effect algebras with state operator state operators on convex effect algebras in particular effect algebras of unital jc algebras jw algebras and convex mv algebras are studied and their relations with conditional expectations in algebraic sense as well as in the sense of probability on mv algebras are shown

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

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

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

convergence and quotient convergence of iterative methods for solving singular linear equations with index one singular systems with index one arise in many applications such as markov chain modelling in this paper we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one we give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems next we show that for the stationary iterative method the convergence and the quotient convergence are equivalent

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

linear differential equations on the riemann sphere and representations of quivers our interest in this article is a generalization of the additive deligne simpson problem which was originally defined for fuchsian differential equations on the riemann sphere we extend this problem to differential equations having an arbitrary number of unramified irregular singular points and we determine the existence of solutions of the generalized additive deligne simpson problems moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the riemann sphere namely open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces furthermore we detail the connectedness of the moduli spaces

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

over and underrelaxation for linear systems with weakly cyclic jacobi matrices of index p abstract d young s results from 1954 concerning the application of the successive overrelaxation sor method to linear systems ax b with matrices possessing property a were generalized by r s varga in 1959 to systems with jacobi matrices ja which are weakly cyclic of index p if the eigenvalues of jpa are nonnegative then for the optimal parameter opt there holds 1 opt 1 1 p 1 here a different proof of varga s results is given and it is shown that 1 1 p 1 opt 1 holds if the eigenvalues of jpa are nonpositive further exact intervals for those parameters which yield convergence of the sor method are given in terms of chebyshev polynomials

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

linear differential equations on p1 and root systems abstract in this paper we study the euler transform on linear ordinary differential operators on p 1 the spectral type is the tuple of integers which count the multiplicities of local formal solutions with the same leading terms we compute the changes of spectral types under the action of the euler transform and show that the changes of spectral types generate a transformation group of a z lattice which is isomorphic to a quotient lattice of a kac moody root lattice with the weyl group as the transformation group

a young eidson s type algorithm for complex p cyclic sor spectra abstract in a recent work of ours we have solved the problem of the minimization of the spectral radius of the iteration matrix of a p cyclic successive overrelaxation sor method for the solution of the linear system ax b when the matrix a is block p cyclic consistently ordered for what is known as the one point problem for any p 3 particular cases of the one point problem were solved by young varga kjellberg kredell russell and others in the present work we develop a theory using the results of our previous one and solve first the two point problem special cases of which were solved by wrigley eiermann niethammer ruttan noutsos and others secondly we generalize and extend our theory to cover the many point problem and develop a young eidson s type algorithm for its solution as possible application areas we mention among others the best block p cyclic repartitioning for the sor method are the solution of large scale systems arising in queueing network problems in markov analysis

maxwell stefan diffusion asymptotics for gas mixtures in non isothermal setting abstract a mathematical model is proposed where the classical maxwell stefan diffusion model for gas mixtures is coupled to an advection type equation for the temperature of the physical system this coupled system is derived from first principles in the sense that the starting point of our analysis is a system of boltzmann equations for gaseous mixtures we perform an asymptotic analysis on the boltzmann model under diffuse scaling to arrive at the proposed coupled system

comparisons between multiplicative and additive schwarz iterations in domain decomposition methods we consider the multiplicative and additive schwarz methods for solving linear systems of equations and we compare their asymptotic rate of convergence moreover we compare the multiplicative schwarz method with the weighted restricted additive schwarz method we prove that the multiplicative schwarz method is the fastest method among these three our comparisons can be done in the case of exact and inexact subspaces solves in addition we analyse two ways of adding a coarse grid correction multiplicatively or additively

sums of orbits of algebraic groups i abstract let g be a simple algebraic group which is defined and split over a field k and let l be a corresponding lie algebra further let r be the corresponding root system we prove here that if char k 0 or a very good prime for r and k r then there exists an orbit o l l k with respect to the adjoint action of g g k such that l o o this is an analogue of the corresponding result for chevalley groups the thompson problem see e w ellers n gordeev on the conjecture of j thompson and o ore trans amer math soc 350 1998 3657 3671 we also prove that if char k 2 for r b r c r f 4 and char k 3 for r g 2 then the zariski closure of a sum of any r r b r r 3 d r e r f 4 r 1 r a r b 3 g 2 2r r c r g orbits of elements of l z l coincides with l for the group g sl 2 k where k is an algebraically closed field of characteristic zero we list all cases of rational k g modules v which have an orbit o v such that o o v

measure free conditioning and extensions of additive measures on finite mv algebras using the well known representation of any finite mv algebra as a product of finite mv chains as factors we obtain a representation of its canonical extension as a girard algebra product of the canonical extensions of the mv chain factors based on this representation and using the results from our last paper we characterize the additive measures on any finite mv algebra resp the weakly and the strongly additive measures on its canonical girard algebra extension and that as convex combinations of the corresponding measures on the respective factors after that we apply the results to measure free defined conditional events which for this reason are considered as elements of the canonical extension of a finite mv algebra of events this definition for conditional events is naturally equivalent to the well known definition as lattice intervals of events here the weak additivity is considered as a reasonable notion for a global additivity property of a measure extension these extensions are essentially different from the measure extension which has a form analogous to the classical conditional probability but which is only levelwise additive moreover the measure free approach of this paper is completely different from the classical approach in which conditional measures of events given a fixed condition event are considered based on additive measures states of events without having defined conditional events

optimal successive overrelaxation iterative methods for p cyclic matrices we consider linear systems whose associated block jacobi matricesj p are weakly cyclic of indexp in a recent paper pierce hadjidimos and plemmons 13 proved that the block two cyclic successive overrelaxation sor iterative method is numerically more effective than the blockq cyclic sor method 2 2 finally as a counterpart to the result of pierce hadjidimos and plemmons we show that under this more general assumption the two cyclic sor method is not always superior to theq cyclic sor method 2 q p

representing uncertainty on set valued variables using belief functions a formalism is proposed for representing uncertain information on set valued variables using the formalism of belief functions a set valued variable x on a domain w is a variable taking zero one or several values in w while defining mass functions on the frame 2 2 w is usually not feasible because of the double exponential complexity involved we propose an approach based on a definition of a restricted family of subsets of 2 w that is closed under intersection and has a lattice structure using recent results about belief functions on lattices we show that most notions from dempster shafer theory can be transposed to that particular lattice making it possible to express rich knowledge about x with only limited additional complexity as compared to the single valued case an application to multi label classification in which each learning instance can belong to several classes simultaneously is demonstrated

optimal p cyclic sor for complex spectra abstract we consider the successive overrelaxation sor method for the solution of a linear system ax b when the matrix a has a block p p partitioned p cyclic form and its associated block jacobi matrix jp is weakly cyclic of index p following the pioneering work by young and varga in the 1950s many researchers have considered various cases for the spectrum jp and have determined optimal values for the relaxation factor 0 2 so that the sor method converges as fast as possible after the most recent work on the best block p cyclic repartitioning and that on the solution of large scale systems arising in queueing network problems in markov analysis the optimization of the convergence of the p cyclic sor for more complex spectra jp has become more demanding here we state the one point problem for the general p cyclic complex sor case the existence and the uniqueness of its solution are established by analyzing and developing further the theory of the associated hypocycloidal curves for the determination of the optimal parameter s an algorithm is presented and a number of illustrative numerical examples are given

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

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

computing generalized inverses of matrices by iterative methods based on splittings of matrices in this paper we study the iterative methods for computing the generalized inverses of the form a 2 t s we prove the result that if a has the splitting a m n and if the generalized inverses a 2 t s and m 2 t s exist then the iterationx j 1 m 2 t snx j m 2 t sconverges to a 2 t s for any x 0 if and only ift t s s email protected 2 t sn 1 or equivalently m 2 t email protected 2 t sn 1 the splitting a m n is called t s splitting of a if m satisfies email protected s c m where t and s are subspaces of c n and c m respectively with dimt dims we also present a specific method to construct convergent t s splitting of a apply these two results to the most of commonly used generalized inverses such as a a d a d w a h k we can get criteria of convergence for the corresponding iterations many old and new splittings and specific choice of the related convergent splittings

towards the determination of the optimal p cyclic ssor abstract suppose that a c n n is a block p cyclic consistently ordered matrix and let b and s denote the block jacobi and the block symmetric successive overrelaxation ssor iteration matrices associated with a respectively extending previous work by hadjidimos and neumann the present authors have determined the exact regions of convergence of the ssor method in the b plane for any p 3 under the further assumption that the eigenvalues of b p are real of the same sign in the present work the investigation goes on further several questions are raised and among others the problem of the determination of the optimal regions of convergence in the spirit of niethammer and varga as well as that of the optimal relaxation factor are examined

two stage iterative methods for consistent hermitian positive semidefinite systems abstract we study stationary and nonstationary two stage iterative methods for the solution of consistent and singular linear systems ax b where a is a hermitian positive semidefinite matrix when the outer splitting is p regular and inner splitting is convergent we prove the convergence of stationary two stage iterative methods with inner iteration number q being a positive even number conditions are given to ensure the convergence of stationary iterative methods for any positive inner iteration number q also we present some theoretical results on the convergence of nonstationary two stage methods a comparison theorem is obtained for hermitian positive semidefinite matrix and some applications to it are discussed moreover we obtain a monotonicity result on the stationary two stage iterative methods

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

iterative and semi iterative methods for computing stationary probability vectors of markov operators iterative and semi iterative methods for computing stationary prob ability vectors of markov type operators are proposed and their convergence properties are analyzed the methods studied apply to certain classes of prob lems in infinite dimensional spaces as well as to classical n x n stochastic ma trices

projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport we investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport the matrices of the corresponding linear systems are symmetric with a positive semi definite real part and an imaginary part with a compatible nullspace we discuss well posedness the symmetry of generalized inverses and cholesky methods we investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems as an application we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field

block iterative algorithms for stochastic matrices abstract the theory of regular splittings for singular m matrices is used to derive the necessary and sufficient conditions for the convergence of iterative decomposition and aggregation techniques in the computation of the perron frobenius eigenvector of a stochastic irreducible matrix these conditions turn out to be less restrictive than those which must be satisfied by the block jacobi or the block gauss seidel method

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

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 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

equivalent conditions for the convergence of extrapolated method for linear systems abstract we further investigate the convergence of extrapolated method for solving linear systems combined with the results in song and wang y song l wang on the semiconvergence of extrapolated iterative methods for singular linear systems appl numer math 44 2003 401 413 equivalent conditions are obtained the new conditions are based on the field of values of normal matrices which are easier to estimate some numerical examples are given to illustrate our results

semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices in this paper we simultaneously study iterations and extrapolated iterations for the consistent rectangular or sigular linear systems 1 1 ax b in the case that a is rectangular let a m n be a subproper splitting of a and let the generalized inverse m t s 1 2 exist and in the case that a is singular let a m n is a usual splitting of a consider the iteration 2 3 x j 1 gx j c where g m t s 1 2 nandc m t s 1 2 b if a is rectangular and g m 1n and c m 1b if a is singular and the extrapolated iteration 2 19 x j 1 g wx email protected where email protected r w 0 1 and 2 20 g w email protected email protected we establish the semiconcovergence criteria of the iterations 2 3 and 2 19 and present the efficient and convenient construction methods of the semiconvergent iteration matrices g and g w and the choices of the extrapolated parameter w

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

torus orbits on homogeneous varieties and kac polynomials of quivers in this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the kac polynomials of supernova quivers which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line we also prove that these polynomials can be expressed as a specialization of tutte polynomials of certain graphs providing a new way to compute them we also discuss connections with the gel fand macpherson correspondence

semiconvergence of block sor method for singular linear systems with p cyclic matrices in this paper we discuss semiconvergence of the block sor method for solving singular linear systems with p cyclic matrices some sufficient conditions for the semiconvergence of the block sor method for solving a general p cyclic singular system are proved

semiconvergence of two stage iterative methods for singular linear systems we study the semiconvergence of two stage iterative methods for solving nonsymmetric singular linear systems the main tool we used to analyze their convergence is the so called r a local p regular splitting which is introduced in this paper

an efficient solver for the incompressible navier stokes equations in rotation form we consider preconditioned iterative methods applied to discretizations of the linearized navier stokes equations in two and three dimensional bounded domains both unsteady and steady flows are considered the equations are linearized by picard iteration we make use of the rotation form of the momentum equations which has several advantages from the linear algebra point of view we focus on a preconditioning technique based on the hermitian skew hermitian splitting of the resulting nonsymmetric saddle point matrix we show that this technique can be implemented efficiently when the rotation form is used we study the performance of the solvers as a function of mesh size reynolds number time step and algorithm parameters our results indicate that fast convergence independent of problem parameters is achieved in many cases the preconditioner appears to be especially attractive in the case of low viscosity and for unsteady problems

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

perturbation analysis of singular linear systems with arbitrary index a new perturbation result for singular but consistent linear systems with coefficient matrix of arbitrary index however if the system is inconsistent we also derive a perturbation bound of the generalized linear least squares solution our treatment extends a classical perturbation result in numerical linear algebra

conditional probability on σ mv algebras using the loomis sikorski representation of s mv algebras and dedekind s complete groups a conditional probability and expectation on these structures are defined as a natural generalization of the usual kolmogorovian conditioning

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

on adi like iteration method for fractional diffusion equations abstract the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal times toeplitz matrices in this paper the alternating direction implicit adi like method based on the classical adi iteration is proposed to solve the discretized linear system theoretical analyses show that the adi like iteration method is convergent each iteration of this method requires the solutions of two linear systems with hessenberg coefficient matrices these two systems can be solved effectively by hessenberg lu factorization numerical examples are presented to illustrate the effectiveness of the adi like iteration method

iterative aggregation disaggregation method for computing stationary probability vectors of markov type operators a global convergence of a class of iterative aggregation disaggregation methods is presented and some applications as well as some conclusions of numerical experiments performed are shown

quantized multiplicative quiver varieties abstract beginning with the data of a quiver q and its dimension vector d we construct an algebra d q d q mat d q which is a flat q deformation of the algebra of differential operators on the affine space mat d q the algebra d q is equivariant for an action by a product of quantum general linear groups acting by conjugation at each vertex we construct a quantum moment map for this action and subsequently define the hamiltonian reduction a d q of d q with moment parameter we show that a d q is a flat formal deformation of lusztig s quiver varieties and their multiplicative counterparts for all dimension vectors satisfying a flatness condition of crawley boevey indeed the product on a d q yields a fedosov quantization the of symplectic structure on multiplicative quiver varieties as an application we give a description of the category of representations of the spherical double affine hecke algebra of type a n 1 and its generalization constructed by etingof oblomkov and rains in terms of a quotient of the category of equivariant d q modules by a serre subcategory of aspherical modules

local convergence of the exact and inexact iterative aggregation method for linear systems and markov operators the iterative aggregation method for the solution of linear systems is extended in several directions to operators on banach spaces to the method with inexact correction i e to methods where the inner linear system is in turn solved iteratively and to the problem of finding stationary distributions of markov operators local convergence is shown in all cases convergence results apply to the particular case of stochastic matrices moreover an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled markov chains as well as for markov chains with other zero nonzero structures

multiplicative preprojective algebras middle convolution and the deligne simpson problem abstract we introduce a family of algebras which are multiplicative analogues of preprojective algebras and their deformations as introduced by m p holland and the first author we show that these algebras provide a natural setting for the middle convolution operation introduced by n m katz in his book rigid local systems and put in an algebraic setting by m dettweiler and s reiter and h volklein we prove a homological formula relating the dimensions of hom and ext spaces study varieties of representations of multiplicative preprojective algebras and use these results to study simple representations we apply this work to the deligne simpson problem obtaining a sufficient and conjecturally necessary condition for the existence of an irreducible solution to the equation a 1 a 2 a k 1 with the a i in prescribed conjugacy classes in gl n c

on necessary conditions for convergence of stationary iterative methods for hermitian semidefinite linear systems abstract in an earlier paper frommer et al 2008 5 we gave sufficient conditions in terms of an energy seminorm for the convergence of stationary iterations for solving linear systems whose coefficient matrix is hermitian and positive semidefinite in this paper we show in which cases these conditions are also necessary and show that they are not necessary in others

rounding error analysis of two stage iterative methods for large linear systems we study the finite precision behavior of two stage iterative methods for solving a linear system ax b which is consistent in case a is singular i e email protected r a the range of a using the rounding error analysis technique presented in accuracy and stability of numerical algorithms siam philadelphia pa 1996 we deduce conditions under which a two stage iterative method is forward stable or backward stable

acute perturbation of the group inverse abstract for any n n complex matrix a let a g be the group inverse of a when a is singular a matrix b a e is said to be an acute perturbation of a if e is small and the spectral radius b g b a g a 1 the acute perturbation coincides with the stable perturbation of the group inverse if the matrix b satisfies condition r b n a 0 n b r a 0 which was introduced by castro gonzalez et al 2008 8 furthermore several examples are provided to illustrate the acute perturbation of the group inverse

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

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

projected iterative algorithms with application to multicomponent transport abstract we investigate projected iterative algorithms for solving constrained symmetric singular linear systems we discuss the symmetry of generalized inverses and investigate projected standard iterative methods as well as projected conjugate gradient algorithms using a generalization of stein s theorem for singular matrices we obtain a new proof of keller s theorem we also strengthen a result from neumann and plemmons about the spectrum of iteration matrices as an application we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients in multicomponent gas mixtures we obtain low cost accurate approximate expressions for the transport coefficients that can be used in multicomponent flow models typical examples for the species diffusion coefficients and the volume viscosity are presented

convergence of block iterative methods applied to sparse least squares problems abstract recently special attention has been given in the mathematical literature to the problems of accurately computing the least squares solutions of very large scale overdetermined systems of linear equations such as those arising in geodetical network problems in particular it has been suggested that one solve such problems iteratively by applying the block sor successive overrelaxation iterative method to a consistently ordered block jacobi matrix that is weakly cyclic of index 3 here we obtain new results theorem 1 giving the exact convergence and divergence domains for such iterative applications it is then shown how these results extend and correct the literature on such applications in addition analogous results theorem 2 are given for the case when the eigenvalues of the associated block jacobi matrix are nonnegative

moduli spaces of meromorphic connections and quiver varieties abstract we describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the riemann sphere with at most one unramified irregular singularity and arbitrary number of simple poles as nakajima s quiver varieties this result enables us to solve partially the additive irregular deligne simpson problem

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

the spectral properties of the preconditioned matrix for nonsymmetric saddle point problems in this paper on the basis of matrix splitting two preconditioners are proposed and analyzed for nonsymmetric saddle point problems the spectral property of the preconditioned matrix is studied in detail when the iteration parameter becomes small enough the eigenvalues of the preconditioned matrices will gather into two clusters one is near 0 0 and the other is near 2 0 for the ppss preconditioner no matter whether a is hermitian or non hermitian and for the phss preconditioner when a is a hermitian or real normal matrix numerical experiments are given to illustrate the performances of the two preconditioners

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

block cyclic sor for markov chains with p cyclic infinitesimal generator we introduce a new application of p cyclic iterations for arbitrary p 2 the block sor method for the computation of the steady state distribution of finite markov chains that possess p cyclic infinitesimal generators is considered it is shown that convergence in a sense more general than the usual may be obtained even if the sor iteration violates the usual conditions for semiconvergence necessary and sufficient conditions for convergence in this extended sense are derived they are then applied in the case where the pth power of the associated jacobi matrix of the system to be solved possesses only nonnegative eigenvalues exact convergence intervals and the optimal values are derived for this case in addition to the usual optimal in the interval 1 p p 1 other values that yield convergence in the extended sense are found to achieve the same optimal convergence rate numerical tests indicate that small perturbations of around the optimal value affect the convergence factor much less if these newly introduced optimal values are used

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

conditional probability on mv algebras an appropriate definition of a conditional probability on an mv algebra was an open problem mentioned in riecan and mundici handbook of measure theory north holland amsterdam 2002 we propose some concept of conditional probability state on a complete mv algebra with product its basic properties will be proven and it will also be demonstrated that the conditional probability in classical probability theory is a special case of this definition moreover the paper contains also a discussion of the interpretation of fuzzy sets and conditioning in fuzzy probability theory

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

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

representations for the drazin inverse of the sum p q r s and its applications abstract let p q r and s be complex square matrices and m p q r s a quadruple p q r s is called a pseudo block decomposition of m if pq qp 0 ps sq qr rp 0 and r d s d 0 where r d and s d are the drazin inverses of r and s respectively we investigate the problem of finding formulae for the drazin inverse of m the explicit representations for the drazin inverses of m and p q r are developed under some assumptions as its application some representations are presented for 2 2 block matrices a b 0 c and a b d c where the blocks a and c are square matrices several results of this paper extend the well known representation for the drazin inverse of a b 0 c given by hartwig and shoaf meyer and rose in 1977 an illustrative example is given to verify our new representations

some explicit solutions of the additive deligne simpson problem and their applications abstract in this paper we construct three infinite series and two extra triples e8 and e 8 of complex matrices b c and a b c of special spectral types associated to simpson s classification in amer math soc proc 1 1992 157 and magyar et al classification in adv math 141 1999 97 this enables us to construct fuchsian systems of differential equations which generalize the hypergeometric equation of gauss riemann in a sense they are the closest relatives of the famous equation because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions in all the cases except for e8 we also explicitly construct scalar products such that a b and c are self adjoint with respect to them in the context of fuchsian systems these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions when the eigenvalues of a b and c are real the matrices and the scalar products become real as well we find inequalities on the eigenvalues of a b and c which make the scalar products positive definite as proved by klyachko spectra of three hermitian or real symmetric matrices b c and a b c form a polyhedral convex cone in the space of triple spectra he also gave a recursive algorithm to generate inequalities describing the cone the inequalities we obtain describe non recursively some faces of the klyachko cone

finite precision behavior of stationary iteration for solving singular systems a stationary iterative method for solving a singular system ax b converges for any starting vector if limi gi exists where g is the iteration matrix and the solution to which it converges depends on the starting vector we examine the behavior of stationary iteration in finite precision arithmetic a pertubation bound for singular systems is derived and used to define forward stability of a numerical method a rounding error analysis enables us to deduce conditions under which a stationary iterative method is forward stable or backward stable the component of the forward error in the null space of a can grow linearly with the number of iterations but it is innocuous as long as the iteration converges reasonably quickly as special cases we show that when a is symmetric positive semidefinite the richardson iteration with optimal parameter is forward stable and if a also has unit diagonal and property a then the gauss seidel method is both forward and backward stable two numerical examples are given to illustrate the analysis

the ostrowski reich theorem for sor iterations extensions to the rank deficient case the ostrowski reich theorem gives the necessary and sufficient condition of convergence of the sor method for hermitian and positive definite matrices ortega and plemmons have generalized the theorem to non hermitian matrices for a general nonsingular matrix a they have given necessary and sufficient conditions of convergence of splitting methods and have studied the convergence of the sor method this note is a generalization of the ortega and plemmons theorems to singular matrices some necessary and sufficient conditions of semi convergence for singular matrices are given

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

successive overrelaxation sor and related methods abstract covering the last half of the 20th century we present some of the basic and well known results for the sor theory and related methods as well as some that are not as well known most of the earlier results can be found in the excellent books by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 young iterative solution of large linear systems academic press new york 1971 and berman and plemmons nonnegative matrices in the mathematical sciences siam philadelphia pa 1994 while some of the most recent ones are given in the bibliography of this paper in this survey both the point and the block sor methods are considered for the solution of a linear system of the form ax b where a c n n and b c n 0 some general results concerning the sor and related methods are given and also some more specific ones in cases where a happens to possess some further property e g positive definiteness l m h matrix property p cyclic consistently ordered property etc

new convergence results for alternating methods abstract the alternating methods for solving the large system of linear equations ax b are investigated the convergence and the monotone convergence theories for the alternating method are formulated when the coefficient matrix is an h matrix or a monotone matrix sufficient conditions are established for the induced splitting by the alternating method to be a regular splitting furthermore new comparison theorems which improve previous comparison theorems are proved and several concrete applications are given

a complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite mv chain we consider extensions of the unique additive measure on a finite mv chain to uncertainty measures on its canonical girard algebra extension if the underlying mv chain has more than two non trivial elements in a previous paper we have proved the non existence of strongly additive measure extensions where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so called divisible disjoint unions this negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all mv subalgebras of the canonical girard algebra extension we obtain a characterization of all such mv subalgebras which are in fact mv chains and using this result we come up to a characterization of all weakly additive measure extensions moreover we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions except for underlying mv chains with one or two or four non trivial elements

optimal p cyclic sor our purpose in this paper is to extend p cyclic sor theory to consistent singular systems and to apply the results to the solution of large scale systems arising em e g in queueing network problems in markov analysis markov chains and queueing models lead to structured singular linear systems and are playing an increasing role in the understanding of complex phenomena arising in computer communication and transportation systems for certain important classes of singular problems we develop a convergence theory for p cyclic sor and show how to repartition for optimal convergence results by kontovasilis plemmons and stewart on the concept of convergence of sor in an em extended sense are rigorously analyzed and applied to the solution of periodic markov chains with period p 2

convergence and comparison theorems for double splittings of matrices in this paper some convergence theorems for the double splitting of a monotone matrix or a hermitian positive definite matrix are presented two comparison theorems for two double splittings of a monotone matrix are obtained meanwhile we establish a new sufficient condition for convergence of the gauss seidel double sor method for an h matrix

a logic for reasoning about the probability of fuzzy events in this paper we present the logic fp l n l which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a mv algebra this logic is defined starting from a basic idea exposed by hajek metamathematics of fuzzy logic kluwer dordrecht 1998 two kinds of semantics have been introduced namely the class of weak and strong probabilistic models the main result of this paper is a completeness theorem for the logic fp l n l w r t both weak and strong models we also present two extensions of fp l n l the first one is the logic fp l n rpl obtained by expanding the fp l n l language with truth constants for the rationals in 0 1 while the second extension is the logic fcp l n email protected allowing to reason about conditional states

optimality relationships forp cyclic sor the optimality question for blockp cyclic sor iterations discussed in young and varga is answered under natural conditions on the spectrum of the block jacobi matrix in particular it is shown that repartitioning a blockp cyclic matrix into a blockq cyclic form q p results in asymptotically faster sor convergence for the same amount of work per iteration as a consequence block 2 cyclic sor is optimal under these conditions