iterative solution of fuzzy linear systems abstract linear systems have important applications to many branches of science and engineering in many applications at least some of the parameters of the system are represented by fuzzy rather than crisp numbers so it is immensely important to develop numerical procedures that would appropriately treat general fuzzy linear systems will be denoted by fls and solve them in this paper firstly a general fuzzy linear system using the embedding approach has been investigated and then several well known numerical algorithms for solving system of linear equations such as richardson extrapolated richardson jacobi jor gaus seidel egs sor aor esor ssor ussor ema and msor are extended for solving fls the iterative methods are followed by convergence theorems and the presented algorithms are tested by solving some numerical examples hackbusch noticed that gauss spelling is less correct than gaus w hackbusch iterative solution of large sparse systems of equations springer verlag inc new york 1994 9

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

solvability of the quaternionic monge ampere equation on compact manifolds with a flat hyperkahler metric abstract a quaternionic version of the calabi problem was formulated by alesker and verbitsky 2010 6 it conjectures a solvability of a quaternionic monge ampere equation on a compact hkt manifold hkt stays for hyperkahler with torsion in this paper this problem is solved under the extra assumption that the manifold admits a flat hyperkahler metric compatible with the underlying hypercomplex structure the proof uses the continuity method and a priori estimates

dirichlet duality and the nonlinear dirichlet problem we study the dirichlet problem for fully nonlinear degenerate elliptic equations of the form f hess u 0 on a smoothly bounded domain n in our approach the equation is replaced by a subset f sym2 n of the symmetric n n matrices with f f 0 we establish the existence and uniqueness of continuous solutions under an explicit geometric f convexity assumption on the boundary we also study the topological structure of f convex domains and prove a theorem of andreotti frankel type two key ingredients in the analysis are the use of subaffine functions and dirichlet duality associated to f is a dirichlet dual set f that gives a dual dirichlet problem this pairing is a true duality in that the dual of f is f and in the analysis the roles of f and f are interchangeable the duality also clarifies many features of the problem including the appropriate conditions on the boundary many interesting examples are covered by these results including all branches of the homogeneous monge ampere equation over and equations appearing naturally in calibrated geometry lagrangian geometry and p convex riemannian geometry and all branches of the special lagrangian potential equation 2008 wiley periodicals inc

on sparse block factorization iterative methods a synthetic formalism is proposed for the description of sparse block factorization iterative methods it is used to develop existence convergence and comparison theorems and to compare block factorization schemes against point factorization schemes

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

exploiting sparsity in adaptive filters this paper studies a class of algorithms called natural gradient ng algorithms the least mean square lms algorithm is derived within the ng framework and a family of lms variants that exploit sparsity is derived this procedure is repeated for other algorithm families such as the constant modulus algorithm cma and decision directed dd lms mean squared error analysis stability analysis and convergence analysis of the family of sparse lms algorithms are provided and it is shown that if the system is sparse then the new algorithms will converge faster for a given total asymptotic mse simulations are provided to confirm the analysis in addition bayesian priors matching the statistics of a database of real channels are given and algorithms are derived that exploit these priors simulations using measured channels are used to show a realistic application of these algorithms

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

qualitative analysis of matrix splitting methods abstract qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal stieltjes toeplitz matrices are studied two particular splittings the so called symmetric tridiagonal splittings and the bidiagonal splittings are considered and conditions for qualitative properties like nonnegativity and shape preservation are shown for them special attention is paid to their close relation to the well known splitting techniques like regular and weak regular splitting methods extensions to block tridiagonal matrices are given and their relation to algebraic representations of domain decomposition methods is discussed the paper is concluded with illustrative numerical experiments

positive cones and convergence conditions for iterative methods based on splittings the study of convergence conditions to solve large and sparse linear systems ax b by iterative methods has been discussed by many authors in this paper by using the partial order induced by positivity cone of matrices and conditions on the matrices and splittings we obtain the convergence of the iterative method the usual partial orders of nonnegative matrices and nonnegative definite matrices satisfy the conditions presented for this new order and hence we unify the theory moreover we also show that the new conditions apply to other partial orders as well thus extending the usual convergence theory to a new situation

a survey of the iterative methods for the solution of linear systems by extrapolation relaxation and other techniques abstract for the iterative solution of linear systems of algebraic equations ax b 1 with a c n n x b c n and det a 0 numerous methods exist although a classification of them seems not to be possible one may note that the first step for the construction of an iterative method usually begins with a splitting of a in 1 thus a is written as a m n where det m 0 and m is easily inverted so that 1 is equivalent to x tx c 2 t m 1n c m 1 the discussion will be restricted to the so called linear stationary iterative methods although some nonstationary ones will be mentioned and emphasis will be given to those which fall into the categories of extrapolation relaxation and similar ones for this it will be assumed that the spectrum t of t in 2 is contained in a well defined compact region r whose complement with respect to the complex plane is simply connected and 1 r under these assumptions and for each specific class of methods described each time an attempt will be made to present the optimum one the optimum is the one out of the class of methods for which the sequence of vectors yielded converges asymptotically to the unique solution of 1 as fast as possible

relationship of eigenvalues for usaor iterative method applied to a class of p cyclic matrices a new relationship of eigenvalues between the jacobi iterative matrix and the usaor iterative matrix for a block p cyclic case is derived many known results are its special cases

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

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

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

viscosity solutions to degenerate complex monge ampere equations we develop an alternative approach to degenerate complex monge ampere equations on compact kahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones we generalize to the kahler case a theorem due to dinew and zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous

the continuity and range of the quaternionic monge ampere operator on quaternionic space in this paper we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic monge ampere operator by introducing a new quaternionic capacity we prove a sufficient condition which implies the weak convergence of quaternionic monge ampere measures triangle u j n rightarrow triangle u n we also obtain an equivalent condition of convergence in c n 1 capacity by using methods from xing proc am math soc 124 2 457 467 1996 as an application the range of the quaternionic monge ampere operator is discussed

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

subsolution theorem and the dirichlet problem for the quaternionic monge ampere equation in this paper we obtain the existence of solution to the dirichlet problem for quaternionic monge ampere equation after showing a stability theorem we prove the subsolution theorem for quaternionic monge ampere equation by combining our stability theorem and kolodziej s approach for the complex case

optimal parameters for 2 cyclic aor assume email protected c n x n is a 2 cyclic consistently ordered matrix and j is denoted as its associated block jacobi iteration matrix we consider the 2 cyclic aor method for solving the consistent linear systems ax b in the case that s j 2 is either nonnegative or nonpositive we give detailed discussion and derive definite expressions on optimal parameters and spectral radius by efficient method moreover we give some numerical examples

quasicontinuity and maximality of quaternionic plurisubharmonic functions abstract in this paper we establish the quaternionic versions of several results in the complex pluripotential theory by using the quaternionic closed positive currents which were introduced in 24 we show that quasicontinuity one of the most important properties of complex plurisubharmonic functions holds also for quaternionic plurisubharmonic functions in h n moreover we prove an equivalent characterization of the maximality which is central to the pluripotential theory a locally bounded plurisubharmonic function u is maximal if and only if it satisfies the homogeneous quaternionic monge ampere equation u n 0

letter to the editor a note on the preconditioned gauss seidel gs method for linear systems wen li j comput appl math 182 2005 81 90 asserted that there are some errors in article by hiroshi niki kyouji harada munenori morimoto and michio sakakihara j comput appl math 164 165 2004 587 600 and li presented a new proof for the corresponding results in h niki et al in this paper we point out some errors in li s assertion moreover we show that a new proof presented by li is imperfect

complexity reduction of the nlms algorithm via selective coefficient update this article proposes an algorithm for partial update of the coefficients of the normalized least mean square nlms finite impulse response fir adaptive filter it is shown that while the proposed algorithm reduces the complexity of the adaptive filter it maintains the closest performance to the full update nlms filter for a given number of updates analysis of the mse convergence and steady state performance for independent and identically distributed i i d signals is provided for the extreme case of one update iteration

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

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

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

partial update lms algorithms partial updating of lms filter coefficients is an effective method for reducing computational load and power consumption in adaptive filter implementations this paper presents an analysis of convergence of the class of sequential partial update lms algorithms s lms under various assumptions and shows that divergence can be prevented by scheduling coefficient updates at random which we call the stochastic partial update lms algorithm spu lms specifically under the standard independence assumptions for wide sense stationary signals the s lms algorithm converges in the mean if the step size parameter spl mu is in the convergent range of ordinary lms relaxing the independence assumption it is shown that s lms and lms algorithms have the same sufficient conditions for exponential stability however there exist nonstationary signals for which the existing algorithms s lms included are unstable and do not converge for any value of spl mu on the other hand under broad conditions the spu lms algorithm remains stable for nonstationary signals expressions for convergence rate and steady state mean square error of spu lms are derived the theoretical results of this paper are validated and compared by simulation through numerical examples

viscosity solutions to complex hessian equations we study viscosity solutions to complex hessian equations in the local case we consider a bounded domain in cn the standard kahler form in cn and 1 m n under some suitable conditions on f g we prove that the equation ddc m n m f x n g on admits a unique viscosity solution modulo the existence of subsolution and supersolution if moreover the datum is holder continuous then so is the solution in the global case let x be a compact hermitian homogeneous manifold where is an invariant hermitian metric not necessarily kahler we prove that the equation ddc m n m f x n has a unique viscosity solution under some natural conditions on f

a survey of aor and tor methods abstract this paper makes a survey on the aor and tor methods we present the recent results about convergence and determination of the optimum parameters mainly those obtained by chinese authors

holder continuous solutions to quaternionic monge ampere equations abstract we prove the holder continuity of the unique solution to quaternionic monge ampere equation with densities in l p p 2 on a bounded strictly pseudoconvex domains

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

convergence theory of restricted multiplicative schwarz methods convergence results for the restricted multiplicative schwarz rms method the multiplicative version of the restricted additive schwarz ras method for the solution of linear systems of the form ax b are provided an algebraic approach is used to prove convergence results for nonsymmetric m matrices several comparison theorems are also established these theorems compare the asymptotic rate of convergence with respect to the amount of overlap the exactness of the subdomain solver and the number of domains moreover comparison theorems are given between the rms and ras methods as well as between the rms and the classical multiplicative schwarz method

rational approximations for ill conditioned equation systems in this study we present iterative methods using rational approximations e g pade approximants which work very well for strongly ill conditioned systems in principle all methods of the family are convergent one type of those methods has the advantage that their convergence behavior is very fast without additional a priori information on the optimal relaxation parameter

the quaternionic monge ampere operator and plurisubharmonic functions on the heisenberg group many fundamental results of pluripotential theory on the quaternionic space mathbb h n are extended to the heisenberg group we introduce notions of a plurisubharmonic function the quaternionic monge ampere operator differential operators d 0 and d 1 and a closed positive current on the heisenberg group the quaternionic monge ampere operator is the coefficient of d 0d 1u n we establish the chern levine nirenberg type estimate the existence of quaternionic monge ampere measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic monge ampere operator unlike the tangential cauchy riemann operator overline partial b on the heisenberg group which behaves badly as partial b overline partial b ne overline partial b partial b the quaternionic counterpart d 0 and d 1 satisfy d 0d 1 d 1d 0 this is the main reason that we have a good theory for the quaternionic monge ampere operator than partial b overline partial b n

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

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

viscosity solutions to second order partial differential equations on riemannian manifolds we prove comparison uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations f x u du d 2 u 0d ef ined on afi nite dimensional riemannian manifold m finest results with hypothesis that require the function f to be degenerate elliptic that is nonincreasing in the second order derivative variable and uniformly continuous with respect to the variable x are obtained under the assumption that m has nonnegative sectional curvature while if one additionally requires f to depend on d 2 u in a uniformly continuous manner then comparison results are established with no restrictive assumptions on curvature

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

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

a viscosity approach to the dirichlet problem for complex monge ampere equations we present a viscosity approach to the dirichlet problem for the complex monge ampere equation det u bar j k f x u our approach differs from previous viscosity approaches to this equation in several ways it is based on contact set techniques the alexandrov bakelman pucci estimate on extensive applications of sup inf convolutions and on a relation between real and complex hessians more specifically this paper includes a notion of viscosity solutions a comparison principle and a solvability theorem the equivalence between viscosity and pluripotential solutions an estimate of the modulus of continuity of a solution in terms of that of a given subsolution and of the right hand side f and an alexandrov bakelman pucci type of l estimate

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

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

on perron frobenius property of matrices having some negative entries we extend the theory of nonnegative matrices to the matrices that have some negative entries we present and prove some properties which give us information when a matrix possesses a perron frobenius eigenpair we apply also this theory by proposing the perron frobenius splitting for the solution of the linear system ax 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 given and proved

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

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

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

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

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 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

on the convergence of some generalized iterative methods abstract this paper is concerned with the generalized accelerated overrelaxation gaor method which constitutes a generalization of the basic iterative methods for the solution of linear systems a number of new theoretical results are presented concerning the convergence theory of the gaor method and special cases of it much attention is given to linear systems with positive definite coefficient matrices although the problem of determining the various optimum parameters of the gaor method in the general case is very difficult to solve theoretically and is still an open one the numerical examples which are presented in this paper show that a suitable exploitation of it may give much better results than the other basic iterative methods

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

hessian measures ii in our previous paper on this topic we introduced the notion of k hessian measure associated with a continuous k convex function in a domain q in euclidean n space k 1 n and proved a weak continuity result with respect to local uniform convergence in this paper we consider k convex functions not necessarily continuous and prove the weak continuity of the associated k hessian measure with respect to convergence in measure the proof depends upon local integral estimates for the gradients of k convex functions

viscosity solutions to quaternionic monge ampere equations abstract quaternionic monge ampere equations have recently been studied intensively using methods from pluripotential theory we present an alternative approach by using the viscosity methods we study the viscosity solutions to the dirichlet problem for quaternionic monge ampere equations d e t f f q f with boundary value f g on here is a bounded domain on the quaternionic space h n g c and f q t is a continuous function on r r which is non decreasing in the second variable we prove a viscosity comparison principle and a solvability theorem moreover the equivalence between viscosity and pluripotential solutions is shown

dirichlet duality and the nonlinear dirichlet problem on riemannian manifolds in this paper we study the dirichlet problem for fully nonlin ear second order equations on a riemannian manifold as in our previous paper hl4 we define equations via closed subsets of the 2 jet bundle where each equation has a natural dual equation basic existence and uniqueness theorems are established in a wide variety of settings however the emphasis is on starting with a constant coefficient equation as a model which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model for example this covers all branches of the homogeneous complex monge ampere equation on an almost complex hermitian manifold x in general for an equation f on a manifold x and a smooth domain x we give geometric conditions which imply that the dirichlet problem on is uniquely solvable for all continuous boundary functions we begin by introducing a weakened form of comparison which has the advantage that local implies global we then introduce two fundamental concepts the first is the notion of a monotonicity cone m for f if x carries a global m subharmonic function then weak comparison implies full compar ison the second notion is that of boundary f convexity which is defined in terms of the asymptotics of f and is used to define bar riers in combining these notions the dirichlet problem becomes uniquely solvable as claimed this article also introduces the notion of local affine jet equiva lence for subequations it is used in treating the cases above but gives results for a much broader spectrum of equations on manifolds including inhomogeneous equations and the calabi yau equation on almost complex hermitian manifolds a considerable portion of the paper is concerned with specific examples they include a wide variety of equations which make sense on any riemannian manifold and many which hold univer sally on almost complex or quaternionic hermitian manifolds or topologically calibrated manifolds

leaky lms algorithm mse analysis for gaussian data despite the widespread usage of the leaky lms algorithm there has been no detailed study of its performance this paper presents an analytical treatment of the mean square error mse performance for the leaky lms adaptive algorithm for gaussian input data the common independence assumption regarding w n and x n is also used exact expressions that completely characterize the second moment of the coefficient vector and algorithm steady state excess mse are developed rigorous conditions for mse convergence are also established analytical results are compared with simulation and are shown to agree well

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

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

hessian measures iii abstract in this paper we continue previous investigations into the theory of hessian measures we extend our weak continuity result to the case of mixed k hessian measures associated with k tuples of k convex functions on domains in euclidean n space k 1 2 n applications are given to capacity quasicontinuity and the dirichlet problem with inhomogeneous terms continuous with respect to capacity or combinations of dirac measures