phase diagrams of random field ising systems we show that the random field ising model may be tuned to obtain two distinct scenarios of phase diagram topology explicit evidence for this is presented in a numerically exact analysis in three dimensions and on a bethe lattice which allows us to investigate the effects of temperature coordination number and asymmetry in the field distribution

random field ising model on the bethe lattice low temperature series expansions have been derived for the random field ising model with a delta function distribution on a bethe lattice by two independent methods a the finite cluster method which uses graph embeddings and appropriate weighting functions b the use of a recursion relation specific to the bethe lattice numerical values have been evaluated when the coordination number q 3 4 and the coefficients analysed to assess critical behaviour for small fields and temperatures near to tco the critical exponent of the magnetisation seems to retain its mean field value but there is clear evidence of a change in critical behaviour at some point on the critical curve it is argued that when q 3 a tricritical point is indicated as found by aharony in his mean field solution

block sor preconditioned projection methods for kronecker structured markovian representations kronecker structured representations are used to cope with the state space explosion problem in markovian modeling and analysis currently an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of markov chains mcs underlying such representations this paper proposes a block successive overrelaxation bsor preconditioner for hierarchical markovian models hmms throughout the paper the hmm acronym stands for hierarchical markovian models and should not be confused with the hmm that is sometimes used for hidden markov models that are composed of multiple low level models and a high level model that defines the interaction among low level models the kronecker structure of an hmm yields nested block partitionings in its underlying continuous time mc which may be used in the bsor preconditioner the computation of the bsor preconditioned residual in each iteration of a preconditioned projection method becomes the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning the proposed bsor preconditioner solves these systems using sparse lu or real schur factors of diagonal blocks the fill in of sparse lu factorized diagonal blocks is reduced using the column approximate minimum degree colamd ordering a set of numerical experiments is presented to show the merits of the proposed bsor preconditioner

circulant preconditioners for stochastic automata networks stochastic automata networks sans are widely used in modeling communication systems manufacturing systems and computer systems the san approach gives a more compact and efficient representation of the network when compared to the stochastic petri nets approach to find the steady state distribution of sans it requires solutions of linear systems involving the generator matrices of the sans very often direct methods such as the lu decomposition are inefficient because of the huge size of the generator matrices an efficient algorithm should make use of the structure of the matrices iterative methods such as the conjugate gradient methods are possible choices however their convergence rates are slow in general and preconditioning is required we note that the milu and minv based preconditioners are not appropriate because of their expensive construction cost in this paper we consider preconditioners obtained by circulant approximations of sans they have low construction cost and can be inverted efficiently we prove that if only one of the automata is large in size compared to the others then the preconditioned system of the normal equations will converge very fast numerical results for three different sans solved by cgs are given to illustrate the fast convergence of our method

iterative aggregation disaggregation techniques for nearly uncoupled markov chains iterative aggregation disaggregation methods provide an efficient approach for computing the stationary probability vector of nearly uncoupled also known as nearly completely decomposable markov chains three such methods that have appeared in the literature recently are considered and their similarities and differences are outlined specifically it is shown that the method of takahashi corresponds to a modified block gauss seidel step and aggregation whereas that of vantilborgh corresponds to a modified block jacobi step and aggregation the third method that of koury et al is equivalent to a standard block gauss seidel step and iteration for each of these methods a lemma is established which shows that the unique fixed point of the iterative scheme is the left eigenvector corresponding to the dominant unit eigenvalue of the stochastic transition probability matrix in addition conditions are established for the convergence of the first two of these methods convergence conditions for the third having already been established by stewart et al all three methods are shown to have the same asymptotic rate of convergence

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

technical note a markov chain partitioning algorithm for computing steady state probabilities this paper presents a partitioning algorithm for recursively computing the steady state probabilities for a finite irreducible markov chain or a markov process the algorithm contains a matrix reduction routine followed by a vector enlargement routine the matrix reduction routine repeatedly partitions the transition matrix for the markov chain creating a sequence of smaller reduced transition matrices the vector enlargement routine computes the components of the steady state probability vector by starting with the smallest reduced matrix and working sequentially toward the original transition matrix this procedure produces an exact solution for the steady state probabilities no special structure is required for the markov chain in theory the procedure imposes no limit on the size of the largest markov chain to which the partitioning algorithm can be applied in practice roundoff errors may require modifications to the method

regenerative analysis and steady state distributions for markov chains we apply regenerative theory to derive certain relations between steady state probabilities of a markov chain these relations are then used to develop a numerical algorithm to find these probabilities the algorithm is a modification of the gauss jordan method in which all elements used in numerical computations are nonnegative as a consequence the algorithm is numerically stable

newton s method solver for high speed viscous separated flowfields a new method for calculating the 2 d laminar navier stokes equations is presented the method uses newton s method for nonlinear systems of equations to find steady state solutions the navier stokes equations are approximated by finite differences using roe s flux difference splitting second order accuracy is attained by using spekreijse s interpolation with van albada s limiter the exact newton s method jacobian matrix is inverted by using recent sparse matrix routines the symbolic manipulation package macsyma is used to develop and write the fortran code

low temperature behaviour of the random field ising model a relationship is established between the ising model in a random positive and negative magnetic field on a lattice with p 1 2 at t 0 each percolation cluster gives rise to a first order transition at a different value of the field there is a significant difference in behaviour between lattices for which pc 1 2 which do not by considering the bethe lattice and allowing the coordination number to become large the results of the mean field approximation are reproduced the above considerations do not apply to a gaussian distribution of fields and the absence of a first order transition can be understood in this case since large clusters overturn for small fields there are clear indications of metastable behaviour

circulant approximation for preconditioning in stochastic automata networks stochastic automata networks sans are widely used in modeling practical systems such as queueing systems communication systems and manufacturing systems for the performance analysis purposes one needs to calculate the steady state distributions of sans usually the steady state distributions have no close form solutions and cannot be obtained efficiently by direct methods such as lu decomposition due to the huge size of the generator matrices an efficient numerical method should make use of the tensor structure of sans generator matrices the generalized conjugate gradient cg methods are possible choices though their convergence rates are slow in general to speed up the convergence rate preconditioned cg methods are considered in this paper in particular circulant based preconditioners for the sans are constructed the preconditioners presented in this paper are easy to construct and can be inverted efficiently numerical examples of practical sans are also given to illustrate the fast convergence rate of the method

toeplitz circulant preconditioners for toeplitz systems and their applications to queueing networks with batch arrivals the preconditioned conjugate gradient method is employed to solve toeplitz systems t n bf x bf b where the generating functions of the n by n toeplitz matrices t n are functions with zeros in this case circulant preconditioners are known to give poor convergence whereas band toeplitz preconditioners offer only linear convergence and can handle only real valued functions with zeros of even orders we propose here preconditioners which are products of band toeplitz matrices and circulant matrices the band toeplitz matrices are used to cope with the zeros of the given generating function and the circulant matrices are used to speed up the convergence rate of the algorithm our preconditioner can handle complex valued functions with zeros of arbitrary orders we prove that the preconditioned toeplitz matrices have singular values clustered around 1 for large n we apply our preconditioners to solve the stationary probability distribution vectors of markovian queueing models with batch arrivals we show that if the number of servers is fixed independent of the queue size n then the preconditioners are invertible and the preconditioned matrices have singular values clustered around 1 for large n numerical results are given to illustrate the fast convergence of our methods

on the convergence of infinite products of matrices abstract in the normed metric space of n square complex matrices convergence of i ai implies convergence of i u i a i with u i 1 i if and only if i q u i converges for any integer q the limit of i u i a i may be considered a function u a of the summable sequence a a i the function u is parametrized by u u i the sequences of summable matrices a a i constitute a normed metric space e with norm a e i a i the family of functions u a i u i a i is equicontinuous at a if a e u i

on the growth of infinite products of slowly varying primitive matrices abstract let a t be a sequence of bounded nonnegative matrices each with perron root t satisfying a condition of uniform primitivity and having nonzero entries bounded away from 0 in a generalization of a result on powers of primitive matrices it is shown that if the matrices a t vary slowly enough i e a t 1 a t e for all t and some small e then as soon as the entries p i j t of the backward product a t a t 1 a 1 are positive the ratios p i j t p i j t 1 are equal to t h i j t where for all t 0 each h i j t 0 is small in a precisely quantified manner the larger t 0 is and the slower the probability normed perron vectors v t of a t have changed in the recent past preceding t 0 the smaller h i j t 0 will be other similar results are derived the resulting approximation p i j t p i j t 1 t is illustrated with two numerical examples

relative error bounds for the lu decomposition via the gth algorithm recently the author showed that the grassmann taksar heyman gth algorithm computes the steady state distribution of a finite state markov chain with low relative error here it is shown that the lu decomposition computed in the course of the gth algorithm also has low relative error the proof requires a refinement of the methods used in the earlier paper

iterative methods for overflow queuing models ii preconditioned conjugate gradient methods are employed to find the steady state probability distribution of markovian queuing networks that have overflow capacity different singular preconditioners that can be handled by separation of variables are discussed the resulting preconditioned systems are nonsingular numerical results show that the number of iterations required for convergence grows very slowly with the queue sizes

a theorem on products of matrices a new result on products of matrices is proved in the following theorem let mi i 1 2 be a bounded sequence of square matrices and k be the l u b of the spectral radii mi then for any positive number e there is a constant a and an ordering p j j 1 2 of the matrices such that j 1nmp j a k e n n 1 2 the ordering is well defined by p j a one to one mapping on the set of positive integers in general the inequality does not hold for any ordering p j a counterexample is provided however some sufficient conditions are given for the result to remain true irrespective of the order of the matrices

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

an iterative aggregation disaggregation algorithm for solving linear equations a general procedure is given for solving large sets of linear equations by first rewriting them in a form suitable for aggregation of both the variables and equations followed by disaggregation a computational algorithm which iteratively aggregates and disaggregates is shown to converge geometrically to the exact solution provided the original problem has a structure suitable for such aggregation the algorithm exhibits fast computation times small main memory requirements and robustness to the starting point a rigorous foundation for aggregation and disaggregation is provided by the equations employed by this algorithm

triangular and skew symmetric splitting method for numerical solutions of markov chains in this paper a theorem is presented to indicate that there exists a nonnegative constant e 0 such that the matrix a q email protected is a positive definite matrix where email protected r n x n is an identity matrix and q email protected r n x n is a matrix with positive diagonal and nonpositive off diagonal elements then a class of triangular and skew symmetric splitting tss iteration method is applied to solve the positive definite linear system ax b for obtaining the stationary probability vector of an irreducible markov chain theoretical analyses show that the tss iteration method converges unconditionally to the unique solution of the linear system with the upper bound of its contraction factor dependent only on the spectrum of the triangular part and independent of the eigenvectors of the matrices involved moreover the inexact triangular and skew symmetric splitting itss iteration method which employs certain krylov subspace methods as the inner iteration processes at each step of the outer tss iteration method is proposed to accelerate the convergence of the tss iteration method numerical experiments are used to illustrate the effectiveness of the tss and itss iteration methods

optimal control of production rate in a failure prone manufacturing system we address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost where certain cost rates are specified for both positive and negative inventories and there is a constant demand rate for the commodity produced the underlying theoretical problem is the optimal control of a continuous time system with jump markov disturbances with an infinite horizon discounted cost criterion we use two complementary approaches first proceeding informally and using a combination of stochastic coupling linear system arguments stable and unstable eigenspaces renewal theory parametric optimization etc we arrive at a conjecture for the optimal policy then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides singularity of the optimal control problem smoothness and validity of the dynamic programming equation etc to give a rigorous proof of optimality of the conjectured policy it is hoped that both approaches will find uses in other such problems also we obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number which we call the optimal inventory level if the current inventory level exceeds the optimal one should not produce at all if less one should produce at the maximum rate while if exactly equal one should produce exactly enough to meet demand we also give a simple explicit formula for the optimal inventory level

multigrid algorithm from cyclic reduction for markovian queueing networks a multigrid method based on cyclic reduction strategy is proposed to solve huge nonsymmetric singular linear systems arising from markovian queueing networks a simple way to construct the matrix dependent prolongation and restriction operators is presented in this paper numerical results for multiple queues are given to illustrate the efficiency and robustness of our methods

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

strong ergodicity for perturbed dual semigroups and application to age dependent population dynamics abstract in this paper we first summarize the perturbation theory in a sun reflexive banach space next we shall introduce the concept of strong ergodicity for the evolutionary system and then we investigate conditions under which the evolutionary system generated by the perturbation of a c 0 semigroup in a sun reflexive banach space becomes strongly ergodic subsequently we construct the evolutionary system corresponding to lotka s renewal equation and apply it to prove strong ergodicity of the age structured population with time dependent vital rates finally we study the controllability of the age structured population controlled by changing its total fertility rate

smoothed aggregation multigrid for markov chains a smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse markov chain it is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for markov chains that have been proposed in the literature the proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems supplemented with a new lumping technique that assures well posedness of the coarse level problems the coarse level operators are singular m matrices on all levels resulting in strictly positive coarse level corrections on all levels numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems both when geometric and algebraic aggregation strategies are used

compact global attractors of discrete inclusions abstract the paper is dedicated to the study of the problem of the existence of compact global attractors of discrete inclusions and to the description of its structure we consider a family of continuous mappings of a metric space w into itself and w f i i i is the family of discrete dynamical systems on the metric space w we consider a discrete inclusion 1 u t 1 f u t associated with m f i i i where f u f u f m for all u w we give sufficient conditions the family of maps m is contracting in the extended sense for the existence of a compact global attractor of 1 if the family m consists of a finite number of maps then the corresponding compact global attractor is chaotic we study this problem in the framework of non autonomous dynamical systems cocyles

numerical methods in markov chain modeling this paper describes and compares several methods for computing stationary probability distributions of markov chains the main linear algebra problem consists of computing an eigenvector of a sparse nonsymmetric matrix associated with a known eigenvalue it can also be cast as a problem of solving a homogeneous singular linear system we present several methods based on combinations of krylov subspace techniques single vector power iteration relaxation procedures and acceleration techniques we compare the performance of these methods on some realistic problems

lumpable continuous time stochastic automata networks abstract the generator matrix of a continuous time stochastic automata network san is a sum of tensor products of smaller matrices which may have entries that are functions of the global state space this paper specifies easy to check conditions for a class of ordinarily lumpable partitionings of the generator of a continuous time san in which aggregation is performed automaton by automaton when there exists a lumpable partitioning induced by the tensor representation of the generator it is shown that an efficient aggregation iterative disaggregation algorithm may be employed to compute the steady state distribution the results of experiments with two san models show that the proposed algorithm performs better than the highly competitive block gauss seidel in terms of both the number of iterations and the time to converge to the solution

the numerical solution of stochastic automata networks abstract stochastic automata networks san s have recently received attention in the literature as an efficient means of modelling parallel systems such as communicating processes concurrent processors shared memory etc the advantage that the san approach has over generalized stochastic petri nets and indeed over any markovian analysis that requires the generation of a transition matrix is that its representation remains compact even as the number of states in the underlying markov chain begins to explode our concern in this paper is with the numerical issues that are involved in solving san networks we introduce stochastic automata and consider the numerical difficulties that result from their interaction we examine how the product of a vector with a compact san descriptor may be formed for this operation is the basis of all iterative solution methods we describe possible solution methods including the power method the method of arnoldi and gmres and show that the two latter methods greatly out perform the power method the method usually used in conjunction with stochastic automata networks finally we consider one possible means of preconditioning but conclude that further research is needed

circulant preconditioners for failure prone manufacturing systems this paper studies the application of preconditioned conjugate gradient methods in solving for the steady state probability distribution of manufacturing systems we consider the optimal hedging policy for a failure prone one machine system the machine produces one type of product and its demand has finite batch arrival the machine states and the inventory levels are modeled as markovian processes we construct the generator matrix for the machine inventory system the preconditioner is constructed by taking the circulant approximation of the near toeplitz structure of the generator matrix we prove that the preconditioned linear system has singular values clustered around one when the number of inventory levels tends to infinity hence conjugate gradient methods will converge very fast when applied to solving the preconditioned linear system numerical examples are given to verify our claim the average running cost for the system can be written in terms of the steady state probability distribution the optimal hedging point can then be obtained by varying different values of the hedging point

exact algorithms for singular tridiagonal systems with applications to markov chains in this paper we propose two exact algorithms to solve the steady state probability distributions of irreducible markov chains whose generator matrices have tridiagonal structure the first exact algorithm is based on divide and conquer procedure and the second one is a parallel algorithm examples on random walks and queuing networks are given to demonstrate the usefulness of the algorithms

stochastic dynamic job shops and hierarchical production planning this paper presents an asymptotic analysis of hierarchical production planning in a general manufacturing system consisting of a network of unreliable machines producing a variety of products the concept of a dynamic job shop is introduced by interpreting the system as a directed graph and the structure of the system dynamics is characterized for its use in the asymptotic analysis the optimal control problem for the system is a state constrained problem since the number of parts in any buffer between any two machines must remain nonnegative a limiting problem is introduced in which the stochastic machine capacities are replaced by corresponding equilibrium mean capacities as the rate of change in machine states approaches infinity the value function of the original problem is shown to converge to that of the limiting problem and the convergence rate is obtained furthermore near optimal controls for the original problem are constructed from near optimal controls of the limiting problem and an error estimate is obtained on the near optimality of the constructed controls

boundary value methods for solving transient solutions of markovian queueing networks in this paper we consider boundary value methods bvms for finding transient solutions of markovian queueing networks algebraic multigrid amg methods with modified restriction operator are applied to solve the resulting system of linear equations numerical examples are given to demonstrate the efficiency of our proposed method