asymptotic properties of solutions of difference equations with several delays and volterra summation equations we study a scalar linear difference equation with several delays by transforming it to a system of volterra equations without delays the results obtained for this system are then used to establish oscillation criteria and asymptotic properties of solutions of the considered equation

on regular splittings of an m matrix abstract let a m n r n be a splitting we call it a regular splitting if m 1 0 and n 0 in this paper we answer questions 5 1 and 5 4 and we generalize theorem 4 5 of schneider linear algebra appl 58 407 429 1984

block gauss elimination followed by a classical iterative method for the solution of linear systems in the last two decades many papers have appeared in which the application of an iterative method for the solution of a linear system is preceded by a step of the gauss elimination process in the hope that this will increase the rates of convergence of the iterative method this combination of methods has been proven successful especially when the matrix a of the system is an m matrix the purpose of this paper is to extend the idea of one to more gauss elimination steps consider other classes of matrices a e g p cyclic consistently ordered and generalize and improve the asymptotic convergence rates of some of the methods known so far

improving jacobi and gauss seidel iterations abstract when convergent jacobi or gauss seidel iterations can be applied to solve systems of linear equations a natural question is how convergence rates are affected if the original system is modified by performing some gaussian elimination we prove that if the initial iteration matrix is nonnegative then such elimination improves convergence our results extend those contained in 4

improving jacobi methods for nonnegative h matrices linear systems abstract in this paper we consider a preconditioned iterative method for solving linear system ax b which is a generalization of method proposed by zhang et al y zhang t z huang x p liu modified iterative methods for nonnegative matrices and m matrices linear systems comput math appl 50 2005 1587 1602 and prove its convergence for the case when a is a nonnegative h matrix numerical examples are also given which show the effectiveness of our method

letter to the editor comment on a comparison theorem of the sor iterative method in this work we point out that there are incorrect assertions in the article by li ying sun li ying sun a comparison theorem for the sor iterative method j comput appl math 181 2005 336 341 moreover we answer an open problem

more on modifications and improvements of classical iterative schemes for m matrices in the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an m matrix in order to improve on the convergence rates of the classical iterative schemes jacobi gauss seidel etc the present work is a contribution towards the generalization of the most common preconditioners used so far

lp boundedness properties for volterra difference equations this work deals with l p boundedness of solutions of a class of volterra functional difference equation the effect of different kinds of the nonhomogeneous term lipschitz local lipschitz completely continuous are taken into consideration

incomplete blockwise factorizations of block h matrices the use of incomplete blockwise factorizations as preconditioners in conjugate gradient like methods has become more and more popular in recent years most of the theory concerning existence and applicability of these factorizations has been limited to m matrices so far here we introduce a more general definition of block h matrices robert 8 and we extend the theory to this class of matrices

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

existence and estimate of solutions of some nonlinear volterra difference equations in hilbert spaces abstract in this paper we consider the volterra difference equation x n f n k 0 n a n k g k x k in the context of hilbert spaces and give sufficient conditions so that the solutions exist and show a bounded behavior

erratum to a note on the preconditioned gauss seidel method for m matrices j comput appl math 219 1 2008 59 71 in this note some errors in a recent article by niki et al h niki t kohno m morimoto the preconditioned gauss seidel method faster than the sor method j comput appl math 219 2008 59 71 are pointed out and a new proof for the corresponding result is presented

the convergence of the modified gauss seidel methods for consistent linear systems in this paper we present a convergence analysis for the modified gauss seidel methods given in gunawardena et al linear algebra appl 154 156 1991 125 and kohno et al linear algebra appl 267 1997 113 for consistent linear systems we prove that the modified gauss seidel method converges for some values of the parameters in the preconditioned matrix

comparison results on preconditioned sor type iterative method for z matrices linear systems in this paper we present some comparison theorems on preconditioned iterative method for solving z matrices linear systems comparison results show that the rate of convergence of the gauss seidel type method is faster than the rate of convergence of the sor type iterative method

scheduled relaxation jacobi method elliptic partial differential equations epdes appear in a wide variety of areas of mathematics physics and engineering typically epdes must be solved numerically which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution the scheduled relaxation jacobi srj is a promising class of methods atypical for combining simplicity and efficiency that has been recently introduced for solving linear poisson like epdes the srj methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances the efficiency in the reduction of the residual increases with the number of levels employed in the algorithm applying the original methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal srj schemes as the mixed non linear algebraic differential system of equations from which they result becomes notably stiff here we present a new methodology for obtaining the parameters of srj schemes that overcomes the limitations of the original algorithm and provide parameters for srj schemes with up to 15 levels and resolutions of up to 215 points per dimension allowing for acceleration factors larger than several hundreds with respect to the jacobi method for typical resolutions and in some high resolution cases close to 1000 most of the success in finding srj optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations furthermore we extend the original algorithm to apply it to certain systems of non linear epdes we compute new optimal parameters of the scheduled relaxation jacobi method the new parameters are calculated for srj schemes with p 6 to p 15 levels we reduce the stiffness in the computation of optimal srj parameters analytically we provide a grid of optimal parameters for different p and numerical resolutions we benchmark srj methods against other algorithms to solve linear systems

letter to the editor a simple estimation for the spectral radius of block h matrices simple bounds for the spectral radius of an h matrix and a block h matrix are presented respectively i e a 2maxi aii for an h matrix a aij and a maxi aii aii 1 for a block h matrix a aij where a denote the spectral radius

on modified jacobi linear operators abstract by means of successive partial substitutions new fixed point linear equations can be obtained from old ones the jacobi method applied to a system in the sequence thus obtained constitutes a partial gauss seidel method applied to the original one and we analyze the behavior of the sequence of spectral radii of the successive iteration matrices the modified jacobi operators we do this under the assumption that the starting operator is nonnegative with respect to a proper cone and has spectral radius less or greater than 1 our main result is that if the jacobi operator obtained after k substitutions is irreducible then the following one either is the same or has strictly smaller or greater spectral radius this result implies that the whole sequence of spectral radii is monotone

linearized stability analysis of discrete volterra equations abstract various different types of stability are defined in a unified framework for discrete volterra equations of the type x n f n n j 0 k n j x n n 0 under appropriate assumptions stability results are obtainable from those valid in the linear case k n j x n b n j x j and a linearized stability theory is studied here by using the fundamental and resolvent matrices several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of banach space b the elements of which are sequences of vectors x n f n e d b n j e d e d n j 0 etc we show that the theory including a number of new results as well as results already known can be presented in a systematic framework in which results parallel corresponding results for classical volterra integral equations of the second kind

existence of solutions for a two point boundary value problem on time scales let t be a time scale let f 0 s 1 xr 2 r be continuous we study the existence of solutions for the nonlinear two point boundary value problems on time scalex t f t x t x t email protected 0 1 x 0 0 x s 1 0 we do not need any growth restrictions on f

comparison results for solving preconditioned linear systems in this paper we present some comparison theorems between two different modified gauss seidel mgs methods the second preconditioning based on the first preconditioning is also discussed in this paper

asymptotic properties of solutions of some volterra difference equations and second order difference equations abstract we give sufficient conditions for the existence of a bounded resp convergent solution of a class of difference equations and volterra difference equation

almost periodic solutions of discrete volterra equations for discrete volterra equations with or without delay we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions we also investigate the relations among solutions of equations discussed and give an example to illustrate our results

an extended gs method for dense linear systems davey and rosindale k davey i rosindale an iterative solution scheme for systems of boundary element equations internat j numer methods engrg 37 1994 1399 1411 derived the gsor method which uses an upper triangular matrix w in order to solve dense linear systems by applying functional analysis the authors presented an expression for the optimum w moreover davey and bounds k davey s bounds a generalized sor method for dense linear systems of boundary element equations siam j comput 19 1998 953 967 also introduced further interesting results in this note we employ a matrix analysis approach to investigate these schemes and derive theorems that compare these schemes with existing preconditioners for dense linear systems we show that the convergence rate of the gauss seidel method with preconditioner p g is superior to that of the gsor method moreover we define some splittings associated with the iterative schemes some numerical examples are reported to confirm the theoretical analysis we show that the egs method with preconditioner p g c o p t produces an extremely small spectral radius in comparison with the other schemes considered

a survey stability and boundedness of volterra difference equations abstract volterra difference equations arise in the modeling of many real phenomena in this survey stability and boundedness problems of some volterra nonlinear difference equations are investigated stability conditions and boundedness are formulated in terms of the characteristics of equations

study on the preconditioners i sm kotakemori et al 2002 2 have reported that the convergence rate of the iterative method with a preconditioner p m i s m a x was superior to one of the modified gauss seidel methods under a special condition the authors derived a theorem comparing the gauss seidel method to remove the requirement for this condition morimoto et al 2004 4 have proposed the preconditioner p s m i s s m however it is pointed out that there exists a special matrix that does not satisfy this comparison theorem to overcome this problem kohno et al 2009 3 have proposed some preconditioners in this note we present a new preconditioner and from numerical results we show that the convergence rate of the proposed method is better than that of the gauss seidel method with other preconditioners in addition we presented the comparison theorem for the proposed preconditioner we succeeded to overcome two drawbacks mentioned above

an efficient algorithm based on splitting for the time integration of the schrodinger equation we present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the schrodinger equation when the hamiltonian operator is either time independent or changes slowly with time in the later case the evolution operator can be effectively approximated in a step by step manner first divide the time integration interval in sufficiently short subintervals and then successively solve a schrodinger equation with a different time independent hamiltonian operator in each of these subintervals when discretized in space the schrodinger equation with the time independent hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous hamiltonian system corresponding to a system of coupled harmonic oscillators the particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals sharp local error bounds are obtained for the solution of the linear autonomous hamiltonian system considered in each time subinterval our schemes can be considered in this setting as polynomial approximations to the matrix exponential in a similar way as methods based on chebyshev and taylor polynomials the theoretical analysis supported by numerical experiments performed for different time independent hamiltonians indicates that the new methods are more efficient than schemes based on chebyshev polynomials for all tolerances and time interval lengths the algorithm we present automatically selects for each time subinterval the most efficient splitting scheme among several new optimized splitting methods for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the hamiltonian operator

upper and lower solution method for a fourth order four point boundary value problem on time scales in this work we consider a fourth order four point boundary value problem on time scales we establish criteria for the existence of a solution by developing the upper and lower solution method and the monotone iterative technique

a comparison theorem for the iterative method with the preconditioner i s max in 1991 gunawardena et al linear algebra appl 154 156 1991 123 have reported the modified gauss seidel method with a preconditioner i s in this article we propose to use a preconditioner i smax instead of i s here smax is constructed by only the largest element at each row of the upper triangular part of a by using the lemma established neumann and plemmons linear algebra appl 88 89 1987 559 we get the comparison theorem for the proposed method simple numerical examples are also given

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

lu decomposition of m matrices by elimination without pivoting abstract it is shown that if a or a is a singular m matrix satisfying the generalized diagonal dominance condition y t a 0 for some vector y 0 then a can be factored into a lu by a certain elimination algorithm where l is a lower triangular m matrix with unit diagonal and u is an upper triangular m matrix the existence of lu decomposition of symmetric permutations of a and for irreducible m matrices and symmetric m matrices follow as colollaries this work is motivated by applications to the solution of homogeneous systems of linear equations ax 0 where a or a is an m matrix these applications arise e g in the analysis of markov chains input output economic models and compartmental systems a converse of the theorem metioned above can be established by considering the reduced normal form of a

the preconditioned gauss seidel method faster than the sor method in recent years a number of preconditioners have been applied to linear systems a d gunawardena s k jain l snyder modified iterative methods for consistent linear systems linear algebra appl 154 156 1991 123 143 t kohno h kotakemori h niki m usui improving modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x j comput appl math 145 2002 373 378 h kotakemori h niki n okamoto accelerated iteration method for z matrices j comput appl math 75 1996 87 97 m usui h niki t kohno adaptive gauss seidel method for linear systems internat j comput math 51 1994 119 125 10 since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix the preconditioning effect is not observed on the nth row of matrix a in the present paper in order to deal with this drawback we propose a new preconditioner in addition the convergence and comparison theorems of the proposed method are established simple numerical examples are also given and we show that the convergence rate of the proposed method is better than that of the optimum sor

limiting periodicity of the solutions for some volterra difference equations conditions of the limiting of periodicity of the solutions for some volterra difference equations are derived these conditions are formulated immediately in terms of the coefficients of equations and their resolvent using the second liapunov method and fixed point theorem

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

an estimation of the spectral radius of a product of block matrices let c r c ij r 1 2 r be block m x m matrices where c ij r are nonnegative n i x n j matrices for i j 1 2 m let be a consistent matrix norm denote for each r by b r c ij r an m m matrix the relation of the spectral radii r r 1 c r and r r 1 b r is studied in this paper it is shown with two proofs that r r 1 c r r r 1 b r as shown in one of the proofs r r 1 b r can be reduced so that it gives a better estimation of r r 1 c r

on optimal improvements of classical iterative schemes for z matrices many researchers have considered preconditioners applied to linear systems whose matrix coefficient is a z or an m matrix that make the associated jacobi and gauss seidel methods converge asymptotically faster than the unpreconditioned ones such preconditioners are chosen so that they eliminate the off diagonal elements of the same column or the elements of the first upper diagonal milaszewicz laa 93 1987 161 170 gunawardena et al laa 154 156 1991 123 143 in this work we generalize the previous preconditioners to obtain optimal methods good jacobi and gauss seidel algorithms are given and preconditioners that eliminate more than one entry per row are also proposed and analyzed moreover the behavior of the above preconditioners to the krylov subspace methods is studied

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

dynamic boundary value problems of the second order bernstein nagumo conditions and solvability abstract this article analyzes the solvability of second order nonlinear dynamic boundary value problems bvps on time scales new bernstein nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the bvps under consideration topological methods are then employed to gain solvability

convergence analysis of the preconditioned gauss seidel method for h matrices in 1997 kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 proved that the convergence rate of the preconditioned gauss seidel method for irreducibly diagonally dominant z matrices with a preconditioner i s a is superior to that of the basic iterative method in this paper we present a new preconditioner i k b which is different from the preconditioner given by kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 and prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an h matrix meanwhile two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given

about the behavior of solutions for volterra difference equations with infinite delay we study the existence of discrete almost automorphic solutions and asymptotic behavior for non linear volterra difference equations of convolution type with infinite delay where the nonlinear perturbation is considered not necessarily globally lipschitz the results are a consequence of application of different fixed point theorems namely the contraction principle the leray schauder the krasnoselkii and matkowski s fixed point technique

modified sor type iterative method for z matrices abstract in this paper we present the convergence analysis for some modified sor type iterative method for solving linear systems and provide a comparison of spectral radius among the sor iterative method and these modified methods comparison results and the numerical example show that the rate of convergence of the gauss seidel method is faster than the rate of convergence of the sor iterative method

modified gauss seidel type methods and jacobi type methods for z matrices abstract in this paper we present the convergence analysis for some modified gauss seidel and jacobi type iterative methods and provide a comparison of spectral radius among the gauss seidel iterative method and these modified methods some recent results are improved

triple positive solutions for some second order boundary value problem on a measure chain in this paper we study the existence of three positive solutions for the second order two point boundary value problem on a measure chain x d d t p t f t x s t x d t 0 email protected t 1 t 2 a 1x t 1 a 2x d t 1 0 a 3x s t 2 a 4x d s t 2 0 where f t 1 s t 2 x 0 xr 0 is continuous and p t 1 s t 2 0 a nonnegative function that is allowed to vanish on some subintervals of t 1 s t 2 of the measure chain the method involves applications of a new fixed point theorem due to bai and ge z b bai w g ge existence of three positive solutions for some second order boundary value problems comput math appl 48 2004 699 707 the emphasis is put on the nonlinear term f involved with the first order delta derivative x d t

the survey of preconditioners used for accelerating the rate of convergence in the gauss seidel method several preconditioned iterative methods reported in the literature have been used for improving the convergence rate of the gauss seidel method in this article on the basis of nonnegative matrix comparisons between some splittings for such preconditioned matrices are derived simple numerical examples are also given

convergence analysis of the two preconditioned iterative methods for m matrix linear systems in this paper we present the two preconditioners i s and i s r for solving m matrix linear systems and discuss the convergence of the two preconditioned iterative methods meanwhile we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of m matrix linear systems by preconditioned krylov subspace methods numerical experiments are given to validate the performance of the preconditioners