entrywise relative perturbation bounds for exponentials of essentially non negative matrices a real square matrix is said to be essentially non negative if all of its off diagonal entries are non negative we establish entrywise relative perturbation bounds for the exponential of an essentially non negative matrix our bounds are sharp and contain a condition number that is intrinsic to the exponential function as an application we study sensitivity of continuous time markov chains

weight enumerators of extremal singly even 60 30 12 codes conway and sloane 1990 have listed the possible weight enumerators for extremal self dual codes up to length 72 in this correspondence we construct extremal singly even self dual 60 30 12 codes whose weight enumerator does not appear in this list in addition we present the possible weight enumerators for extremal self dual codes of length 60

sensitivity analysis of discrete markov chains via matrix calculus abstract markov chains are often used as mathematical models of natural phenomena with transition probabilities defined in terms of parameters that are of interest in the scientific question at hand sensitivity analysis is an important way to quantify the effects of changes in these parameters on the behavior of the chain many properties of markov chains can be written as simple matrix expressions and hence matrix calculus is a powerful approach to sensitivity analysis using matrix calculus we derive the sensitivity and elasticity of a variety of properties of absorbing and ergodic finite state chains for absorbing chains we present the sensitivities of the moments of the number of visits to each transient state the moments of the time to absorption the mean number of states visited before absorption the quasistationary distribution and the probabilities of absorption in each of several absorbing states for ergodic chains we present the sensitivity of the stationary distribution the mean first passage time matrix the fundamental matrix and the kemeny constant we include two examples of application of the results to demographic and ecological problems

markov chain sensitivity measured by mean first passage times abstract the purpose of this article is to present results concerning the sensitivity of the stationary probabilities for an n state time homogeneous irreducible markov chain in terms of the mean first passage times in the chain

exponential synchronizationlike criterion for state dependent impulsive dynamical networks this paper focuses on the problem of the exponential synchronizationlike criteria for state dependent impulsive dynamical networks sidns two types of sufficient conditions which are applied to ensure every solution intersecting each impulsive surface exactly once are derived for each type of collision conditions combining with comparison principle and inequality techniques some sufficient conditions are obtained to ensure local exponential synchronizationlike for sidn moreover a quiet different impulsive strategy concerning the trigger rules of impulsive instants is proposed finally an example is given to demonstrate the effectiveness of our results

on a direct method for the solution of nearly uncoupled markov chains this note is concerned with the accuracy of the solution of nearly uncoupled markov chains by a direct method based on the lu decomposition it is shown that plain gaussian elimination may fail in the presence of rounding errors a modification of gaussian elimination with diagonal pivoting and correction of small pivots is proposed and analyzed it is shown that the accuracy of the solution is affected by two condition numbers associated with aggregation and the coupling respectively

the effect of network topology on the stability of discrete state models of genetic control boolean networks have been proposed as potentially useful models for genetic control an important aspect of these networks is the stability of their dynamics in response to small perturbations previous approaches to stability have assumed uncorrelated random network structure real gene networks typically have nontrivial topology significantly different from the random network paradigm to address such situations we present a general method for determining the stability of large boolean networks of any specified network topology and predicting their steady state behavior in response to small perturbations additionally we generalize to the case where individual genes have a distribution of expression biases and we consider a nonsynchronous update as well as extension of our method to non boolean models in which there are more than two possible gene states we find that stability is governed by the maximum eigenvalue of a modified adjacency matrix and we test this result by comparison with numerical simulations we also discuss the possible application of our work to experimentally inferred gene networks

new perturbation bounds for denumerable markov chains this paper is devoted to perturbation analysis of denumerable markov chains bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain where the bounds are given by the weighted supremum norm in addition bounds for the perturbed stationary probabilities are established furthermore bounds on the norm of the asymptotic decomposition of the perturbed stationary distribution are provided where the bounds are expressed in terms of the norm of the ergodicity coefficient or the norm of a special residual matrix refinements of our bounds for doeblin markov chains are considered as well our results are illustrated with a number of examples

combinatorial fiber tracking of the human brain this paper presents a novel fiber tracking algorithm termed combinatorial tracking which uses stochastic process modeling and global optimization algorithm for tractography combinatorial tracking is a probabilistic tracking algorithm that transforms the brain s white matter into a grid in which each voxel has 26 weighted connections with adjacent voxels we model the random walk on this graph using a markov chain model and suggest two approaches for fiber reconstruction in the first approach we find the most probable paths between two voxels with prior connectivity knowledge using a shortest path algorithm in the second approach the all pairs mean first passage time mfpt matrix m or hitting time as referred to in the spectral graph theory literature is calculated analytically we suggest that m can be interpreted as a global connectivity matrix and use it for fiber reconstruction we also introduce a simulation framework that can be used to calculate specific elements of the matrix m and show how it can be employed to select the target of a fiber in a high resolution diffusion tensor imaging dti dataset because any source and any target voxel can be connected combinatorial tracking permits true connectivity analysis overcoming the limitations of conventional tracking especially stopping criteria e g low fa we applied combinatorial tracking to a standard dti dataset and demonstrated the reconstruction of the cortico thalamic pathway the pyramidal decussation and the medial cerebellar peduncle fibers while the dti ellipsoid served as input for the algorithms any diffusion imaging based orientation density function odf can be used this framework can potentially turn diffusion imaging tractography into a true connectivity measure

extremal self dual codes with an automorphism of order 2 a method to design binary self dual codes with an automorphism of order two without fixed points is presented extremal self dual codes with lengths 40 42 44 54 58 68 are constructed many of them have weight enumerators for which extremal codes were previously not known to exist

computing singular values of diagonally dominant matrices to high relative accuracy for a row diagonally dominant matrix if all of its off diagonal entries and its diagonally dominant parts which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off diagonal entries in that row are accurately known we develop an algorithm that computes all the singular values including zero ones if any with relative errors in the order of the machine precision when the matrix is also symmetric with positive diagonals i e a symmetric positive semi definite diagonally dominant matrix our algorithm computes all eigenvalues to high relative accuracy rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm

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

multiple eigenvalue sensitivity analysis abstract this paper surveys the development of the sensitivity analysis of multiple eigenvalues and the associated invariant subspaces of two kinds of matrix eigenvalue problems in which the matrices are analytically dependent on several parameters

entrywise perturbation theory for diagonally dominant m matrices with applications this paper introduces a new perspective on the study of computational problems related to diagonally dominant m matrices by establishing some new entrywise perturbation results if a diagonally dominant m matrix is perturbed in a way that each off diagonal entry and its row sums i e the quantities of diagonal dominance have small relative errors we show that its determinant cofactors each entry of the inverse and the smallest eigenvalue all have small relative errors the error bounds are given and they do not depend on any condition number applying this result to the studies of electrical circuits and tail probabilities of a queue whose embedded markov chains is of gi m 1 type we discuss the relative sensitivity of the operating speed of circuits and of the percentile of the queue length respectively

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

entrywise perturbation theory and error analysis for markov chains grassmann taksar and heyman introduced a variant of gaussian climination for computing the steady state vector of a markov chain in this paper we prove that their algorithm is stable and that the problem itself is well conditioned in the sense of entrywise relative error thus the algorithm computes each entry of the steady state vector with low relative error even the small steady state probabilities are computed accurately the key to our analysis is to focus on entrywise relative error in both the data and the computed solution rather than making the standard assessments of error based on norms our conclusions do not depend on any condition numbers for the problem

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

on the first and second order derivatives of the perron vector abstract in a previous work 5 the authors developed formulas for the second order partial derivatives of the perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix these formulas which involve the group generalized inverse of an associated m matrix were used to investigate the concavity and convexity of the perron root as a function of the entries the authors now combine the above results together with an approach taken in an earlier joint paper 6 of the second author with l elsner and c johnson and they develop formulas for the second order derivatives of an appropriately normalized perron vector with respect to the matrix entries which again are given in terms the group generalized inverse of an associated m matrix convexity properties of the perron vector as a function of the entries of the matrix are then examined in addition formulas for the first derivative of the perron vector resulting from different normalizations of this eigenvector are also given a by product of one of these formulas yields that the group generalized inverse of a singular and irreducible m matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

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

newton noda iteration for finding the perron pair of a weakly irreducible nonnegative tensor we present a newton noda iteration nni for computing the perron pair of a weakly irreducible nonnegative mth order tensor mathscr a a by combining the idea of newton s method with the idea of the noda iteration the method requires the selection of a positive parameter theta k k in the kth iteration and produces a scalar sequence approximating the spectral radius of mathscr a a and a positive vector sequence approximating the perron vector we propose a halving procedure to determine the parameters theta k k starting with theta k 1 k 1 for each k such that the scalar sequence is monotonically decreasing convergence of this sequence to the spectral radius of mathscr a a and convergence of the vector sequence to the perron vector is guaranteed for any initial positive unit vector as long as the sequence theta k k so chosen is bounded below by a positive constant in this case we always have theta k 1 k 1 near convergence and the convergence is quadratic very often the halving procedure will return theta k 1 k 1 i e no halving is actually used for each k if the tensor is semisymmetric m ge 4 m 4 and theta k 1 k 1 then the computational work in the kth iteration of nni is roughly the same as that for one iteration of the ng qi zhou algorithm which is linearly convergent for the smaller class of weakly primitive tensors

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

synchronization of dynamical hypernetworks dimensionality reduction through simultaneous block diagonalization of matrices we present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems we are able to reduce the dimensionality of the problem by using simultaneous block diagonalization of matrices we obtain necessary and sufficient conditions for stability of the synchronous solution in terms of a set of lower dimensional problems and test the predictions of our low dimensional analysis through numerical simulations under certain conditions this technique may yield a substantial reduction of the dimensionality of the problem for example for a class of dynamical hypernetworks analyzed in the paper we discover that arbitrarily large networks can be reduced to a collection of subsystems of dimensionality no more than 2 we apply our reduction technique to a number of different examples including the class of undirected unweighted hypermotifs with 3 nodes

finding equilibrium probabilities of qbd processes by spectral methods when eigenvalues vanish abstract in this paper we discuss the use of spectral or eigenvalue methods for finding the equilibrium probabilities of quasi birth death processes for the case where some eigenvalues are zero since this leads to multiple eigenvalues at zero a difficult problem to analyze we suggest to eliminate such eigenvalues to accomplish this the dimension of the largest jordan block must be established and some initial equations must be eliminated the method is demonstrated by two examples one dealing with a tandem queue the other one with a shorter queue problem

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

group inverses of m matrices associated with nonnegative matrices having few eigenvalues abstract we continue here earlier investigations of the structure of group generalized inverses a of singular and irreducible m matrices a ri m where m is an n n nonnegative and irreducible matrix whose spectral radius is r and where m is assumed to belong to one of several special classes of nonnegative matrices we now focus on cases where m has only a few distinct eigenvalues we are particularly interested in instances of such matrices m which cause all or almost all of the off diagonal entries of the corresponding a to be nonpositive we then apply our results to the study of the sign pattern of the off diagonal entries of the group generalized inverse of a ri m where m comes from one of the following families of nonnegative matrices 1 the magic square of order n 4k generated by matlab 2 the doubly regular tournament matrices the latter type is an example of a special type of a nonnegative matrix m for which the group inverse of the associated m matrix satisfies a a t for some 0

some restrictions on weight enumerators of singly even self dual codes in this correspondence we give some restrictions on weight enumerators of singly even self dual n n 2 d codes whose shadows have minimum weight d 2 as a consequence we determine the weight enumerators for which there is an extremal singly even self dual 40 20 8 code and an optimal singly even self dual 50 25 10 code

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

a direct projection method for markov chains a direct method based on oblique projections is adapted to compute the stationary distribution vector of a finite markov chain the algorithm can also be used to compute the group inverse of the corresponding generator matrix it is shown how to update the stationary vector and other quantities of interest when one row of the transition probability matrix is modified a gth like variant that appears to compute the stationary probabilities to high relative accuracy is developed

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

some new extremal codes of lengths 76 and 78 we find new extremal 76 38 14 and 78 39 14 binary self dual codes the code of length 76 is the first known self dual code with these parameters it is shadow extremal and the shortest known self dual code with minimal distance 14 it has an automorphism of order 19 without fixed coordinates the code of length 78 has a new weight enumerator and an automorphism of order 19 with two fixed coordinates

perron frobenius theory over real closed fields and fractional power series expansions abstract some of the main results of the perron frobenius theory of square nonnegative matrices over the reals are extended to matrices with elements in a real closed field we use the results to prove the existence of a fractional power series expansion for the perron frobenius eigenvalue and normalized eigenvector of real square nonnegative irreducible matrices which are obtained by perturbing a possibly reducible nonnegative matrix further we identify a system of equations and inequalities whose solution yields the coefficients of these expansions for irreducible matrices our analysis assures that any solution of this system yields a fractional power series with a positive radius of convergence

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

accurate svds of weakly diagonally dominant m matrices we present a new o n3 algorithm which computes the svd of a weakly diagonally dominant m matrix to high relative accuracy the algorithm takes as an input the offdiagonal entries of the matrix and its row sums

solving matrix polynomial equations arising in queueing problems abstract the matrix equation i 0 n a i x i 0 where the a i s are m m matrices is encountered in the numerical solution of markov chains which model queueing problems we provide here a unifying framework in terms of mobius mapping to relate different resolution algorithms having a quadratic convergence this allows us to compare algorithms like logarithmic reduction lr and cyclic reduction cr which extend graeffe s iteration to matrix polynomials and the invariant subspace is approach which extends cardinal s algorithm we devise new iterative techniques having quadratic convergence and present numerical experiments

spectral simplex method we develop an iterative optimization method for finding the maximal and minimal spectral radius of a matrix over a compact set of nonnegative matrices we consider matrix sets with product structure i e all rows are chosen independently from given compact sets row uncertainty sets if all the uncertainty sets are finite or polyhedral the algorithm finds the matrix with maximal minimal spectral radius within a few iterations it is proved that the algorithm avoids cycling and terminates within finite time the proofs are based on spectral properties of rank one corrections of nonnegative matrices the practical efficiency is demonstrated in numerical examples and statistics in dimensions up to 500 some generalizations to non polyhedral uncertainty sets including euclidean balls are derived finally we consider applications to spectral graph theory mathematical economics dynamical systems and difference equations

on a two level multigrid solution method for finite markov chains abstract we study a two level algebraic multigrid scheme for computing the stationary distribution of a homogeneous markov chain with a finite state space we reveal its relationship to an iterative aggregation disaggregation method provide an error analysis and prove its local convergence moreover we discuss the stochastic meaning of a block sor procedure that can be used as iterative smoother of the multigrid scheme

a method to calculate steady state distributions of large markov chains by aggregating states this paper develops an efficient iterative algorithm to calculate the steady state distribution of nearly all irreducible discrete time markov chains computational experiences suggest that for large markovian systems more than 130 states the proposed algorithm can be ten times faster than standard gaussian elimination in finding solutions to an accuracy of 0 1 the proposed algorithm is developed in three stages first we develop a very efficient algorithm for determining steady state distributions of a restricted class of markovian systems a second result establishes a relationship between a general irreducible markovian system and a system in the restricted class of markovian systems finally we combine the two results to produce an efficient iterative algorithm to solve markov systems the paper concludes with a discussion of the observed performance of the algorithm

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

m matrix generalized inverses of m matrices abstract let p be an n n nonnegative irreducible and stochastic matrix and consider the associated singular and irreducible m matrix q i p we show that without further restrictive spectral or combinatorial assumptions on p it is reasonable to expect that the group inverse q of q will be an m matrix only when p lies in a small wedge about a rank 1 nonnegative matrix a wedge whose existence is itself dependent on p we go on to consider the question of when q has a nonzero generalized inverse not necessarily the group inverse which is an m matrix we give examples of all sorts of classes of p for which such is the case but point out that the general problem remains unsolved

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

lu decomposition of m matrices by elimination without pivoting abstract it is shown that if a or a is a singular m matrix satisfying the generalized diagonal dominance condition y t a 0 for some vector y 0 then a can be factored into a lu by a certain elimination algorithm where l is a lower triangular m matrix with unit diagonal and u is an upper triangular m matrix the existence of lu decomposition of symmetric permutations of a and for irreducible m matrices and symmetric m matrices follow as colollaries this work is motivated by applications to the solution of homogeneous systems of linear equations ax 0 where a or a is an m matrix these applications arise e g in the analysis of markov chains input output economic models and compartmental systems a converse of the theorem metioned above can be established by considering the reduced normal form of a

folding algorithm a computational method for finite qbd processes with level dependent transitions this paper presents a new computational method for steady state analysis of finite quasi birth death qbd processes with level dependent transitions the qbd state space is defined in two dimension with n phases and k levels instead of formulating solutions in matrix geometric form the folding algorithm provides a technique for direct computation of spl pi p 0 where p is the qbd generator which is an nk spl times nk matrix taking a finite sequence of fixed cost binary reduction steps the k level matrix p is eventually reduced to a single level matrix from which a boundary vector is obtained each step halves the matrix size but keeps the qbd form the solution spl pi is expressed as a product of the boundary vector and a finite sequence of expansion factors the time and space complexity for solving spl pi p 0 is therefore reduced from o n sup 3 k and o n sup 2 k to o n sup 3 log sub 2 k and o n sup 2 log sub 2 k respectively the folding algorithm has a number of highly desirable advantages when it is applied to queueing analysis first the algorithm handles the multilevel control problem in finite buffer systems second its total independence of the phase structure allows the algorithm to apply to more elaborate multiple state markovian sources its computational efficiency numerical stability and superior error performance are also distinctive advantages

the computation of the mean first passage times for markov chains abstract a survey of a variety of computational procedures for finding the mean first passage times in markov chains is presented the author recently developed a new accurate computational technique an extended gth procedure hunter 2016 17 similar to that developed by kohlas 1986 20 in addition the author recently developed a variety of new perturbation techniques for finding key properties of markov chains including finding the mean first passage times hunter 2016 18 these recently developed procedures are compared with other procedures including the standard matrix inversion technique using the fundamental matrix kemeny and snell 1960 19 some simple generalized matrix inverse techniques developed by hunter 2007 15 and some modifications to the fund technique of heyman 1995 7 matlab is used to compute errors and estimate computation times when the techniques are used on some test problems that have been used in the literature together with some large sparse state space cases for accuracy a preference for the procedure of the author is exhibited for the test problems however it appears that the procedure as presented requires longer computational times

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

perturbation theory for the ldu factorization and accurate computations for diagonally dominant matrices we present a structured perturbation theory for the ldu factorization of row diagonally dominant matrices and we use this theory to prove that a recent algorithm of ye math comp 77 264 2195 2230 2008 computes the l d and u factors of these matrices with relative errors less than 14n 3 u where u is the unit roundoff and n n is the size of the matrix the relative errors for d are componentwise and for l and u are normwise with respect the max norm a m max ij a ij these error bounds guarantee that for any diagonally dominant matrix a we can compute accurately its singular value decomposition and the solution of the linear system ax b for most vectors b independently of the magnitude of the traditional condition number of a and in o n 3 flops

accurate solutions of m matrix sylvester equations this paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an m matrix sylvester equation ax xb c by which we mean both a and b have positive diagonal entries and nonpositive off diagonal entries and p i m otimes a b rm t otimes i n is a nonsingular m matrix and c is entrywise nonnegative it is proved that small relative perturbations to the entries of a b and c introduce small relative errors to the entries of the solution x thus the smaller entries of x do not suffer bigger relative errors than its larger entries unlikely the existing perturbation theory for general sylvester equations we then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute x as accurately as the input data deserve

cluster and group synchronization in delay coupled networks we investigate the stability of synchronized states in delay coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics using a master stability approach we find that the master stability function shows a discrete rotational symmetry depending on the number of groups the coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum which helps to simplify the evaluation of the master stability function our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times multiple coupling functions but also with multiple kinds of local dynamics in the networks nodes we illustrate our results by calculating stability in the example of delay coupled semiconductor lasers and in a model for neuronal spiking dynamics

blockwise perturbation theory for nearly uncoupled markov chains and its application abstract let p be the transition matrix of a nearly uncoupled markov chain the states can be grouped into aggregates such that p has the block form p p ij i j 1 k where p ii is square and p ij is small for i j let t be the stationary distribution partitioned conformally as t 1 t k t in this paper we bound the relative error in each aggregate distribution i t caused by small relative perturbations in p ij the error bounds demonstrate that nearly uncoupled markov chains usually lead to well conditioned problems in the sense of blockwise relative error as an application we show that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve such structured backward errors and compute each aggregate distribution with high relative accuracy

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

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

beam tracking for interference alignment in slowly fading mimo interference channels a perturbations approach under a linear framework in this paper the beam design for signal space interference alignment in slowly fading multiuser multiple input multiple output mimo interference channels is considered based on a linear formulation for interference alignment a predictive beam tracking algorithm is proposed using matrix perturbation theory the proposed algorithm based on a mixture of iteration and update computes interference aligning beamforming vectors at the current time by updating the previous beam vectors based on the channel difference between the two time steps during the predictively updating phase and yields significant reduction in computational complexity compared with existing methods recalculating beams at each time step the tracking performance of the algorithm is analyzed in terms of mean square error and sum rate loss between the predictively updating approach and the recalculating approach and the impact of imperfect channel knowledge is also investigated under the state space channel model numerical results show that the proposed algorithm has almost the same performance as non predictive methods in sum rate thus the proposed algorithm provides a very efficient way to realize interference alignment in a realistic slowly fading mimo channel environment

network synchronization of groups in this paper we study synchronized motions in complex networks in which there are distinct groups of nodes where the dynamical systems on each node within a group are the same but are different for nodes in different groups both continuous time and discrete time systems are considered we initially focus on the case where two groups are present and the network has bipartite topology i e links exist between nodes in different groups but not between nodes in the same group we also show that group synchronous motions are compatible with more general network topologies where there are also connections within the groups

stochastic complementation uncoupling markov chains and the theory of nearly reducible systems a concept called stochastic complementation is an idea which occurs naturally although not always explicitly in the theory and application of finite markov chains this paper brings this idea to the forefront with an explicit definition and a development of some of its properties applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of markov chains furthermore the role of stochastic complementation in the development of the classical simon ando theory of nearly reducible system is presented

computable error bounds for aggregated markov chains abstract this paper describes a method for computing the steady state probability vector of a nearly completely decomposable markov chain the method is closely related to one proposed by simon and ando and developed by courtois however the method described here does not require the determination of a completely decomposable stochastic approximation to the transition matrix and hence it is applicable to matrices other than stochastic an error analysis of the procedure is given which results in effectively computable error bounds author

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

highly accurate doubling algorithm for quadratic matrix equation from quasi birth and death process abstract a highly accurate doubling algorithm to solve the most fundamental quadratic matrix equation in the quasi birth and death qbd process is developed it follows from the general framework of the doubling algorithm for the first standard form sf1 but can be implemented to compute the minimal nonnegative solution with high entrywise relative accuracy for all entries large or tiny the algorithm is globally and quadratically convergent except for qbd equations in the critical case where convergence is linear with the linear rate 1 2 numerical examples are presented to demonstrate and confirm our claims the development here parallels the recent work of xue and li 2017 3 on the m matrix algebraic riccati equation

the state reduction and related algorithms and their applications to the study of markov chains graph theory and the optimal stopping problem abstract we discuss the state reduction gth grassmann taksar heyman algorithm for recursively finding invariant measure we demonstrate the relationship between this algorithm and the freidlin wentzell tree decomposition approach to study the characteristics of markov chains the structure of the state reduction gth algorithm suggests the natural idea for finding the distribution of a markov chain at the moment of first visit to a given set and some similar characteristics we study the possible range of such algorithms we also present a new algorithm for solving the classical problem of optimal stopping of a markov chain based on a similar idea of sequential elimination of some states we give shorter and more transparent proofs of some previously known results and improve the bounds of freidlin wentzell in the perturbation theory of markov chains some applications to graph theory are also discussed

nonnegative alternating circulants leading to m matrix group inverses abstract let c be the set of all an n n nonnegative irreducible alternating circulant matrices we characterize a subset c of c such that if b c and the perron root of b is r then the group inverse ri b of the singular and irreducible m matrixri b is also an m matrix this is equivalent to the fact that for each such b the perron root at b is a concave function in each of the off diagonal entries the characterization for the case when n is odd presents more difficulties than for the case when n is even so the two cases are treated separately

the computation of key properties of markov chains via perturbations abstract computational procedures for the stationary probability distribution the group inverse of the markovian kernel and the mean first passage times of a finite irreducible markov chain are developed using perturbations the derivation of these expressions involves the solution of systems of linear equations and structurally inevitably the inverses of matrices by using a perturbation technique starting from a simple base where no such derivations are formally required we update a sequence of matrices formed by linking the solution procedures via generalised matrix inverses and utilising matrix and vector multiplications four different algorithms are given some modifications are discussed and numerical comparisons are made using a test example the derivations are based upon the ideas outlined by hunter 14

componentwise bounds for nearly completely decomposable markov chains using stochastic comparison and reordering this paper presents an improved version of a componentwise bounding algorithm for the state probability vector of nearly completely decomposable markov chains and on an application it provides the first numerical results with the type of algorithm discussed the given two level algorithm uses aggregation and stochastic comparison with the strong stochastic st order in order to improve accuracy it employs reordering of states and a better componentwise probability bounding algorithm given st upper and lower bounding probability vectors results in sparse storage show that there are cases in which the given algorithm proves to be useful

accurate computation of the smallest eigenvalue of a diagonally dominant m matrix if each off diagonal entry and the sum of each row of a diagonally dominant m matrix are known to certain relative accuracy then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers in this paper we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms

the markov chain tree theorem in commutative semirings and the state reduction algorithm in commutative semifields we extend the markov chain tree theorem to general commutative semirings and we generalize the state reduction algorithm to general commutative semifields this leads to a new universal algorithm whose prototype is the state reduction algorithm which computes the markov chain tree vector of a stochastic matrix

aggregation algorithms for perturbed markov chains with applications to networks modeling a powerly perturbed markov chain mc is a family of finite mcs given by transition probability matrices p varepsilon p ij varepsilon or generators for continuous time mcs such that p ij varepsilon delta ij varepsilon d ij o varepsilon d ij varepsilon rightarrow0 in the simplest case we have popular nearly completely decomposable ncd chains it has been shown by directed forest expansions that solutions of linear systems related to p varepsilon have a form x i varepsilon alpha i varepsilon a i o varepsilon a i for some vectors boldsymbol alpha alpha i and mathbf a a i many characteristics of mcs such as stationary distributions mean passage times or limiting matrices can be defined in this way we present efficient combinatorial aggregation algorithms for computing mathbf a and boldsymbol alpha the algorithms rely on grouping the states of mcs in such a way that the probability of changing the state inside the group is of a greater order of magnitude than the interactions between groups in contrast to existing methods our approach is based on a combinatorial framework and can be seen as an algorithmization of the famous matrix tree theorem we establish some preliminary results on the complexity of our algorithms numerical experiments on several network models such as queueing systems or the google problem show the potential applicability of the method in real life problems

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen