on the modified successive overrelaxation method new forms of the modified successive overrelaxation msor mksor mksor1 and mksor2 are introduced these forms depend on the updated successive overrelaxation method ksor and the combination of the sor and the ksor methods we present analysis of the convergence of the mksor methods with fixed parameters for solving linear system of equations ax b where a belongs to a subclass of consistently ordered matrices these matrices appear in the numerical treatment of elliptic partial differential equations corresponding to the red black ordering also functional relations for the eigenvalues of the iteration matrices of the jacobi the mksor methods and the relaxation parameters are established a general representative numerical example illustrating this treatment is considered different values of the relaxation parameters are used on each set of equations we have implemented our treatment using computer algebra software mathematica 8 0 new results on this subject are presented and compared with those of young this approach is promising and will help in the numerical treatment of boundary value problems other extensions and applications for further work are mentioned

optimal parameters for linear second degree stationary iterative methods in this paper we show that the optimal parameters for linear second degree stationary iterative methods applied to nonsymmetric linear systems can be found by solving the same minimax problem used to find optimal parameters for the chebyshev iteration in fact the chebyshev iteration is asymptotically equivalent to a linear seconnd degree stationary method the method of finding optimal parameters for the chebyshev iteration given in manteuffel numer math 28 1977 pp 307 327 can be used to find optimal parameters for the sitationary method as well

p cyclic matrices and the symmetric successive overrelaxation method abstract in this paper the new functional equation 1 2 p 1 p 2 2 2 p p which connects the eigenvalues of a particular weakly cyclic of index p jacobi matrix b to the eigenvalues of its associated symmetric successive overrelaxation ssor matrix s is derived this functional equation is then applied to the problem of determining bounds for the intervals of convergence and divergence of the ssor iterative method for classes of h matrices

general k part stationary iterative solutions to linear systems the solution of i b x b is sought where b is a n times n matrix i is the identity matrix b is an n vector of known constants and x is the unknown vector when n is large a direct solution of the equation is not feasible one instead seeks an approximation to x this paper discusses stationary k part methods for finding sequences of approximations to x after presenting a generalization of the companion matrix criteria for the convergence of k part stationary operator coefficient algorithms two theorems are proved which relate the spectrum of a particular form of the companion matrix to the roots of a meromorphic function evaluated at the eigenvalues of b a corollary which results from these theorems gives the convergence criteria for the most general k part stationary constant coefficient algorithm in terms of the image of the unit disc under a rational function which is easily related to the stationary algorithm it is then shown that the asymptotic rate of convergence for this most gener

a method for the improvement of the convergence rates of stationary iterative schemes for the solution of complex linear systems abstract in this paper a number of theorems and lemmas are stated and proved which can be used as criteria for the evaluation of some unknown eigenvalues of the coefficient matrix of a complex linear system in connection with a consistent with the linear system stationary iterative scheme the aforementioned eigenvalues can be used together with or without the iterative approximations to improve the convergence rates of the iterative scheme numerical examples show the validity of the theory developed

convergence theory of extrapolated iterative methods for a certain class of non symmetric linear systems a variety of iterative methods considered in 3 are applied to linear algebraic systems of the formau b where the matrixa is consistently ordered 12 and the iteration matrix of the jacobi method is skew symmetric the related theory of convergence is developed and the optimum values of the involved parameters for each considered scheme are determined it reveals that under the aforementioned assumptions the extrapolated successive underrelaxation method attains a rate of convergence which is clearly superior over the successive underrelaxation method 5 when the jacobi iteration matrix is non singular

convergence of preconditioned iterative methods abstract relations of the eigenvalues and the eigenvectors of some preconditioned iterative matrices such as bpsd bpj bmpsd and bmpj with those of the jacobi iterative matrix are established under the condition that the coefficient matrix a of the system of algebraic equations considered is a t q r matrix the convergence of these iterative methods is analysed when a is a t 1 1 t 1 2 or h matrix

a practical choice of parameters in improved sor newton method with orderings abstract in this paper on the basis of the results of ishihara et al 1997 we first discuss global convergence theorems for the improved sor newton and block sor newton methods with orderings applied to a system of mildly nonlinear equations which includes as a special case the discretized version of the dirichlet problem for the equation u p x ux q y uy f x y u where f is continuously differentiable and fu x y u 0 moreover we propose a practical choice of the multiple relaxation parameters i for them numerical examples are also given

oscillation theorems for forced neutral difference equations abstract some oscillation criteria are established for forced neutral difference equations of the form x n px n k q n x n f n and 2 x n px n k q n x n f n the results remain valid even when f n 0

oscillation of some nonlinear difference equations we investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations several explicit oscillation criteria are established our main results are supported by illustrative examples

on hybrid semi iterative methods given a large sparse system of linear algebraic equations in fixed point form bf x t bf x bf c one way to solve this system is to apply a semi iterative method sim to the basic iteration method bf x m t bf x m 1 bf c it is known that if the spectrum sigma t of t is contained in some compact subset omega of the complex plane where 1 notin omega then there are asymptotically optimal sims associated with the basic iteration and omega whose asymptotic rates of convergence are best possible for the class of all matrices t with sigma t subseteq omega however for a given compact set omega an asymptotically optimal sim for omega is usually not well suited for efficient numerical computation unless this sim can be generated from a k step recurrence formula in this paper hybrid semi iterative methods are investigated which consist of two independent steps first bf x t bf x bf c is transformed into a consistent linear syst

on the construction of semi iterative methods given a nonsingular linear system a bf x bf b a splitting a m n leads to the one step iteration 1 bf x m t bf x m 1 bf c with t m 1 n and c m 1 bf b we investigate semiiterative methods with respect to 1 under the assumption that the eigenvalues of t are contained in some compact set u subset mathbb c 1 notin u using results about maximal convergence of polynomials and uniformly distributed nodes from approximation theory we describe semiiterative methods which are asymptotically optimal with respect to u under mildly restrictive assumptions on u we construct semiiterative methods which are asymptotically optimal with respect to u and which can be interpreted as one step nonstationary methods

optimal semi iterative methods for complex sor with results from potential theory we consider the application of semi iterative methods sim to the standard sor method with complex relaxation parameter under the following two assumptions 1 the associated jacobi matrix j is consistently ordered and weakly cyclic of index 2 and 2 the spectrum j of j belongs to a compact subset of the complex plane mathbb c which is symmetric with respect to the origin by using results from potential theory we determine the region of optimal choice of omega in mathbb c for the combination sim sor and settle for a large class of compact sets the classical problem of characterising completely all the cases for which the use of the sim sor is advantageous over the sole use of sor under the hypothesis that sigma j subset sigma in particular our results show that unless the outer boundary of is an ellipse sim sor is always better and furthermore one of the best possible choices is an asymptotically optimal sim applied to the gauss seidel method in addition we derive the optimal complex sor parameters for all ellipses which are symmetric with respect to the origin our work was motivated by recent results of m eiermann and r s varga

the impact of eigenvalue locality on the convergence behavior of the psd method for two cyclic matrices abstract in this paper we analyse the convergence of the preconditioned simultaneous displacement psd method applied to linear systems of the form au b where a is a two cyclic matrix convergence conditions and optimum values of the parameters of the method are determined in the cases where the eigenvalues of the associated jacobi iteration matrix are either all real or all imaginary it is shown that the convergence behavior of the psd method is greatly affected by the locality of the eigenvalues of the associated jacobi iteration matrix in particular it is shown that when these eigenvalues are real the psd method degenerates into the extrapolated gauss seidel method whereas when they are imaginary its convergence is increased by an order of magnitude and becomes equivalent to the extrapolated sor method finally a comparison with the ssor method reveals that the psd method possesses a better convergence behavior in all cases

higher nakayama algebras i construction we introduce higher dimensional analogues of the nakayama algebras from the viewpoint of iyama s higher auslander reiten theory more precisely for each nakayama algebra a and each positive intege

some recent results on the modified sor theory abstract for the solution of the linear system ax i t x c 1 where t is a weakly cyclic of index p 2 matrix of a special structure the block sor method with different relaxation factors associated with the row blocks of a is considered first a well known relationship connecting the eigenvalue spectra of the block jacobi matrix t of a and its associated modified sor msor matrix is proved for all p 3 via an approach due to varga niethammer and cai next it is shown that the matrix analogue of the eigenvalue relationship holds at least for p 3 this together with the facts that the matrix analogue holds true for p 2 and also for any p 3 provided all relaxation factors coincide suggests a more general validity of the aforementioned matrix identity then based on the matrix relationship an equivalence is established between the msor method and a stationary p parameter p step iterative one for the solution of 1 so one can study convergence properties and determine optimal parameters of both methods by studying the simpler of the two that is the p step one finally some applications of the theory developed when p 2 are presented and a brief discussion concerning comparisons of the optimum msor and aor methods in a specific case is given

the analysis ofk step iterative methods for linear systems from summability theory using the theory of euler methods from summability theory we investigate general iterative methods for solving linear systems of equations in particular for a given euler method a regions of the complex plane is determined such that ak step iterative method converges if the eigenvalues of an iteration operatort are contained ins for a givens optimal methods are described and upper and lower bounds are derived for the associated asymptotic rate of convergence special attention is given to two step methods with complex parameters

domains of divergence of the ussor method applied on p cyclic matrices using a recently derived classical type general functional equation relating the eigenvalues of a weakly cyclic jacobi iteration matrix to the eigenvalues of its associated unsymmetric successive overrelaxation ussor iteration matrix we obtain bounds for the convergence of the ussor method when applied to systems with ap cyclic coefficient matrix

is modified psd equivalent to modified sor for two cyclic matrices in this paper we study the convergence analysis of the modified preconditioned simultaneous displacement mpsd method when a is a two cyclic matrix convergence conditions and optimum values of the parameters are determined in case the eigenvalues of the associated jacobi iteration matrix are either all real or all imaginary

on the convergence of some generalized preconditioned iterative methods this paper generalizes the preconditioning techniques introduced by evans 2 for the numerical solution of the linear system au b and defines two extrapolated versions of the gauss siedel method as well as an extrapolated version of the successive overrelaxation method finally it establishes some convergence theorems for the considered iterative schemes when a has particular properties such as being consistently ordered positive definite having weak diagonal dominance etc

on an ssor matrix relationship and its consequences a matrix relationship connecting the jacobi and the symmetric successive overrelaxation ssor matrices associated with a k cyclic consistently ordered matrix a is presented next the equivalence of the ssor method and a certain monoparametric k step one for the solution of the linear algebraic system ax b is established the aforementioned equivalence can be exploited to derive regions of convergence optimum parameters involved etc of the two iterative methods above this is done by studying the simplest of the two methods that is the monoparametric k step one to show how the idea works the case k 2 is very briefly discussed

a note on the ssor and ussor iterative methods applied top cyclic matrices the purpose of this note is threefold i to derive the new functional equation begin gathered left lambda left 1 omega right left 1 hat omega right right p lambda k left lambda omega hat omega omega hat omega right left zeta l right k left lambda hat omega omega omega hat omega right left zeta u right k hfill cdot left omega hat omega omega hat omega right 2k mu p hfill end gathered which couples the nonzero eigenvalues of the ussor iteration matrix t omega hat omega with the eigenvalues of the associated block jacobi matrixb in thep cyclic case ii to interpret the exponentk in this equation by means of graph theory andiii to connect the above equation with known results in the literature

solution of elliptic pdes by fast poisson solvers using a local relaxation factor abstract using a local discrete fourier analysis one and two step iterative procedures are developed to solve a large class of 2 d and 3 d nonseparable elliptic partial differential equations pdes the one step procedure is a modified d yakanov gunn iterative procedure in which the relaxation factor is grid point dependent the two step procedure is designed to accelerate the one step procedure both are easy to implement and applicable to a variety of boundary conditions they are also computationally efficient as indicated by the results of numerical comparisons with other established methods furthermore these new algorithms possess two important properties which the traditional iterative methods lack i e i the convergence rate is relatively insensitive to grid cell size and aspect ratio and ii the convergence rate can be easily estimated using the coefficient of the pde being solved for a set of constant coefficient model problems it is show theoretically that the two step vs one step computational efficiency ratio ranges between 1 and 2 the higher ratio is realized whenever the need to accelerate the one step procedure is greater i e whenever its convergence rate is lower it is also shown numerically that the two step procedure can substantially outperform the one step procedure in the numerical solution of many pdes with variable coefficients

an ad hoc sor method abstract an ad hoc method is presented for determining a sequence of successive overrelaxation factors that although having no rigorous foundations at the moment appears to produce a highly competitive iteration scheme for several classes of difference equations a different is chosen for each difference equation

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

on the convergence of the maor method abstract in order to solve a linear system ax b hadjidimos et al 1992 defined a class of modified aor maor method whose special case implies the msor method in this paper some sufficient and or necessary conditions for convergence of the maor and msor methods will be achieved when a is a two cyclic matrix and when a is a hermitian positive definite matrix an h l or m matrix and a strictly or irreducibly diagonally dominant matrix the convergence results on the msor method are better than some known theorems the optimum parameters and the optimum spectral radii of the maor and msor methods are obtained which also answers the open problem given by hadjidimos et al

convergence domains of the ssor method for a class of generalized consistently ordered matrices abstract the functional equation relating the eigenvalues of the block symmetric successive overrelaxation ssor iteration matrix with those of the block jacobi iteration matrix found by chong and cai 1985 is used in order to obtain precise domains of convergence of the block ssor iteration method associated with a class of generalized consistently ordered gco k p k matrices a p 2 k 1 p 1 we show that the domain of convergence depends only on the relaxation parameter the spectral radius of the block jacobi iteration matrix and the value of l k p unlike the case k p k 1 p 1 p 3 which was studied in an earlier paper by the authors 1989 beyond certain critical values of l the domain of convergence does not grow monotonically with l and manifests various sorts of behavior however as in the case k p k 1 p 1 it is shown that the intersection of the convergence domains taken over all pairs p k coincides with the exact domain of convergence of the point ssor iteration method associated with nonsingular h matrices a that is matrices which can be diagonally scaled to being strictly diagonally dominant

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

maximal τd rigid pairs abstract let t be a 2 calabi yau triangulated category t a cluster tilting object with endomorphism algebra consider the functor t t t mod it induces a bijection from the isomorphism classes of cluster tilting objects to the isomorphism classes of support tilting pairs this is due to adachi iyama and reiten the notion of d 2 angulated categories is a higher analogue of triangulated categories we show a higher analogue of the above result based on the notion of maximal d rigid pairs

d abelian quotients of d 2 angulated categories abstract let t be a triangulated category if t is a cluster tilting object and i add t is the ideal of morphisms factoring through an object of add t then the quotient category t i is abelian this is an important result of cluster theory due to keller reiten and konig zhu more general conditions which imply that t i is abelian were determined by grimeland and the first author now let t be a suitable d 2 angulated category for an integer d 1 if t is a cluster tilting object in the sense of oppermann thomas and i add t is the ideal of morphisms factoring through an object of add t then we show that t i is d abelian the notions of d 2 angulated and d abelian categories are due to geiss keller oppermann and jasso they are higher homological generalisations of triangulated and abelian categories which are recovered in the special case d 1 we actually show that if end t t is the endomorphism algebra of t then t i is equivalent to a d cluster tilting subcategory of mod in the sense of iyama this implies that t i is d abelian moreover we show that is a d gorenstein algebra more general conditions which imply that t i is d abelian will also be determined generalising the triangulated results of grimeland and the first author

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

a comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented linear systems in this paper we study the impact of two types of preconditioning on the numerical solution of large sparse augmented linear systems the first preconditioning matrix is the lower triangular part whereas the second is the product of the lower triangular part with the upper triangular part of the augmented system s coefficient matrix for the first preconditioning matrix we form the generalized modified extrapolated successive overrelaxation gmesor method whereas the second preconditioning matrix yields the generalized modified preconditioned simultaneous displacement gmpsd method which is an extrapolated form of the symmetric successive overrelaxation method we find sufficient conditions for each aforementioned iterative method to converge in addition we develop a geometric approach for determining the optimum values of their parameters and corresponding spectral radii it is shown that both iterative methods studied gmesor and gmpsd attain the same rate of convergence numerical results confirm our theoretical expectations

n angulated categories we dene n angulated categories by modifying the axioms of triangulated categories in a natural way we show that heller s parametrization of pre triangulations extends to pre n angulations we obtain a large class of examples of n angulated cate gories by considering n 2 cluster tilting subcategories of triangulated categories which are stable under the n 2 nd power of the suspension functor as an application we show how n angulated calabi yau categories yield triangulated calabi yau categories of higher calabi yau dimension finally we sketch a link to algebraic geometry and string theory

on local relaxation methods and their application to convection diffusion equations abstract this paper discusses local relaxation lr methods which can be regarded as generalizations of the successive overrelaxation sor method the difference is that within an lr method the relaxation factor is allowed to vary from equation to equation a number of existing methods are found to be in fact special lr methods moreover based on sor theory a new lr method is developed the performance of lr methods is illustrated by applying them to central difference approximations of convection diffusion equations it is found that equations with small diffusion coefficients can be handled without difficulty for equations with strongly varying coefficients and for nonlinear equations a properly selected lr method can be significantly more efficient than the optimum sor method as a special example a 16 16 driven cavity problem for a reynolds number of 10 6 can be solved in just a few seconds on a modern computer

k step iterative methods for solving nonlinear systems of equations let n n be frechet differentiable and let the equation 1 x phi left x right have at least one fixed point we considerk step stationary iterative methods 2 y m mu 0 phi left y m 1 right mu 1 y m 1 mu k y m k m geqq k with 0 1 k 1 using results for an affine mapping n n it is proven that 2 may converge locally even in cases where the usual iterationx m x m 1 belonging to 1 diverges these results are extended to nonstationary methods of type 2 and to cyclic mappings

the relationship between the jacobi and the successive overrelaxation sor matrices of a k cyclic matrix abstract let a be a k l l generalized consistently ordered matrix with t and l its associated jacobi and sor matrices whose eigenvalues and satisfy the well known relationship 1 k k k k 1 for a subclass of the above matrices a we prove that the matrix analogue of the previous relationship holds exploiting the matrix relationship we show that the sor method is equivalent to a certain monoparametric k step iterative one when used for the solution of the fixed point problem x tx c