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

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

an rqp algorithm using a differentiable exact penalty function for inequality constrained problems in this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function the algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an armijo type line search procedure that avoids the need to evaluate second order derivatives of the problem functions we prove that the algorithm possesses global and superlinear convergence properties numerical results are reported

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

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

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

comparison theorems of preconditioned gauss seidel methods for m matrices in this paper a new preconditioner for the gauss seidel method is proposed for solving linear systems whose coefficient matrix is an m matrix several comparison theorems are shown for the proposed method with several preconditioners it follows from the comparison results that our preconditioner is one of the best preconditioners in the sense of convergence rate finally numerical examples are given to illustrate our theoretical results two conjectures are proposed as well based on our numerical tests

gaussian process surrogates for failure detection an important task of uncertainty quantification is to identify the probability of undesired events in particular system failures caused by various sources of uncertainties in this work we consider the construction of gaussian process surrogates for failure detection and failure probability estimation in particular we consider the situation that the underlying computer models are extremely expensive and in this setting determining the sampling points in the state space is of essential importance we formulate the problem as an optimal experimental design for bayesian inferences of the limit state i e the failure boundary and propose an efficient numerical scheme to solve the resulting optimization problem in particular the proposed limit state inference method is capable of determining multiple sampling points at a time and thus it is well suited for problems where multiple computer simulations can be performed in parallel the accuracy and performance of the proposed method is demonstrated by both academic and practical examples

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

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

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

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

optimal bayesian design by inhomogeneous markov chain simulation we consider decision problems defined by a utility function and an underlying probability model for all unknowns the utility function quantifies the decision maker s preferences over consequences the optimal decision maximizes the expected utility function where the expectation is taken with respect to all unknowns that is future data and parameters in many problems the solution is not analytically tractable for example the utility function might involve moments that can be computed only by numerical integration or simulation also the nature of the decision space i e the set of all possible actions might have a shape or dimension that complicates the maximization the motivating application for this discussion is the choice of a monitoring network when the optimization is performed over the high dimensional set of all possible locations of monitoring stations possibly including choice of the number of locations we propose an approach to optimal bayesian design based on inhomogeneous markov

a recursive quadratic programming algorithm that uses differentiable exact penalty functions in this paper a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied the line search functions used are approximations to fletcher s differentiable exact penalty function global convergence and local superlinear convergence results are proved and some numerical results are given

decision theoretic sensitivity analysis for complex computer models when using a computer model to inform a decision it is important to investigate any uncertainty in the model and determine how that uncertainty may impact on the decision in probabilistic sensitivity analysis model users can investigate how various uncertain model inputs contribute to the uncertainty in the model output however much of the literature focuses only on output uncertainty as measured by variance the decision problem itself often is ignored even though uncertainty as measured by variance may not equate to uncertainty about the optimum decision consequently traditional variance based measures of input parameter importance may not correctly describe the importance of each input we review a decision theoretic framework for conducting sensitivity analysis that addresses this problem because computation of these decision theoretic measures can be impractical for complex computer models we provide efficient computational tools using gaussian processes we give an illustration in the fiel

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

two new modified gauss seidel methods for linear system with m matrices in 2002 h kotakemori et al proposed the modified gauss seidel mgs method for solving the linear system with the preconditioner p i s m a x h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x j comput appl math 145 2002 373 378 since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix the preconditioning effect is not observed on the nth row in the present paper to deal with this drawback we propose two new preconditioners the convergence and comparison theorems of the modified gauss seidel methods with these two preconditioners for solving the linear system are established the convergence rates of the new proposed preconditioned methods are compared in addition numerical experiments are used to show the effectiveness of the new mgs methods

improvements of preconditioned aor iterative method for l matrices in this paper we improve the preconditioned aor method of linear systems considered by evans et al the aor iterative method for new preconditioned linear systems comput appl math 132 2001 461 466 in evans paper the coefficient matrix of linear system have to be an l matrix with a i i 1a i 1 i 0 i 1 n 1 and 0

a successive quadratic programming algorithm with global and superlinear convergence properties this paper presents a successive quadratic programming algorithm for solving general nonlinear programming problems in order to avoid the maratos effect direction finding subproblems are derived by modifying the second order approximations to both objective and constraint functions of the problem we prove that the algorithm possesses global and superlinear convergence properties

a note on the preconditioner pm i sm kotakemori et al h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x journal of computational and applied mathematics 145 2002 373 378 have reported that the convergence rate of the iterative method with a preconditioner p m i s m was superior to one of the modified gauss seidel method under the condition these authors derived a theorem comparing the gauss seidel method with the proposed method however through application of a counter example wen li wen li a note on the preconditioned gaussseidel gs method for linear systems journal of computational and applied mathematics 182 2005 81 91 pointed out that there exists a special matrix that does not satisfy this comparison theorem in this note we analyze the reason why such a to counter example may be produced and propose a preconditioner to overcome this problem

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

a comparison theorem of the improving gauss seidel method for h matrix and its comparison matrix in 1997 kohno et al have reported that the improving modified gauss seidel method which was referred to as the imgs method was superior to the classical iterative methods h matrix is a widely used matrix in this paper we apply this method to h matrix and its comparison matrix with a more general splitting we present the corresponding convergence results and the comparison results of the spectral radius the convergence rates are as well as compared some recent results are improved

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

on upper triangular block weak regular splittings of a singular m matrix abstract let a be in the frobenius normal form and let a m n be a weak regular splitting we call this weak regular splitting an upper triangular block weak regular splitting if m and n are upper triangular block matrices partitioned in the same way as a first we study weak regular splittings of a singular irreducible m matrix and obtain some interesting results we also investigate the index of the spectral radius of the iteration matrix m 1 n when the splitting a m n is assumed to be upper triangular weak regular and we obtain ind 1 m 1 n ind 0 a by this result we improve the main results obtained by h schneider

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

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

bayesian optimal design via interacting particle systems we propose a new stochastic algorithm for bayesian optimal design in nonlinear and high dimensional contexts following peter muller we solve an optimization problem by exploring the expected utility surface through markov chain monte carlo simulations the optimal design is the mode of this surface considered a probability distribution our algorithm relies on a particle method to efficiently explore high dimensional multimodal surfaces with simulated annealing to concentrate the samples near the modes we first test the method on an optimal allocation problem for which the explicit solution is available to compare its efficiency with a simpler algorithm we then apply our method to a challenging medical case study in which an optimal protocol treatment needs to be determined for this case we propose a formalization of the problem in the framework of bayesian decision theory taking into account physicians knowledge and motivations we also briefly review further improvements and alternatives

probabilistic sensitivity analysis of complex models a bayesian approach summary in many areas of science and technology mathematical models are built to simulate complex real world phenomena such models are typically implemented in large computer programs and are also very complex such that the way that the model responds to changes in its inputs is not transparent sensitivity analysis is concerned with understanding how changes in the model inputs influence the outputs this may be motivated simply by a wish to understand the implications of a complex model but often arises because there is uncertainty about the true values of the inputs that should be used for a particular application a broad range of measures have been advocated in the literature to quantify and describe the sensitivity of a model s output to variation in its inputs in practice the most commonly used measures are those that are based on formulating uncertainty in the model inputs by a joint probability distribution and then analysing the induced uncertainty in outputs an approach which is known as probabilistic sensitivity analysis we present a bayesian framework which unifies the various tools of prob abilistic sensitivity analysis the bayesian approach is computationally highly efficient it allows effective sensitivity analysis to be achieved by using far smaller numbers of model runs than standard monte carlo methods furthermore all measures of interest may be computed from a single set of runs

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

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

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

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

a new iterative method for solving linear systems abstract a new iterative method for solving linear systems is derived it can be considered as a modification of the gauss seidel method the modified method can be two times faster than the original one

preconditioned aor iterative methods for m matrices linear systems with m matrices often appear in a wide variety of areas in this paper we give general preconditioners for solving the systems with nonsingular m matrix we show that our preconditioners increase the convergence rate of aor iterative methods numerical results for corresponding preconditioned gmres methods are also given

invariant probabilistic sensitivity analysis in evaluating opportunities investors wish to identify key sources of uncertainty we propose a new way to measure how sensitive model outputs are to each probabilistic input e g revenues growth idiosyncratic risk parameters we base our approach on measuring the distance between cumulative distributions risk profiles using a metric that is invariant to monotonic transformations thus the sensitivity measure will not vary by alternative specifications of the utility function over the output to measure separation we propose using either kuiper s metric or kolmogorov smirnov s metric we illustrate the advantages of our proposed sensitivity measure by comparing it with others most notably the contribution to variance measures our measure can be obtained as a by product of a monte carlo simulation we illustrate our approach in several examples focusing on investment analysis situations this paper was accepted by peter wakker decision analysis

bayesian inference for the uncertainty distribution of computer model outputs we consider a problem of inference for the output of a computationally expensive computer model we suppose that the model is to be used in a context where the values of one or more inputs are uncertain so that the input configuration is a random variable we require to make inference about the induced distribution of the output this distribution is called the uncertainty distribution and the general problem is known to users of computer models as uncertainty analysis to be specific we develop bayesian inference for the distribution and density functions of the model output modelling the output as a function of its inputs as a gaussian process we derive expressions for the posterior mean and variance of the distribution and density functions based on data comprising observed outputs at a sample of input configurations we show that direct computation of these expressions may encounter numerical difficulties we develop an alternative approach based on simulating approximate realisations from the posterior distribution of the output function two examples are given to illustrate our methods

on regular splittings and graph compatible splittings of an m matrix abstract let a m n be a regular splitting of an m matrix we give some equivalent properties which are equivalent to graph compatible splittings we also give some interesting conditions such that regular splittings of an irreducible singular m matrix are convergent

stationary splitting iterative methods for the matrix equation axb c abstract stationary splitting iterative methods for solving a x b c are considered in this paper the main tool to derive our new method is the induced splitting of a given nonsingular matrix a m n by a matrix h such that i h 1 exists convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants in particular for certain surface fitting applications our method is much more efficient than the progressive iterative approximation pia a conventional iterative method often used in computer aided geometric design cagd

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

on the local convergence of quasi newton methods for constrained optimization we consider the application of a general class of quasi newton methods to the solution of the classical equality constrained nonlinear optimization problem specifically we develop necessary and sufficient conditions for the q superlinear convergence of such methods and present a companion linear convergence theorem the essential conditions relate to the manner in which the hessian of the lagrangian function is approximated

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

a general preconditioner for linear complementarity problem with an m matrix abstract in this paper we first present a general preconditioner p for solving linear complementarity problem lcp associated with an m matrix a and a vector f and prove that the lcp a f is equivalent to the lcp p a p f then based on this general preconditioner p two preconditioned ssor methods for solving the linear complementarity problems are proposed we show that this general preconditioner p accelerates the convergence of two ssor methods under the assumption that p a is a z matrix in addition we also give a practically concrete choice for the preconditioner p satisfying aforementioned assumption numerical examples are used to illustrate the theoretical results obtained

comparison results of the preconditioned aor methods for l matrices abstract in this paper we first provide comparison results of several types of the preconditioned aor paor methods for solving a linear system whose coefficient matrix is an l matrix satisfying some weaker conditions than those used in the recent literature next we propose an application of paor method to a preconditioner of krylov subspace method lastly numerical results are provided to show that krylov subspace method with the paor preconditioner performs quite well as compared with the ilu 0 preconditioner

a preconditioned two step modulus based matrix splitting iteration method for linear complementarity problem abstract in this paper a preconditioned two step modulus based matrix splitting iteration method for linear complementarity problems associated with an m matrix is proposed the convergence analysis of the presented method is given in particular we provide a comparison theorem between preconditioned two step modulus based gauss seidel ptmgs iteration method and two step modulus based gauss seidel tmgs iteration method which shows that ptmgs method improves the convergence rate of original tmgs method for linear complementarity problem numerical tested examples are used to illustrate the theoretical analysis

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

transformations and invariance in the sensitivity analysis of computer experiments summary monotonic transformations are widely employed in statistics and data analysis in computer experiments they are often used to gain accuracy in the estimation of global sensitivity statistics however one faces the question of interpreting results that are obtained on the transformed data back on the original data the situation is even more complex in computer experiments because transformations alter the model input output mapping and distort the estimators this work demonstrates that the problem can be solved by utilizing statistics which are monotonic transformation invariant to do so we offer an investigation into the families of metrics either based on densities or on cumulative distribution functions that are monotonic transformation invariant and we introduce a new generalized family of metrics numerical experiments show that transformations allow numerical convergence in the estimates of global sensitivity statistics both invariant and not in cases in which it would otherwise be impossible to obtain convergence however one fully exploits the increased numerical accuracy if the global sensitivity statistic is monotonic transformation invariant conversely estimators of measures that do not have this invariance property might lead to misleading deductions

on the relative importance of input factors in mathematical models safety assessment for nuclear waste disposal this article deals with global quantitative sensitivity analysis of the level e model a computer code used in safety assessment for nuclear waste disposal the level e code has been the subject of two international benchmarks of risk assessment codes and monte carlo methods and is well known in the literature we discuss the level e model with reference to two different settings in the first setting the objective is to find the input factor that drives most of the output variance in the second setting we strive to achieve a preestablished reduction in the variance of the model output by fixing the smallest number of factors the emphasis of this work is on how to define the concept of importance in an unambiguous way and how to assess it in the simultaneous occurrence of correlated input factors and non additive models

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

a successive quadratic programming method that uses new corrections for search directions abstract in this paper a successive quadratic programming method for solving general nonlinear programming problems is proposed and studied in order to avoid the maratos effect we present new corrections for the search directions global convergence and local superlinear convergence results are proved and some numerical results are reported

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

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