specified task and 3 a decoder that transforms the internal representation back to an interpretable output format the core used in this study is fuzzy clustering but there are many other operations that are facilitated by the models two schemes for encoding the data and decoding it after clustering are presented one method uses possibility and necessity measures for encoding and several variants of a center of gravity defuzzification method for decoding the second approach uses piecewise linear splines to encode the data and decode the clustering results both procedures are illustrated using two small sets of heterogeneous fuzzy data

sharp error bounds of some krylov subspace methods for non hermitian linear systems1project 19601036 supported by the national natural science foundation of china 1 the convergence of the krylov subspace methods e g full orthogonal method fom and generalized minimal residual method gmres etc for solving large non hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective in particular when its spectrum lies in the open right left half plane or is on the real axis related theoretical error bounds are established which reveal some intrinsic relationships between the convergence properties and the eigen characteristics of the coefficient matrix these results not only generalize all the known ones for the diagonalizable matrices in the literature but also sharp the corresponding estimates in jia acta mathematica sinica new series 14 1998 507 518

some functional inverse santalo inequalities abstract let f r n r be a log concave function and for z r n define f z y inf x r n e x z y z f x for every y r n we discuss the problem of finding a sharp lower bound to the product p f inf z r n r n f x d x r n f z y d y we prove that if n 1 then p f e and characterize the case of equality the same method allows to give a new simple proof of the fact that if f is sign invariant then for all n p f 4 n these inequalities are functional versions with exact lower bounds of the so called inverse santalo inequality for convex bodies that we state and discuss as conjectures

comparing fuzzy probabilistic and possibilistic partitions using the earth mover s distance a number of noteworthy techniques have been put forth recently in different research fields for comparing clusterings herein we introduce a new method for comparing soft fuzzy probabilistic and possibilistic partitions based on the earth mover s distance emd and the ordered weighted average owa the proposed method is a metric depending on the ground distance for all but possibilistic partitions it is extremely flexible due to its emd formulation owa aggregation and abstract concept of ground distance in theory our method is agnostic to the type uncertainty of soft partition clustering algorithm and distance measure used in the clustering algorithm s and it is applicable to the clustering of both object and relational data validation is performed theoretically experimentally as well as in terms of computational complexity emphasis is placed on the set of possibilistic partitions specifically noise and coincident clusters which are important cases that have received little to no attention to date in the comparing clustering literature improvements are reported in terms of metric properties and computational complexity over existing extended concordance discordance e g soft rand and jaccard approaches and improved design and robustness in comparison with existing transportation problem based approaches

the lanczos arnoldi algorithm and controllability we present methods for computing the controllable space of a linear dynamic system the methods are based on orthogonalizing krylov sequences and can use much less storage than other methods with little or no loss of numerical stability

projection methods for large lyapunov matrix equations abstract in the present paper we propose krylov subspace methods for solving large lyapunov matrix equations of the form ax xat bbt 0 where a and b are real n n and n s matrices respectively with s n such problems appear in many areas of control theory such as the computation of hankel singular values model reduction algorithms and others the proposed methods are based on the arnoldi process we show how to extract low rank approximate solutions to lyapunov matrix equations and give some theoretical results finally some numerical tests will be reported to illustrate the effectiveness of the proposed method

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

minimal residual methods augmented with eigenvectors for solving sylvester equations and generalized sylvester equations in this paper we propose minimal residual methods augmented with eigenvectors for solving large sylvester matrix equations ax xb c and generalized sylvester matrix equations axb x c the subspace from which the approximate solution is extracted is the kronecker product subspace of two augmented krylov subspaces numerical experiments report the effectiveness of these methods

on short recurrence krylov type methods for linear systems with many right hand sides block and global krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and several right hand sides these methods are advantageous since they allow to cast the major part of the arithmetic in terms of matrix block vector products and since in the block case they take their iterates from a potentially richer subspace in this paper we consider the most established krylov subspace methods which rely on short recurrences i e bicg qmr and bicgstab we propose modifications of their block variants which increase numerical stability thus at least partly curing a problem previously observed by several authors moreover we develop modifications of the global variants which almost halve the number of matrix vector multiplications we present a discussion as well as numerical evidence which both indicate that the additional work present in the block methods can be substantial and that the new economic versions of the global bicg and qmr method can be considered as good alternatives to the bicgstab variants

a flexible and adaptive simpler block gmres with deflated restarting for linear systems with multiple right hand sides block gmres is one of the most popular algorithms for solving large non hermitian linear systems with multiple right hand sides the simpler block gmres algorithm is a variation of block gmres which avoids the factorization of a block upper hessenberg matrix and it is much simpler to program and requires less work than block gmres however theoretical results given in this paper indicate that it is less stable due to the ill conditioning of the basis used in order to overcome this difficulty we propose an adaptive simpler block gmres algorithm for large linear systems with multiple right hand sides theoretical analysis is made to show the reason why the new algorithm is superior to its original counterpart moreover we consider how to restart and precondition the adaptive simpler block gmres algorithm and propose a flexible and adaptive simpler block gmres algorithm with deflated restarting numerical experiments show the numerical behavior of our new algorithm

implementation of an adaptive algorithm for richardson s method abstract we discuss the implementation of an adaptive algorithm proposed by one of us the algorithm is a hybrid of the gmres method and richardson s method richardson s method rm depends on a set of parameters that are computed by minimizing the l 2 norm of a polynomial over the convex hull of eigenvalues execution of gmres yields not only an approximate solution but also the approximate convex hull rm is used to avoid storing and working with a large number of vectors as gmres often requires this method is also advantageous for the solution of large problems we consider several test problems and compare our algorithm primarily with the conjugate gradient squared algorithm but also with gmres and to cg applied to the normal equations for many test problems our algorithm takes roughly 50 percent more work than the conjugate gradient squared algorithm although if the matrix is either preconditioned or indefinite our algorithm is more efficient however our algorithm currently imposes an undesirable burden on the user who is invited to consider a variety of numerical parameters to manipulate such as the number of steps of rm in order to enhance performance the values we suggest are only empirical

learning based fast iterative convergence of 3 d mom via eigen agmres method three dimensional 3 d full wave electromagnetic simulation using method of moments mom under the framework of fast solver algorithms like fast multipole method fmm is often bottlenecked by the speed of convergence of the krylov subspace based iterative process this is primarily because the electric field integral equation efie matrix even with cutting edge preconditioning techniques often exhibits bad spectral properties arising from frequency or geometry based ill conditioning which render iterative solvers slow to converge or stagnate occasionally in this communication a novel technique to expedite the convergence of mom matrix solution at a specific frequency is proposed by extracting and applying eigen vectors from a previously solved neighboring frequency in an augmented generalized minimum residual agmres iterative framework this technique can be applied in unison with any preconditioner numerical results demonstrate up to 40 speed up in convergence using the proposed eigen agmres method

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

convergence properties of block gmres and matrix polynomials abstract this paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right hand sides a convergence theory is developed based on a representation of the method using matrix valued polynomials relations between the roots of the residual polynomial for block gmres and the matrix e pseudospectrum are derived and illustrated with numerical experiments the role of invariant subspaces in the effectiveness of block methods is also discussed

generalized global conjugate gradient squared algorithm in this paper a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right hand sides is presented the method can be derived by using products of two nearby global bicg polynomials and formal orthogonal polynomials of which global cgs and global bicgstab are just particular cases we also show to apply the method for solving the sylvester matrix equation finally numerical examples are given to illustrate the effectiveness of the proposed method

on fuzzy cluster validity indices cluster analysis aims at identifying groups of similar objects and helps to discover distribution of patterns and interesting correlations in large data sets especially fuzzy clustering has been widely studied and applied in a variety of key areas and fuzzy cluster validation plays a very important role in fuzzy clustering this paper introduces the fundamental concepts of cluster validity and presents a review of fuzzy cluster validity indices available in the literature we conducted extensive comparisons of the mentioned indices in conjunction with the fuzzy c means clustering algorithm on a number of widely used data sets and make a simple analysis of the experimental results

numerical study on incomplete orthogonal factorization preconditioners we design analyse and test a class of incomplete orthogonal factorization preconditioners constructed from givens rotations incorporating some dropping strategies and updating tricks for the solution of large sparse systems of linear equations comprehensive accounts about how the preconditioners are coded what storage is required and how the computation is executed for a given accuracy are presented a number of numerical experiments show that these preconditioners are competitive with standard incomplete triangular factorization preconditioners when they are applied to accelerate krylov subspace iteration methods such as gmres and bicgstab

cimgs an incomplete orthogonal factorization preconditioner a new preconditioner for symmetric positive definite systems is proposed analyzed and tested the preconditioner compressed incomplete modified gram schmidt cimgs is based on an incomplete orthogonal factorization cimgs is robust both theoretically and empirically existing in exact arithmetic for any full rank matrix numerically it is more robust than an incomplete cholesky factorization preconditioner ic and a complete cholesky factorization of the normal equations theoretical results show that the cimgs factorization has better backward error properties than complete cholesky factorization for symmetric positive definite m matrices cimgs induces a regular splitting and better estimates the complete cholesky factor as the set of dropped positions gets smaller cimgs lies between complete cholesky factorization and incomplete cholesky factorization in its approximation properties these theoretical properties usually hold numerically even when the matrix is not an m matrix when the drop set satisfies a mild and easily verified or enforced property the upper triangular factor cimgs generates is the same as that generated by incomplete cholesky factorization this allows the existence of the ic factorization to be guaranteed based solely on the target sparsity pattern

on the block gmres method with deflated restarting abstract in this paper we consider the block gmres method with deflated restarting for solving nonsymmetric linear systems with multiple right hand sides we modify slightly the restarted block gmres method with deflation of eigenvalues proposed by morgan to obtain a new one it is shown that the modified method is mathematically equivalent to the morgan s original method however the error analysis shows the modified version minimizes the numerical errors during a restart and therefore is better suited if the linear systems have to be solved with high precision numerical experiments report the effectiveness of the modified method