comparisons of nonnegative splittings of matrices abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors in this article some new comparison theorems for two nonnegative splittings a splitting a m n is nonnegative if m 1 exist and m 1 n is nonnegative are derived they extend the known results in the literature in addition we also point out three incorrect conditions in a paper by beauwens furthermore we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates this also answers the open question which additional and appropriate conditions should be imposed on miller neumann splittings to obtain strict inequality finally some applications to a class of generalized aor sor and jor iterative methods whose special cases imply block also point aor sor and jor iterative methods for solving linear systems are discussed

cohen macaulay modules co torsion pairs and virtually gorenstein algebras abstract we use torsion pairs in stable categories and cotorsion pairs in modules categories to study in general infinitely generated cohen macaulay modules and a generalization of modules of finite projective or injective dimension over an artin algebra we concentrate our investigation to the study of virtually gorenstein algebras which provide a common generalization of gorenstein algebras and algebras of finite representation or cohen macaulay type this class of algebras on the one hand has rich homological structure and satisfies several representation torsion theoretic finiteness conditions and on the other hand it is closed under various operations for instance derived equivalences and stable equivalences of morita type in addition virtual gorensteinness provides a useful tool for the study of the gorenstein symmetry conjecture and modified versions of the telescope conjecture for module or stable categories

weighted max norms splittings and overlapping additive schwarz iterations summary weighted max norm bounds are obtained for algebraic additive schwarz iterations with overlapping blocks for the solution of ax b when the coefficient matrix a is an m matrix the case of inexact local solvers is also covered these bounds are analogous to those that exist using a norms when the matrix a is symmetric positive definite a new theorem concerningp regular splittings is presented which provides a useful tool for the a norm bounds furthermore a theory of splittings is developed to represent algebraic additive schwarz iterations this representation makes a connection with multisplitting methods with this representation and using a comparison theorem it is shown that a coarse grid correction improves the convergence of additive schwarz iterations when measured in weighted max norm

some comparison theorems for nonnegative splittings of matrices in a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings they extended the corresponding result of miller and neumann and implied the earlier theorems of varga beauwens csordas and varga an open question by miller and neumann which additional and appropriate conditions should be imposed to obtain strict inequality was also answered this article continues to investigate the comparison theorems for nonnegative splittings the new results extend and imply the known theorems by the author miller and neumann

optimal domain splitting for interpolation by chebyshev polynomials polynomial interpolants defined using chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval however the convergence rate can be arbitrarily close to unity if the function has a singularity close to the interval when extended to the complex plane in such cases splitting the interval and doing piecewise interpolation may be more efficient in the total number of nodes than the global interpolant because the convergence rate is determined by bernstein ellipses obtained through a joukowski conformal map relative efficiency of splitting at any point in the interval can be calculated and then optimized over the interval the optimal splitting may be applied recursively the chebfun software project uses a simple rule of thumb without prior singularity information to create a binary search that can be shown to do an excellent job of finding the optimal splitting in most cases however the process can use a large number of intermedi

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

quotients of cluster categories higher cluster categories were recently introduced as a generalization of cluster categories this paper shows that in dynkin types a and d half of all higher cluster categories can be obtained as quotients of cluster categories the other half are quotients of 2 cluster categories the lowest type of higher cluster categories hence in dynkin types a and d all higher cluster phenomena are implicit in cluster categories and 2 cluster categories in contrast the same is not true in dynkin type e

quasi abelian hearts of twin cotorsion pairs on triangulated categories abstract we prove that under a mild assumption the heart h of a twin cotorsion pair s t u v on a triangulated category c is a quasi abelian category if c is also krull schmidt and t u we show that the heart of the cotorsion pair s t is equivalent to the gabriel zisman localisation of h at the class of its regular morphisms in particular suppose c is a cluster category with a rigid object r and x r the ideal of morphisms factoring through x r ker hom c r then applications of our results show that c x r is a quasi abelian category we also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of c

on the perron frobenius theory for complex matrices abstract we extend here the perron frobenius theory of nonnegative matrices to certain complex matrices following the generalization of the perron frobenius theory to matrices that have some negative entries given by noutsos 14 we introduce here two types of extensions of the perron frobenius theory to complex matrices we present and prove here some sufficient conditions and some necessary and sufficient conditions for a complex matrix to have a perron frobenius eigenpair we apply this theory by introducing perron frobenius splittings as well as complex perron frobenius splittings for the solution of complex linear systems a x b by classical iterative methods perron frobenius splittings constitute an extension of the well known regular splittings weak regular splittings and nonnegative splittings convergence and comparison properties are also given and proved

an adaptive partition of unity method for chebyshev polynomial interpolation for a function that is analytic on and around an interval chebyshev polynomial interpolation provides spectral convergence however if the function has a singularity close to the interval the rate of convergence is near one in these cases splitting the interval and using piecewise interpolation can accelerate convergence chebfun includes a splitting mode that finds an optimal splitting through recursive bisection but the result has no global smoothness unless conditions are imposed explicitly at the breakpoints an alternative is to split the domain into overlapping intervals and use an infinitely smooth partition of unity to blend the local chebyshev interpolants a simple divide and conquer algorithm similar to chebfun s splitting mode can be used to find an overlapping splitting adapted to features of the function the algorithm implicitly constructs the partition of unity over the subdomains this technique is applied to explicitly given functions as well as to the solutions of singularly pertur

comparison theorems for splittings of matrices this paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices on the basis of nonnegative matrix theory comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived when the coefficient matrix a is hermitian positive semi definite comparison theorems about two p regular splittings and two parallel multisplitting methods are proved

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

comparison theorems using general cones for norms of iteration matrices we prove comparison theorems for norms of iteration matrices in splittings of matrices in the setting of proper cones in a finite dimensional real space by considering cone linear absolute norms and cone max norms subject to mild additional hypotheses we show that these comparison theorems can hold only for such norms within the class of cone absolute norms finally in a banach algebra setting we prove a comparison theorem for spectral radii without appealing to perron frobenius theory

rigid objects triangulated subfactors and abelian localizations we show that the abelian category mathsf mod text mathcal x of coherent functors over a contravariantly finite rigid subcategory mathcal x in a triangulated category mathcal t is equivalent to the gabriel zisman localization at the class of regular maps of a certain factor category of mathcal t and moreover it can be calculated by left and right fractions thus we generalize recent results of buan and marsh we also extend recent results of iyama yoshino concerning subfactor triangulated categories arising from mutation pairs in mathcal t in fact we give a classification of thick triangulated subcategories of a natural pretriangulated factor category of mathcal t and a classification of functorially finite rigid subcategories of mathcal t if the latter has serre duality in addition we characterize 2 cluster tilting subcategories along these lines finally we extend basic results of keller reiten concerning the gorenstein and the calabi yau property for categories arising from certain rigid not necessarily cluster tilting subcategories as well as several results of the literature concerning the connections between 2 cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors

on algebraic multi level methods for non symmetric systems comparison results abstract we establish theoretical comparison results for algebraic multi level methods applied to non singular non symmetric m matrices we consider two types of multi level approximate block factorizations or amg methods the amli and the mamli method we compare the spectral radii of the iteration matrices of these methods this comparison shows that the spectral radius of the mamli method is less than or equal to the spectral radius of the amli method moreover we establish how the quality of the approximations in the block factorization effects the spectral radii of the iteration matrices we prove comparisons results for different approximations of the fine grid block as well as for the used schur complement we also establish a theoretical comparison between the amg methods and the classical block jacobi and block gauss seidel methods

convergence theorems for parallel alternating iterative methods the parallel multisplitting nonstationary iterative model a was introduced by bru elsner and neumann linear algebra appl 103 1988 175 192 for solving nonsingular linear system ax b using a weak nonnegative multisplitting of the first type in this paper new results using a weak nonnegative multisplitting of the second type are introduced when a is a monotone matrix and using p regular multisplitting when a is a symmetric positive definite matrix combining model a and alternating iterative methods two new models of parallel multisplitting nonstationary iterations are introduced it is shown that when matrix a is monotone and the multisplittings are weak nonnegative of the first or second type both models lead to convergent schemes when matrix a is symmetric positive definite and the multisplittings are p regular the schemes are also convergent

an algebraic convergence theory for restricted additive schwarz methods using weighted max norms convergence results for the restrictive additive schwarz ras method of cai and sarkis siam j sci comput 21 1999 pp 792 797 for the solution of linear systems of the form ax b are provided using an algebraic view of additive schwarz methods and the theory of multisplittings the linear systems studied are usually discretizations of partial differential equations in two or three dimensions it is shown that in the case of a symmetric positive definite the projections defined by the methods are not orthogonal with respect to the inner product defined by a and therefore the standard analysis cannot be used here the convergence results presented are for the class of m matrices and more generally for h matrices using weighted max norms comparison between different versions of the ras method are given in terms of these norms a comparison theorem with respect to the classical additive schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory several ras variants are considered including new ones and two level schemes

on the exponentially slow motion of a viscous shock using formal asymptotic methods we study the internal layer behavior associated with the following viscous shock problem in the limit e 0 the convex nonlinearity f u satisfies f f for the steady problem we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution this indeterminacy resulting from neglecting certain exponentially small effects is eliminated by using the projection method which exploits certain properties of the spectrum associated with the linearized operator for the time dependent problem we show that the viscous shock which is formed from initial data drifts towards the equilibrium solution on an exponentially long time interval of the order o ec e for some c 0 this exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock for burgers equation f u u2 2 the results give an analytical characterization of the slow shock layer motion observed numerically in kreiss and kreiss see 11 we also show that the shock layer behavior is very sensitive to small changes in the boundary operator in addition using a wkb type method the slow viscous shock motion is studied numerically for small e the results comparing favorably with corresponding analytical results finally we relate the slow viscous shock motion to similar slow internal layer motion for the allen cahn equation

from triangulated categories to module categories via localisation we show that the category of finite dimensional modules over the endomorphism algebra of a rigid object in a hom finite triangulated category is equivalent to the gabriel zisman localisation of the category with respect to a certain class of maps this generalises the 2 calabi yau tilting theorem of keller reiten in which the module category is obtained as a factor category to the rigid case

comparison results between jacobi and other iterative methods some comparison results between jacobi iterative method with the modified preconditioned simultaneous displacement mpsd iteration and other iterations for solving nonsingular linear systems are presented it is showed that spectral radius of jacobi iteration matrix b is less than that of several iteration matrices introduced in liu j numer methods comput appl 1 1992 58 under some conditions

conditions for strict inequality in comparisons of spectral radii of splittings of different matrices we present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices but we also obtain new results for nonstrict inequalities of the spectral radii we emphasize two different approaches one combinatorial and the other analytic and discuss their merits in the light of the results obtained we try to get fairly general results and indicate by counter examples that some of our hypotheses cannot be relaxed in certain directions

some comparison theorems for weak nonnegative splittings of bounded operators abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors in this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general banach space and rather general cones and in a hilbert space which extend some of the results obtained by wo nicki japan j indust appl math 11 1994 289 342 and marek and szyld numer math 44 1984 23 35 furthermore we present new theorems also for bounded operator which extend some results by csordas and varga numer math 44 1984 23 35 for weak nonnegative splittings of matrices

a note on abelian quotient categories abstract let c be a triangulated category with a serre functor s and x a non zero contravariantly finite rigid subcategory of c then x is cluster tilting if and only if the quotient category c x is abelian and s x x 2 as an application this result generalizes the work by beligiannis

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

a note on comparison theorems for nonnegative matrices in a recent paper 4 csordas and varga have unified and extended earlier theorems of varga in 10 and wo nicki in 11 on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings the main purpose of this note is to show a connection between the csordas varga paper and a paper by beauwens 1 in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular monotonic norms already used in 1 play an important role in our work here

some results on a generalized alternating iterative method climent and perea j j climent c perea convergence and comparison theorems for a generalized alternating iterative method appl math comput 143 2003 1 14 introduced a generalized alternating iterative method in this paper we establish convergence results for a nonsingular h matrix upper bound to the spectral radius of iterative matrix and comparison theorem for a monotone matrix moreover we give some numerical examples to show our results

comparison theorems for the convergence factor of iterative methods for singular matrices abstract iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix t i e by the quantity t max t 1 where t is the spectrum of t theorems are presented comparing the convergence factor of two iterative methods the comparison is based on the relationship between the matrices of the splittings a cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison although this cone is based on the unknown projection onto the null space of a matrix the characterization provided in the paper allows in specific instances the cone to be readily computable

new comparison results for parallel multisplitting iterative methods abstract the parallel multisplitting nonstationary iterative model a and models 1 2 were introduced by bru et al bru elsner neumann models of parallel chaotic iteration methods linear algebra appl 103 1988 175 192 and climent et al climent perea tortosa zamora convergence theorems for parallel alternating iterative methods appl math comput 148 2004 497 517 respectively using nonnegative splitting theory we present some new comparison theorems between two general parallel multisplittings of model a and models 1 2 when the coefficient matrix is a monotone matrix particularly we establish some sufficient conditions for strict inequalities in comparisons of spectral radii of different iteration matrices

comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same nonsingular matrix have been compared in the works of many authors in this paper new comparison theorems which extend song s results y song numer math 65 1993 245 252 for weak splittings of the same or different types are presented these new comparison theorems are derived by using nonnegativity in respect to a proper cone k n ew converses which partially solve an open question asked by v a miller m neumann numer math 47 1985 427 434 are also presented 1999 elsevier science inc all rights reserved

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

wakamatsu tilting modules abstract we study a generalization of tilting modules to modules of possibly infinite projective dimension introduced by wakamatsu in j algebra 114 1988 106 114 in particular we characterize these modules in terms of suitable subcategories of finitely generated modules and in terms of cotorsion theories

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

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

an effective stationary iterative method via double splittings of matrices abstract in this paper a new stationary iterative method based on two double splittings of coefficient matrix is proposed to solve linear system a x b some convergence results and comparison theorems of the new method are provided theoretical analysis shows that this method is superior to some existing ones under certain conditions numerical examples are given to illustrate the theoretical results and examine the effectiveness of the proposed method

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

convergence and comparison theorems for a generalized alternating iterative method using as a principal tool the convergence results of standard iterative process for the solution of linear systems alternating iterative methods are studied we extend the convergence theorem for the stationary alternating iterative method of benzi and szyld numererische mathematik 76 1997 309 for weak nonnegative splittings of the first type of a monotone matrix to weak nonnegative splitting of the second type on the other hand we introduce a more general method the nonstationary alternating iterative method establishing convergence results for weak nonnegative splittings of a monotone matrix and for p regular splittings of a symmetric positive definite matrix

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway