an iterative method for nonsymmetric systems with multiple right hand sides we propose a method for the solution of linear systems ax b where a is a large possibly sparse nonsymmetric matrix of order n and b is an arbitrary rectangular matrix of order n times s with s of moderate size the method uses a single krylov subspace per step as a generator of approximations a projection process and a richardson acceleration technique it thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right hand sides numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right hand sides

restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

modified vector coding analysis of trunk and lower extremity kinematics during maximum and sub maximum back squats the back squat is a complex movement with significant demands on the lower extremities and trunk to raise an external load the back squat is simultaneously an open and closed kinetic chain movement that requires coordination of the entire body for successful completion of the lift therefore this study aimed to examine coordination of the thigh and shank trunk and thigh and the hip and knee during the concentric phase of maximum supra maximum at 105 max and sub maximum at 80 max back squats fourteen resistance trained adults participated in this study maximum back squat loads were determined using a previously determined progressive load protocol motion capture of the trunk and lower extremities and ground reaction force data were recorded during all squats angle angle plots and modified vector coding were performed to analyze segment and joint coupling angles and knee hip moments overall the concentric phase of back squats depict a transition from early knee dominance to hip dominance as the system ascends interestingly all squats presented with coupling of thigh rising and trunk falling based on the angle angle plots and the modified vector coding results the prolonged coupling of trunk falling and thigh rising likely resulted in too large of a moment arm for the external load for the participants to overcome during supramax conditions

a harmonic restarted arnoldi algorithm for calculating eigenvalues and determining multiplicity abstract a restarted arnoldi algorithm is given that computes eigenvalues and eigenvectors it is related to implicitly restarted arnoldi but has a simpler restarting approach harmonic and regular rayleigh ritz versions are possible for multiple eigenvalues an approach is proposed that first computes eigenvalues with the new harmonic restarted arnoldi algorithm then uses random restarts to determine multiplicity this avoids the need for a block method or for relying on roundoff error to produce the multiple copies

diffusion limit of fair resource control stationarity and interchange of limits we study a resource sharing network where each job requires the concurrent occupancy of a subset of links servers resources and each link s capacity is shared among job classes that require its service the real time allocation of the service capacity among job classes is determined by the so called proportional fair scheme which allocates the capacity among job classes taking into account the queue lengths and the shadow prices of link capacity we show that the usual traffic condition is necessary and sufficient for the diffusion limit to have a stationary distribution we also establish the uniform stability of the prelimit networks and hence the existence of their stationary distributions to justify the interchange of two limits the limit in time and limit in diffusion scaling we identify a bounded workload condition and show it is a sufficient condition to justify the interchange for the stationary distributions and their moments this last result is essential for the validity of the diffu

deflated gmres for systems with multiple shifts and multiple right hand sides abstract we consider solution of multiply shifted systems of nonsymmetric linear equations possibly also with multiple right hand sides first for a single right hand side the matrix is shifted by several multiples of the identity such problems arise in a number of applications including lattice quantum chromodynamics where the matrices are complex and non hermitian some krylov iterative methods such as gmres and bicgstab have been used to solve multiply shifted systems for about the cost of solving just one system restarted gmres can be improved by deflating eigenvalues for matrices that have a few small eigenvalues we show that a particular deflated method gmres dr can be applied to multiply shifted systems in quantum chromodynamics it is common to have multiple right hand sides with multiple shifts for each right hand side we develop a method that efficiently solves the multiple right hand sides by using a deflated version of gmres and yet keeps costs for all of the multiply shifted systems close to those for one shift an example is given showing this can be extremely effective with a quantum chromodynamics matrix

quasi kernel polynomials and their use in non hermitian matrix iterations abstract some of the most efficient iterative algorithms for large sparse hermitian matrix computations are based on orthogonal or kernel polynomials for the case of non hermitian matrices methods based on orthogonal or kernel polynomials are less satisfactory in that the resulting algorithms involve long recurrences consequently it is usually too expensive to run the full algorithms and restarts are necessary a typical example is the generalized minimal residual method gmres for solving non hermitian linear systems where work and storage per iteration grow linearly with the iteration number recently two quasi minimal residual methods qmr for solving non hermitian linear systems have been proposed which unlike gmres are based on short recurrences and hence can be used as true iterative schemes without restarts in this paper the concept of quasi kernel polynomials is introduced some general theory for quasi kernel polynomials is developed such as recurrence relations and a characterization of roots of quasi kernel polynomials as generalized eigenvalues it is pointed out that the qmr approaches are based on two particular instances of quasi kernel polynomials also the use of quasi kernel polynomials for approximating eigenvalues or pseudospectra of large sparse non hermitian matrices is briefly discussed

a hybrid block gmres method for nonsymmetric systems with multiple right hand sides the block gmres bgmres method is a natural generalization of the gmres algorithm for solving large non symmetric systems with multiple right hand sides unfortunately its cost increases significantly per iteration frequently rendering the method impractical in this paper we propose a hybrid block gmres method which offers significant performance improvements over bgmres this method uses the matrix polynomial obtained in the course of a bgmres step and combines the advantages of the block approach with those of successful hybrid methods we discuss the properties and several implementation variants of the method and report results from numerical experiments we also describe how to use these techniques in order to solve multiply shifted systems with multiple right hand sides

a heavy traffic limit theorem for networks of queues with multiple customer types the central result of this paper is a heavy traffic limit theorem for the vector of total station workloads in an open network of queues with multiple customer types under first come first served and priority disciplines the limit process is a regulated brownian motion on the nonnegative orthant with parameters specified from the first two moments of the interarrival and service time distributions and a matrix of reduced routing information through the phenomenon of state space collapse associated limit results for queue length workload and sojourn time processes by customer type are obtained jointly as simple transformations of the total workload limit process diffusion approximations based on the theorem are discussed

stationary distribution convergence for generalized jackson networks in heavy traffic in a recent paper gamarnik and zeevi gamarnik d a zeevi 2006 validity of heavy traffic steady state approximations in open queueing networks ann appl probab 16 1 56 90 it was shown that under suitable conditions stationary distributions of the scaled queue lengths process for a generalized jackson network converge to the stationary distribution of the associated reflected brownian motion in the heavy traffic limit the proof relied on certain exponential integrability assumptions on the primitives of the network in this note we show that the above result holds under much weaker integrability conditions we provide an alternative proof of this result assuming in addition to natural heavy traffic and stability assumptions only standard independence and square integrability conditions on the network primitives that are commonly used in heavy traffic analysis furthermore under additional integrability conditions we establish convergence of moments of stationary distributions

validity of heavy traffic steady state approximations in multiclass queueing networks the case of queue ratio disciplines a class of stochastic processes known as semi martingale reflecting brownian motions srbms is often used to approximate the dynamics of heavily loaded queueing networks in two influential papers bramson bramson m 1998 state space collapse with applications to heavy traffic limits for multiclass queueing networks queueing systems 30 89 148 and williams williams rj 1998b diffusion approximations for open multiclass queueuing networks sufficient conditions involving state space collapse queueing systems 30 27 88 laid out a general and structured approach for proving the validity of such heavy traffic approximations in which an srbm is obtained as a diffusion limit from a sequence of suitably normalized workload processes however for multiclass networks it is still not known in general whether the steady state distribution of the srbm provides a valid approximation for the steady state distribution of the original network in this paper we study the case of queue ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative in addition to standard assumptions made in the literature towards the stability of the pre and post limit processes and the existence of diffusion limits we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate for the special case of static priority networks such linear attraction is known to hold under certain conditions on the network primitives the analysis elucidates interesting connections between stability of the pre and post limit processes their respective fluid models and state space collapse and identifies the respective roles played by all of the above in establishing validity of heavy traffic steady state approximations

truncation strategies for optimal krylov subspace methods optimal krylov subspace methods like gmres and gcr have to compute an orthogonal basis for the entire krylov subspace to compute the minimal residual approximation to the solution therefore when the number of iterations becomes large the amount of work and the storage requirements become excessive in practice one has to limit the resources the most obvious ways to do this are to restart gmres after some number of iterations and to keep only some number of the most recent vectors in gcr this may lead to very poor convergence and even stagnation therefore we will describe a method that reveals which subspaces of the krylov space were important for convergence thus far and exactly how important they are this information is then used to select which subspace to keep for orthogonalizing future search directions numerical results indicate this to be a very effective strategy

adaptively preconditioned gmres algorithms the restarted gmres algorithm proposed by saad and schultz siam j sci statist comput 7 1986 pp 856 869 is one of the most popular iterative methods for the solution of large linear systems of equations ax b with a nonsymmetric and sparse matrix this algorithm is particularly attractive when a good preconditioner is available the present paper describes two new methods for determining preconditioners from spectral information gathered by the arnoldi process during iterations by the restarted gmres algorithm these methods seek to determine an invariant subspace of the matrix a associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved

restarted gmres for shifted linear systems shifted matrices which differ by a multiple of the identity only generate the same krylov subspaces with respect to any fixed vector this fact has been exploited in lanczos based methods like cg qmr and bicg to simultaneously solve several shifted linear systems at the expense of only one matrix vector multiplication per iteration here we develop a variant of the restarted gmres method exhibiting the same advantage and we investigate its convergence for positive real matrices in some detail we apply our method to speed up multiple masses calculations arising in lattice gauge computations in quantum chromodynamics one of the most time consuming supercomputer applications

analysis of projection methods for solving linear systems with multiple right hand sides we analyze a class of krylov projection methods but mainly concentrate on a specific conjugate gradient cg implementation by smith peterson and mittra ieee transactions on antennas and propogation 37 1989 pp 1490 1493 to solve the linear system ax b where a is symmetric positive definite and b is a multiple of right hand sides this method generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems orthogonally onto the generated krylov subspace to get the approximate solutions the whole process is repeated with another unsolved system as a seed until all the systems are solved we observe in practice a superconvergence behavior of the cg process of the seed system when compared with the usual cg process we also observe that only a small number of restarts is required to solve all the systems if the right hand sides are close to each other these two features together make the method particularly effective in this paper we give theoretical proof to justify these observations furthermore we combine the advantages of this method and the block cg method and propose a block extension of this single seed method

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

iterative system solvers for the frequency analysis of linear mechanical systems abstract the paper deals with the numerical treatment of the direct frequency domain dfd analysis of linear mechanical systems attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency strategies for transforming the system into the shifted form t i z d are proposed and discussed where is related to the frequency for two formulations of the shifted system the performance of some krylov sub space iterative solvers is tested and compared to that of a multi frontal direct method advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift frequency numerical experiments on some prototype structural dynamics problems are reported the results shown demonstrate how the devised strategies for the iterative solution can outperform in many cases the direct solver

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

recycling krylov subspaces for sequences of linear systems many problems in science and engineering require the solution of a long sequence of slowly changing linear systems we propose and analyze two methods that significantly reduce the total number of matrix vector products required to solve all systems we consider the general case where both the matrix and right hand side change and we make no assumptions regarding the change in the right hand sides furthermore we consider general nonsingular matrices and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix our methods work well under these general assumptions and hence form a significant advancement with respect to related work in this area we can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems we consider two approaches that allow for the continuous improvement of the recycled subspace at low cost we consider both hermitian and non hermitian problems and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling we also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics materials science and computational physics

a block qmr algorithm for non hermitian linear systems with multiple right hand sides abstract many applications require the solution of multiple linear systems that have the same coefficient matrix but differ in their right hand sides instead of applying an iterative method to each of these systems individually it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously in this paper we propose a block version of freund and nachtigal s quasi minimal residual qmr method for the iterative solution of non hermitian linear systems the block qmr method uses a novel lanczos type process for multiple starting vectors which was recently developed by aliaga boley freund and hernandez to compute suitable basis vectors for the underlying block krylov subspaces we describe the basic block qmr method and also give important implementation details in particular we show how to incorporate deflation to drop converged linear systems and to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences numerical results are reported that illustrate typical features of the block qmr method

arnoldi methods for large sylvester like observer matrix equations and an associated algorithm for partial spectrum assignment abstract we propose an arnoldi based numerical method for solving a sylvester type equation arising in the construction of the luenberger observer given an n n matrix a and an n m matrix g the method simultaneously constructs an m m hessenberg matrix h with a preassigned spectrum and an n m orthonormal matrix x such that ax xh g we consider the case when a is large and sparse so that the standard techniques such as the well known hessenberg schur method for solving a sylvester equation cannot be easily applied as a byproduct we propose an algorithm for the partial pole assignment problem for large matrices

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation