a splitting method for complex symmetric indefinite linear system abstract in this paper not requiring that the hermitian part of the complex symmetric linear system must be hermitian positive definite a class of splitting methods is established by the modified positive negative stable splitting pns of the coefficient matrix and is called the mpns method theoretical analysis shows that the mpns method is absolutely convergent under proper conditions some useful properties of the corresponding mpns preconditioned matrix are obtained numerical experiments are reported to illustrate the efficiency of both the mpns method and the mpns preconditioner

on the simplification of generalized conjugate gradient methods for nonsymmetrizable linear systems abstract the conjugate gradient cc method developed by hestenes and stiefel in 1952 can be effectively used to solve the linear system au b when a is symmetrixable in the sense that za and z are symmetric and positive definite spd for some z a number of generalizations of the cg method have been proposed by the authors and by others for handling the nonsymmetrizable case for many problems the amount of computer memory and computational effort required may be so large as to make the procedures not feasible truncated schemes are often used but in some cases the truncated methods may not converge even though the nontruncated schemes converge however it is well known that if a is symmetric the generalized cg schemes can be greatly simplified even though a is not spd so that the truncated schemes are equivalent to the nontruncated schemes in the present paper it is shown that such a simplification can occur if a nonsingular matrix h is available such that ha ath of course if a at then h can be taken to be the identity matrix it is also shown that such an h always exists however it may not be practical to compute h these results are used to derive three variations of the lanczos method for solving nonsymmetrizable systems two of the forms are well known but the third appears to be new an argument is given for choosing the third form over the other two

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

extrapolation vs projection methods for linear systems of equations it is shown that the four vector extrapolation methods minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm when applied to linearly generated vector sequences are krylov subspace methods and are equivalent to some well known conjugate gradient type methods a unified recursive method that includes the conjugate gradient conjugate residual and generalized conjugate gradient methods is developed finally the error analyses for these methods are unified and some known and some new error bounds for them are given

on predicting incompressible flows by using a stabilized finite difference method with penalty this paper presents a finite difference solution method for the incompressible navier stokes equations in primitive variables form to provide the necessary coupling between continuity and momentum penalty method is introduced into continuity equation differential terms arising in navier stokes equations are discretized by stabilized finite differences of the su type streamline upwind to calculate the convective terms the use of a penalty function method in a finite difference method of staggered grids allows to eliminate the pressure and the result is a system of equations of the velocities arises the aim of this paper is to describe the techniques used in the approximations to calculate each term of the discretized model in particular time integration techniques solution of non linearities associated to convective terms and constitutive laws stabilization procedures and discretization of navier stokes equation via finite differences are explained solving three standard fluid mechanics problems validates the method the flow of a newtonian fluid in a sudden contraction the developing non newtonian fluid in a circular tube and a transient laminar axisymmetric flow trough a tube with a moderate local area reduction comparison with available experimental results is used to examine the accuracy of the method reported a close agreement between the numerical prediction and the experimental data

globally solvable systems of complex vector fields we consider a class of involutive systems of n smooth vector fields on the n 1 dimensional torus we obtain a complete characterization for the global solvability of this class in terms of liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1 form in the minimal covering space

the convergence rate of the chebyshev sim under a perturbation of a complex line segment spectrum abstract the chebyshev semiiterative method chsim is probably the best known and most often used method for the iterative solution of linear system x t x c where the spectrum of t is located in a complex line segment excluding 1 the asymptotic convergence factor acf of the chsim under a perturbation of is considered several formulae for the approximation to the acfs up to the second order of a perturbation are derived this generalizes the results about the sensitivity of the asymptotic rate of convergence to the estimated eigenvalues by hageman and young in the case that both and are real two numerical examples are given to illustrate the theoretical results

a splitting iterative method for the discrete dynamic linear systems this paper is concerned with a splitting iterative method for a class of complex symmetric linear systems from an n degree of freedom n dof discrete system this splitting iterative method is established by the complex symmetric and skew hermitian splitting of the coefficient matrix and is called the css method which is from the classical state space formulation of frequency analysis of the discrete dynamic linear systems the convergence properties of the css method are obtained the corresponding css preconditioner is proposed and some useful properties of the preconditioned matrix are established the presented numerical examples are to illustrate the efficiency of both the css method and the css preconditioner

on the stable implementation of the generalized minimal error method the paper reviews several implementations of the generalized minimal error method gmerr method for solving nonsymmetric systems of linear equations that minimize the euclidean norm of the error in the related generalized krylov subspace we show the relation to the methods in the symmetric indefinite case a new variant of the gmerr method is proposed and the stable implementation based on the householder transformations is discussed numerical stability of the most frequent implementations is analyzed and the theoretical results are illustrated by numerical examples

phase locking in networks of synaptically coupled mckean relaxation oscillators we use geometric dynamical systems methods to derive phase equations for networks of weakly connected mckean relaxation oscillators we derive an explicit formula for the connection function when the oscillators are coupled with chemical synapses modeled as the convolution of some input spike train with an appropriate synaptic kernel the theory allows the systematic investigation of the way in which a slow recovery variable can interact with synaptic time scales to produce phase locked solutions in networks of pulse coupled neural relaxation oscillators the theory is exact in the singular limit that the fast and slow time scales of the neural oscillator become effectively independent by focusing on a pair of mutually coupled mckean oscillators with alpha function synaptic kernels we clarify the role that fast and slow synapses of excitatory and inhibitory type can play in producing stable phase locked rhythms in particular we show that for fast excitatory synapses there is coexistence of a stable synchronous a stable anti synchronous and one stable asynchronous solution for slower synapses the anti synchronous solution can lose stability whilst for even slower synapses it can regain stability the case of inhibitory synapses is similar up to a reversal of the stability of solution branches using a return map analysis the case of strong pulsatile coupling is also considered in this case it is shown that the synchronous solution can co exist with a continuum of asynchronous states

on the class of chebyshev approximation problems which arise in connection with a conjugate gradient type method we investigate approximation problems of the form mathop min limits p p 0 1 mathop max p lambda limits lambda in s p withp ranging over all polynomials of degree k ands any line segment of the complex plane problems of this type appear in connection with error bounds for the minimum residual algorithm for solving linear systems with coefficient matrixa ei di n wheren nh andd besides a few general results on p we present the explicit solution of p for segmentss which are parallel to the imaginary and symmetric to the real axis this result leads to an optimal semi iterative method for the class of matricesa i n wheren nt

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

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

a maximal projection solution of ill posed linear system in a column subspace better than the least squares solution linear equations system column subspace maximal projection solution maximal projection algorithm least squares solution inverse problems a b s t r a c t a better method than the least squares solution is proposed in this paper to solve an n dimensional ill posed linear equations system ax b in an m dimensional column subspace cm which is selected in such a way that each column in cm is in a closer proximity to b we maximize the orthogonal projection of b onto y ax to find an approximate solution x span a cm where a is a nonzero free vector then we can prove that the maximal projection solution mp is better than the least squares solution ls with b axmp b axls numerical examples of inverse problems under a large noise maybe up to 30 are discussed which confirm the efficiency of presently developed mp algorithms mpa and mpa m

some notes on multisplltting methods and m step preconditioners for linear systems abstract to solve the real nonsingular linear system ax b on parallel and vector machines multisplitting methods and m step preconditioners are considered in this work in particular 1 we extend adams and ong s idea and theory of the m step additive preconditioners and ruggiero and galligani s method of the arithmetic mean and we generalize the corresponding convergence results 2 we determine suitable relaxed m step preconditioners and solve the problem of minimizing the related condition number with respect to the relaxation extrapolation parameter involved 3 finally we complete the theoretical solution of the problem of determining the optimum sor additive iterative method for 2 cyclic positive definite matrices

introduction and study of fourth order theta schemes for linear wave equations a new class of high order implicit three time step schemes for semi discretized wave equations is introduced and studied these schemes are constructed using the modified equation approach generalizing the q scheme their stability properties are investigated via an energy analysis which enables us to design super convergent schemes and also optimal stable schemes in terms of consistency errors specific numerical algorithms for the fully discrete problem are tested and discussed showing the efficiency of our approach compared to second order q schemes

an iterative method for large sparse linear systems on a vector computer in this paper we consider thearithmetic mean method for solving large sparse systems of linear equations this iterative method converges for systems with coefficient matrices that are symmetric positive definite or positive real or irreducible l matrices with a strong diagonal dominance the method is very suitable for parallel implementation on a multiprocessor system such as the cray x mp some numerical experiments on systems resulting from the discretization by means of the usual 5 point difference formulae of an elliptic partial differential equation are presented

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

an efficient implementation of a conformal mapping method based on the szego kernel an implementation based on iterative techniques of a method to compute the riemann mapping function is presented the method has been recently introduced by n kerzman and the author it expresses the szego kernel as the solution of an integral equation of the second kind it is shown how to treat symmetric regions the algorithm is tested on five examples the numerical results show that the method is competitive with respect to accuracy stability and efficiency

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

polynomial preconditioned gmres and gmres dr we look at solving large nonsymmetric systems of linear equations using polynomial preconditioned krylov methods we give a simple way to find the polynomial it is shown that polynomial preconditioning can significantly improve restarted gmres for difficult problems and the reasons for this are examined stability is discussed and algorithms are given for increased stability next we apply polynomial preconditioning to gmres with deflated restarting it is shown that this is worthwhile for sparse matrices and for problems with many small eigenvalues multiple right hand sides are also considered

on hybrid semi iterative methods given a large sparse system of linear algebraic equations in fixed point form bf x t bf x bf c one way to solve this system is to apply a semi iterative method sim to the basic iteration method bf x m t bf x m 1 bf c it is known that if the spectrum sigma t of t is contained in some compact subset omega of the complex plane where 1 notin omega then there are asymptotically optimal sims associated with the basic iteration and omega whose asymptotic rates of convergence are best possible for the class of all matrices t with sigma t subseteq omega however for a given compact set omega an asymptotically optimal sim for omega is usually not well suited for efficient numerical computation unless this sim can be generated from a k step recurrence formula in this paper hybrid semi iterative methods are investigated which consist of two independent steps first bf x t bf x bf c is transformed into a consistent linear syst

orthogonal error methods in this paper we discuss the conjugate gradient method generalized to nonsymmetric matrices constructing a class of algorithms that includes both the conjugate gradient algorithms as described in 12 and the orthogonal residual algorithms cf elman 10 we call this class orthogonal error algorithms and characterize the class of matrices for which finite term orthogonal error algorithms exist

global solvability of real analytic involutive systems on compact manifolds part 2 the focus of this work is the smooth global solvability of a linear partial differential operator mathbb l associated to a real analytic closed non exact 1 form b defined on a real analytic closed n manifold that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co rank one we define an appropriate covering projection widetilde m rightarrow m such that the pullback of b has a primitive widetilde b and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of widetilde b are connected as a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator when m is orientable we prove a connection between the global solvability of mathbb l and that of mathbb l n 1 which is the last non trivial operator of the complex in particular we prove that mathbb l is globally solvable if and only if mathbb l n 1 is globally solvable in the smooth category we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a morse 1 form i e when the structure is of mizohata type

the two stage iterative methods for symmetric positive definite matrices1supported by the state major key project for basic researches and doctorial point foundation of china 1 two stage iterative methods for the solution of the symmetric positive definite linear system ax b are considered in this paper in particular strong conditions and feeble conditions on splittings are given so that the two stage methods are convergent for any number of inner iterations and for any nonnegative even number of inner iterations respectively some suitable examples of strong p regular splittings are presented and the convergence of some feasible two stage methods is discussed in which the outer splitting is weak p regular and the inner splitting is strong or weak p regular splittings

solving the time fractional schrodinger equation by krylov projection methods the time fractional schrodinger equation is a fundamental topic in physics and its numerical solution is still an open problem here we start from the possibility to express its solution by means of the mittag leffler function then we analyze some approaches based on the krylov projection methods to approximate this function their convergence properties are discussed together with related issues numerical tests are presented to confirm the strength of the approach under investigation

pulse coupled decentral synchronization synchronous firing of biological oscillators is explained by a mathematical model developed by mirollo and strogatz siam j appl math 50 1990 pp 1645 1662 exactly the same principles may be carried over to achieve decentral synchronization in time division multiple access systems however some extensions of the basic model are necessary in view of technical implementations this paper deals with synchronization in finite time for two oscillators and convergence to synchrony except on a set of initial conditions of measure zero for cliques of oscillators particularly for nondifferentiable e g piecewise linear state functions moreover for oscillator pairs we prove synchronization even if there is a certain delay between transmission and reception of pulses a simulation tool named imdesy is developed to investigate general networks of oscillators with arbitrary topologies and more refined reaction strategies this is demonstrated in two extensive examples

necessary and sufficient conditions for the simplification of generalized conjugate gradient algorithms recent papers by faber and manteuffel characterize the cases where orthodir a generalized conjugate gradient algorithm for solving complex non hermitian linear systems can be defined by a short recurrence formula and thus simplified in the present paper similar results are obtained for two other generalized conjugate gradient algorithms namely orthomin and orthores necessary and sufficient conditions on the coefficient matrix a are given so that simplification is possible illustrative examples are given

boundary value techniques for initial value problems in ordinary differential equations the numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered we discuss a finite difference method which was already investigated by fox in 1954 and fox and mitchell in 1957 hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown we further propose and discuss new jalerkin methods for initial value problems along the lines of the boundary value approach

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

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

projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport we investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport the matrices of the corresponding linear systems are symmetric with a positive semi definite real part and an imaginary part with a compatible nullspace we discuss well posedness the symmetry of generalized inverses and cholesky methods we investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems as an application we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field

intrinsic modulation of pulse coupled integrate and fire neurons intrinsic neuromodulation is observed in sensory and neuromuscular circuits and in biological central pattern generators we model a simple neuronal circuit with a system of two pulse coupled integrate and fire neurons and explore the parameter regimes for periodic firing behavior the inclusion of biologically realistic features shows that the speed and onset of neuronal response plays a crucial role in determining the firing phase for periodic rhythms we explore the neurophysiological function of distributed delays arising from both the synaptic transmission process and dendritic structure as well as discrete delays associated with axonal communication delays bifurcation and stability diagrams are constructed with a mixture of simple analysis numerical continuation and the kuramoto phase reduction technique moreover we show that for asynchronous behavior the strength of electrical synapses can control the firing rate of the system

the two stage arithmetic mean method in several recent works the arithmetic mean method for solving large sparse linear systems has been introduced and analysed each iteration of this method consists of solving two independent systems when we obtain two approximate solutions of these systems by a prefixed number of steps of an iterative scheme we generate an inner outer procedure called two stage arithmetic mean method general convergence theorems are proved for m matrices and for symmetric positive definite matrices in particular we analyze a version of two stage arithmetic mean method for t q r matrices deriving the convergence conditions the method is well suited for implementation on a parallel computer numerical experiments carried out on cray t3d permits to evaluate the effectiveness of the two stage arithmetic mean method

a parallel algorithm for solving block tridiagonal linear systems abstract in this paper we consider a new form of the arithmetic mean method for solving large block tridiagonal linear systems the iterative method converges for systems with coefficient matrices that are symmetric positive definite or positive real or irreducible l matrices with a strong diagonal dominance when the coefficient matrix is symmetric positive definite an additive preconditioner for the conjugate gradient method is derived both the iterative method and the preconditioner are very suitable for parallel implementation on a multivector computer some numerical experiments on systems resulting from the discretization of an elliptic partial differential equation are carried out on the cray y mp

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

on the convergence of krylov subspace methods for matrix mittag leffler functions in this paper we analyze the convergence of some commonly used krylov subspace methods for computing the action of matrix mittag leffler functions as is well known such functions find application in the solution of fractional differential equations we illustrate the theoretical results by some numerical experiments

period focusing induced by network feedback in populations of noisy integrate and fire neurons the population dynamics of an ensemble of nonleaky integrate and fire stochastic neurons is studied the model selected allows for a detailed analysis of situations where noise plays a dominant role simulations in a regime with weak to moderate interactions show that a mechanism of excitatory message interchange among the neurons leads to a decrease in the firing period dispersion of the individual units the dispersion reduction observed is larger than what would be expected from the decrease in the period this period focusing is explained using a mean field model it is a dynamical effect that arises from the progressive decrease of the effective firing threshold as a result of the messages received by each unit from the rest of the population a back of the envelope formula to calculate this nontrivial dispersion reduction and a simple geometrical description of the effect are also provided

global solvability of a class of involutive systems abstract we consider a class of involutive systems of real vector fields on then dimensional torus and prove that global solvability is equivalent to an algebraic condition involving liouville forms

the arithmetic mean method for solving systems of nonlinear equations in finite differences abstract in this paper we consider the application of additive operator splitting methods for solving a finite difference nonlinear system of the form f u i a u u w 0 generated by the discretization of two dimensional diffusion convection problems with neumann boundary conditions existence and uniqueness of a solution of this system has been proved under standard assumptions on the matrix a u and the source term w using the fact that the matrix a u can be decomposed into different splittings we develop a nonlinear arithmetic mean method and a two stage iterative method a fixed point arithmetic mean method for solving the system above the convergence of these methods has been analyzed numerical experiments show that the fixed point arithmetic mean method is rapidly convergent when the diffusion coefficient is weakly nonlinear

experiments with conjugate gradient algorithms for homotopy curvetracking there are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points i e with probability one the essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map hompack is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve and has separate routines for dense and sparse jacobian matrices the hompack algorithms for sparse jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy jacobian matrix a required linear algebra step for homotopy curve tracking here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with craig s preconditioned conjugate gradient method used in hompack the test problems used include actual large scale sparse structural mechanics problems

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

a system of nonsmooth equations solver based upon subgradient method in this paper a subgradient method is developed to solve the system of nonsmooth equations first the system of nonsmooth equations is transformed into a nonsmooth optimization problem with zero minimal objective function value then a subgradient method is applied to solve the nonsmooth optimization problem during the processes the pre known optimal objective function value is adopted to update step sizes the corresponding convergence results are established as well several numerical experiments and applications show that the proposed method is efficient and robust

the superlinear convergence behaviour of gmres abstract gmres is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems it is well known that gmres often has a so called superlinear convergence behaviour i e the rate of convergence seems to improve as the iteration proceeds for the conjugate gradients method this phenomenon has been related to a modest degree of convergence of the ritz values it has been observed in experiments that for gmres too changes in the convergence behaviour seem to be related to the convergence of ritz values in this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by ritz values gmres from then on converges at least as fast as for a related system in which these eigenvalues and their eigenvector components are missing

on iterative solution of linear equations arising in bvps of odes abstract an application of the cimmino projection method to the solution of linear equations arising in the solution of boundary value problems of ordinary differential equations by a difference method is investigated the modified cimmino algorithm converges rapidly to the solution of corresponding linear system ax b provided some practical estimations of the diagonal dominance of aat hold the algorithm is effectively implementable on vector and multiprocessor computers

weighted fom inverse vector iteration method for computing a few smallest largest eigenvalues of pair a b the generalized eigenvalue problem ax lbx has special properties when a b is a symmetric and positive definite pair we have recently developed a new method for computing a few smallest largest eigenvalues of a symmetric positive definite problem ax lbx h saberi najafi a refahi fom inverse vector iteration method for computing a few smallest largest eigenvalues of pair a b appl math in press the method used three parameters which are weighted arnoldi inner and outer iterations parameters in this article those parameters have been optimized to increase the speed of convergence the method has been tested by some numerical examples

a quadtree adaptive multigrid solver for the serre green naghdi equations the serre green naghdi sgn equations also known as the fully nonlinear boussinesq wave equations accurately describe the behaviour of dispersive shoaling water waves this article presents and validates a novel combination of methods for the numerical approximation of solutions to the sgn equations the approach preserves the robustness of the original finite volume saint venant solver in particular for the treatment of wetting drying and equilibrium states the linear system of coupled vector equations governing the dispersive sgn momentum sources is solved simply and efficiently using a generic multigrid solver this approach generalises automatically to adaptive quadtree meshes adaptive mesh refinement is shown to provide orders of magnitude gains in speed and memory when applied to the dispersive propagation of waves during the tohoku tsunami the source code test cases and examples are freely available

numerical solution of heated subsidence mound problem in a porous medium abstract two different numerical models are constructed to solve a two dimensional subsidence mound problem heated along the moving wet dry interface one numerical model is based on cartesian coordinates while the other is based on polar coordinates in both approaches coordinate transformations are used that render the interface stationary the problem involves a system of three coupled equations an elliptic equation for a stream function a parabolic equation for the temperature and a non linear equation for the boundary location good agreement is found between the results of both methods graphic results are presented for the decay of a subsidence mound for different values of the various parameters in the model problem

global solvability of real analytic involutive systems on compact manifolds the focus of this work is the smooth global solvability of a linear partial differential operator mathbb l associated to a real analytic closed non exact 1 form b defined on a real analytic closed n manifold that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co rank one we define an appropriate covering projection widetilde m rightarrow m such that the pullback of b has a primitive widetilde b and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of widetilde b are connected as a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator when m is orientable we prove a connection between the global solvability of mathbb l and that of mathbb l n 1 which is the last non trivial operator of the complex in particular we prove that mathbb l is globally solvable if and only if mathbb l n 1 is globally solvable in the smooth category we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a morse 1 form i e when the structure is of mizohata type

polynomial preconditioning on vector computers abstract this paper is concerned with the computation of the minimal eigenpair of the generalized eigenproblem with the preconditioned conjugate gradient method the ssor polynomial preconditioner and the additive polynomial preconditioner have been analyzed the results of an extensive computer experimentation on cray y mp are reported

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

necessary and sufficient conditions for the existence of a conjugate gradient method we characterize the class cg s of matrices a for which the linear system a bf x bf b can be solved by an s term conjugate gradient method we show that except for a few anomalies the class cg s consists of matrices a for which conjugate gradient methods are already known these matrices are the hermitian matrices a a and the matrices of the form ae i theta di b with b b

changing the norm in conjugate gradient type algorithms a conjugate gradient type method for solving ax b or more precisely an orthogonal error method operatorname oe b c a is determined by a formal inner product matrix b and a left preconditioning matrix c relations are pointed out between quantities such as iterates residuals direction vectors and recurrence coefficients of orthogonal error methods with different but in a particular way related matrices b namely for operatorname oe b ca k c a k 0 1 ldots and for operatorname oe a h c h k b c a k 0 1 ldots the relations for the first sequence of methods have to do with rutishauser s lr algorithm those for the second one are based on a generalization of the schonauer weiss smoothing algorithm relevant for practice are the cases k 0 1 for b ca and b ca h

inexact overlapped block broyden methods for solving nonlinear equations in this paper a parallelizable overlapped block broyden method is presented for solving large systems of nonlinear equations the basic idea is to perform the block broyden iteration described in siam j sci comput 18 1997 1367 for the overlapped blocks and then assemble the overlapping solutions in a weighted average manner at each iteration a family of nonlinear overlapped solvers could be generated combining it with some iterative or direct linear solvers the conditions under which the algorithm is locally convergent are studied and many useful techniques regarding the implementation are also considered

on analysis of projection methods for rational function approximation to the matrix exponential we revisit the analysis of projection methods for rational function approximation to the matrix exponential and present a new result for establishing convergence of the pade plus krylov approximation when a 1

convergence properties of block gmres and matrix polynomials abstract this paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right hand sides a convergence theory is developed based on a representation of the method using matrix valued polynomials relations between the roots of the residual polynomial for block gmres and the matrix e pseudospectrum are derived and illustrated with numerical experiments the role of invariant subspaces in the effectiveness of block methods is also discussed

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

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

on the global solvability of involutive systems abstract we consider a class of involutive systems in t n 1 associated with a closed 1 form defined on the torus t n we prove that under a geometric condition the global solvability of this class is equivalent to a diophantine condition involving liouville forms and the connectedness of all sublevel and superlevel sets of a global primitive associated with the system

conjugate gradient methods for continuation problems ii we study linear conjugate gradient cg methods for large sparse continuation problems first we show how the cg methods can be incorporated as linear solvers in the context of a continuation method next we indicate how the preconditioned generalized minimal residual algorithm may be used to test for simple and multiple bifurcations without solving the bordered linear systems sample numerical results are reported some concluding remarks concerning the performance of three term cg methods are given

analysis of projection methods for rational function approximation to the matrix exponential krylov subspace methods for approximating the action of the matrix exponential exp a on a vector v are analyzed with a hermitian and negative semidefinite our approach is based on approximating the exponential with the commonly employed diagonal pade and chebyshev rational functions which yield a system of equations with a polynomial coefficient matrix we derive optimality properties and error bounds for the convergence of a galerkin type approximation and of a computationally feasible and extensively used alternative as complementary results we theoretically justify the use of a popular a posteriori error estimate and we provide upper bounds for the components of the solution vector our theoretical and numerical results show that this methodology may provide an appropriate framework to devise new strategies such as more powerful acceleration schemes