superior properties of the presb preconditioner for operators on two by two block form with square blocks matrices or operators in two by two block form with square blocks arise in numerous important applications such as in optimal control problems for pdes the problems are normally of very large scale so iterative solution methods must be used thereby the choice of an efficient and robust preconditioner is of crucial importance since some time a very efficient preconditioner the preconditioned square block presb method has been used by the authors and coauthors in various applications in particular for optimal control problems for pdes it has been shown to have excellent properties such as a very fast and robust rate of convergence that outperforms other methods in this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs under certain conditions the condition number of the preconditioned matrix is bounded by 2 or even smaller furthermore under certain assumptions the rate of convergence is superlinear

optimal strong stationary times for random walks on the chambers of a hyperplane arrangement this paper studies markov chains on the chambers of real hyperplane arrangements a model that generalizes famous examples such as the tsetlin library and riffle shuffles we discuss cutoff for the tsetlin library for general weights and we give an exact formula for the separation distance for the hyperplane arrangement walk we introduce lower bounds which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions using similar techniques we also prove a uniform lower bound for the mixing time of glauber dynamics on a monotone system

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

multilevel algorithms for constrained compact fixed point problems in this paper the authors extend the multilevel algorithm of atkinson and brakhage for compact fixed point problems and the projected newton method of bertsekas to create a fast multilevel algorithm for parabolic boundary control problems having bound constraints on the control results are extended from finite dimension on constraint identification this approach permits both adaptive integration in time and inexact evaluation of the cost functional

applications of the brauer complex card shuffling permutation statistics and dynamical systems by algebraic group theory there is a map from the semisimple conjugacy classes of a finite group of lie type to the conjugacy classes of the weyl group picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the weyl group using the brauer complex it is proved that this measure agrees with a second measure on conjugacy classes of the weyl group induced by a construction of cellini using the affine weyl group formulas for cellini s measure in type a are found this leads to new models of card shuffling and has interesting combinatorial and number theoretic consequences an analysis of type c gives another solution to a problem of rogers in dynamical systems the enumeration of unimodal permutations by cycle structure the proof uses the factorization theory of palindromic polynomials over finite fields contact is made with symmetric function theory

infinitely many large amplitude homoclinic orbits for a class of autonomous hamiltonian systems abstract for a class of fourth order autonomous hamiltonian systems we give geometrical conditions that imply the existence of infinitely many homoclinic orbits to a saddle focus equilibrium our results are global in the sense that they do not rely on any transversality assumption they are applied to two example systems one arises in elasticity and in water wave theory the other describes the motion of a particle in a certain rotating potential

on the canonically invariant calculation of maslov indices after a short review of various ways to calculate the maslov index appearing in semiclassical gutzwiller type trace formulae we discuss a coordinate independent and canonically invariant formulation recently proposed by sugita 2000 phys lett a 266 321 2001 ann phys ny 288 227 we give explicit formulae for its ingredients and test them numerically for periodic orbits in several hamiltonian systems with mixed dynamics we demonstrate how the maslov indices and their ingredients can be useful in the classification of periodic orbits in complicated bifurcation scenarios for instance in a novel sequence of seven orbits born out of a tangent bifurcation in the h non heiles system

semisimple orbits of lie algebras and card shuffling measures on coxeter groups abstract random walk on the chambers of hyperplane arrangements is used to define a family of card shuffling measures h w x for a finite coxeter group w and real x 0 by algebraic group theory there is a map from the semisimple orbits of the adjoint action of a finite group of lie type on its lie algebra to the conjugacy classes of the weyl group choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the weyl group for types a b and the identity conjugacy class of w for all types it is proved that for q very good this measure on conjugacy classes is equal to the measure arising from h w q the possibility of refining to a map to elements of the weyl group is discussed

gmres and integral operators in this paper we show how the properties of integral operators and their approximations are reflected in the performance of the gmres iteration and how these properties can be used to smooth the gmres iterates by an implicit application of nystrom interpolation thereby strengthening the norm in which convergence takes place the smoothed iteration has very similar properties to broyden s method we present an example to illustrate the ideas

computing the maslov index of solitary waves part 1 hamiltonian systems on a four dimensional phase space abstract when solitary waves are characterized as homoclinic orbits of a finite dimensional hamiltonian system they have an integer valued topological invariant the maslov index we develop a new robust numerical algorithm to compute the maslov index to understand its properties and to study the implications for the stability of solitary waves the algorithm reported here is developed in the exterior algebra representation which leads to a fast algorithm with some novel properties new results on the maslov index for solitary wave solutions of reaction diffusion equations the fifth order korteweg de vries equation and the long wave short wave resonance equations are presented part 1 considers the case of a four dimensional phase space and part 2 considers the case of a 2 n dimensional phase space with n 2

character theory of monoids over an arbitrary field abstract the basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of munn ponizovski mcalister rhodes and zalcstein in particular mcalister determined the space of functions spanned by the irreducible characters of a finite monoid over c and the ring of virtual characters in this paper we present the corresponding results over an arbitrary field as a consequence we obtain a quick proof of the theorem of berstel and reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid this is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics

superlinear convergence using block preconditioners for the real system formulation of complex helmholtz equations abstract complex valued helmholtz equations arise in various applications and a lot of research has been devoted to finding efficient preconditioners for the iterative solution of their discretizations in this paper we consider the helmholtz equation rewritten in real valued block form and use a preconditioner in a special two by two block form we show that the corresponding preconditioned krylov iteration converges at a mesh independent superlinear rate

fast computation of the maslov index for hyperbolic linear systems with periodic coefficients the maslov index is a topological property of periodic orbits of finite dimensional hamiltonian systems that is widely used in semiclassical quantization quantum chaology stability of waves and classical mechanics the maslov index is determined from the analysis of a linear hamiltonian system with periodic coefficients in this paper a numerical scheme is devised to compute the maslov index for hyperbolic linear systems when the phase space has a low dimension the idea is to compute on the exterior algebra of the ambient vector space where the lagrangian subspace representing the unstable subspace is reduced to a line when the exterior algebra is projectified the lagrangian subspace always forms a closed loop the idea is illustrated by application to hamiltonian systems on a phase space of dimension 4 the theory is used to compute the maslov index for the spectral problem associated with periodic solutions of the fifth order korteweg de vries equation

some properties of the arnoldi based methods for linear ill posed problems in this paper we study some properties of the classical arnoldi based methods for solving infinite dimensional linear equations involving compact operators these problems are intrinsically ill posed since a compact operator does not admit a bounded inverse we study the convergence properties and the ability of these algorithms to estimate the dominant singular values of the operator

a newton gmres approach for the analysis of the postbuckling behavior of the solutions of the von karman equations we propose a newton gmres type algorithm to solve the discrete von karman problem we show that this algorithm is efficient both in memory and computation time and robust in the neighborhood of the singular points of the bifurcation diagrams placing ourselves in the context of the schaeffer and golubitsky theory we use this algorithm to study the postbuckling behavior of a rectangular plate clamped and compressed along its four sides

convergence of krylov subspace solvers with schwarz preconditioner for the exterior maxwell problem the consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties in this paper we are interested in the resolution of an exterior maxwell problem in 3d as a first step we focus on the justification of an algorithm described in literature using an interpretation as a schwarz method the study of the convergence indicates that it depends significantly on the thickness of the domain of computation this analysis suggests the use of the finite element term of schwarz method as a preconditioner for use of krylov iterative solvers an analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach the consideration of the preconditioner suggested by the schwarz method leads to a superlinear convergence of the gmres predicted by the analytical study and verified numerically

iterative methods for large continuation problems abstract the computation of solution paths for continuation problems requires the solution of a sequence of nonlinear systems of equations each nonlinear system can be solved by computing the solution of a succession of linear systems of equations determined by jacobian matrices associated with the nonlinear system of equations points on the solution path where the jacobian matrix is singular are referred to as singular points and require special handling they may be turning points or bifurcation points in order to detect singular points it is essential to monitor the eigenvalues of smallest magnitude of the jacobian matrices generated as the solution path is traversed we describe iterative methods for the computation of solution paths for continuation problems so large that factorization of the jacobian matrices is infeasible or impractical the iterative methods simultaneously solve linear systems of equations determined by the jacobian matrices and compute a few eigenvalue eigenvector pairs associated with the eigenvalues of smallest magnitude of each jacobian a bordering algorithm with a pseudo arclength parametrization is applied in the vicinity of turning points to overcome the singularity of the jacobian a bifurcation perturbation strategy is used to compute solution paths at bifurcation points our iterative methods are based on the block lanczos algorithm and are applicable to problems with large symmetric jacobian matrices

hyperplane arrangements interval orders and trees abstract a hyperplane arrangement is a finite set of hyperplanes in a real affine space an especially important arrangement is the braid arrangement which is the set of all hyperplanes xi xj 1 1 i j n in rn some combinatorial properties of certain deformations of the braid arrangement are surveyed in particular there are unexpected connections with the theory of interval orders and with the enumeration of trees for instance the number of labeled interval orders that can be obtained from n intervals i1 in of generic lengths is counted there is also discussed an arrangement due to n linial whose number of regions is the number of alternating or intransitive trees as defined by gelfand graev and postnikov gelfand i m graev m i and postnikov a 1995 preprint finally a refinement is given related to counting labeled trees by number of inversions of a result of shi shi j y 1986 lecture notes in mathematics no 1179 springer verlag that a certain deformation of the braid arrangement has n 1 n 1 regions

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

condition estimates for pseudo arclength continuation we bound the condition number of the jacobian in pseudo arclength continuation problems and we quantify the effect of this condition number on the linear system solution in a newton gmres solve pseudo arclength continuation solves parameter dependent nonlinear equations g u lambda 0 by introducing a new parameter s which approximates arclength and viewing the vector x u lambda as a function of s in this way simple fold singularities can be computed directly by solving a larger system f x s 0 by simple continuation in the new parameter s it is known that the jacobian f x of f with respect to x u lambda is nonsingular if the path contains only regular points and simple fold singularities we introduce a new characterization of simple folds in terms of the singular value decomposition and we use it to derive a new bound for the norm of f x 1 we also show that the convergence rate of gmres in a newton step for f x s 0 is essentially the same as that of the original problem g u lambda 0 in particular we prove that the bounds on the degrees of the minimal polynomials of the jacobians f x and g u differ by at most 2 we illustrate the effectiveness of our bounds with an example from radiative transfer theory

random walks on quasisymmetric functions abstract conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations spectral results are also obtained several important random walks are now realized this way stanley s qs distribution results from endomorphisms given by evaluation maps a shuffles result from the a th convolution power of the universal character and the tchebyshev operator of the second kind introduced recently by ehrenborg and readdy yields traditional riffle shuffles a conjecture of ehrenborg regarding the spectra for a family of random walks on ab words is proven a theorem of stembridge from the theory of enriched p partitions is also recovered as a special case

computing the maslov index of solitary waves part 2 phase space with dimension greater than four this paper extends the theory of the maslov index of solitary waves in part 1 to the case where the phase space is of dimension greater than four the starting point is hamiltonian pdes in one space dimension and time whose steady part is a hamiltonian ode with a phase space of dimension six or greater this steady hamiltonian ode is the main focus of the paper homoclinic orbits of the steady ode represent solitary waves of the pde and one of the properties of the homoclinic orbits is the maslov index we develop formulae for the maslov index the maslov angle and its subangles in an exterior algebra framework and develop numerical algorithms to compute them in addition a new numerical approach based on a discrete qr algorithm is proposed the maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time dependent problem the theory is applied to a class of reaction diffusion equations the longwave shortwave resonance equations and the seventh order kdv equation

tracing the buckling of a rectangular plate with the block gmres method we investigate bifurcation scenario of the von karman equations with partially clamped boundary conditions defined in rectangular domains first we study how the preconditioned block gmres method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations next we discuss linear stabilities of the von karman equations with partially clamped boundary conditions in particular we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von karman equations via computer algebra the block gmres method is used to solve linear systems and to detect singularity along solution paths of the discrete problem sample numerical results are reported

perverse sheaves over real hyperplane arrangements let h be an arrangement of real hyperplanes in r the complexification of h defines a natural stratification of c we denote by perv c h the category of perverse sheaves on c smooth with respect to this stratification we give a description of perv c h as the category of representations of an explicit quiver with relations whose vertices correspond to real faces of h of all dimensions the relations are of monomial nature they identify some pairs of paths in the quiver they can be formulated in terms of the oriented matroid associated to h

simultaneous solution of large scale linear systems and eigenvalue problems with a parallel gmres method a method for simultaneous solution of large and sparse linearized equation sets and the corresponding eigenvalue problems is presented such problems arise from the discretization and the solution of nonlinear problems with the finite element method and newton iteration the method is based on a parallel version of the preconditioned gmres m by deflation the parallel code exploits the architecture of the computational clusters using the mpi message passing interface the convergence rate the parallel speedup and the memory requirements of the proposed method are reported and evaluated

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

bicgstab vpastab and an adaptation to mildly nonlinear systems the key equations of bicgstab are summarised to show its connections with pade and vector pade approximation these considerations lead naturally to stabilised vector pade approximation of a vector valued function vpastab and an algorithm for the acceleration of convergence of a linearly generated sequence of vectors a generalisation of this algorithm for the acceleration of convergence of a nonlinearly generated system is proposed here and comparative numerical results are given

on the quiver of the descent algebra abstract we study the quiver of the descent algebra of a finite coxeter group w the results include a derivation of the quiver of the descent algebra of types a and b our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to w

mesh independence of the generalized davidson algorithm abstract we give conditions under which the generalized davidson algorithm for eigenvalue computations is mesh independent in this case mesh independence means that the iteration statistics residual norms convergence rates for example of a sequence of discretizations of a problem in a banach space converge the statistics for the infinite dimensional problem we illustrate the result with several numerical examples

superlinear convergence of conjugate gradients we give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other krylov subspace methods we present a new bound on the relative error after n iterations this bound is valid in an asymptotic sense when the size n of the system grows together with the number of iterations the bound depends on the asymptotic eigenvalue distribution and on the ratio n n under appropriate conditions we show that the bound is asymptotically sharp our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials an important tool in our investigations is a constrained energy problem in logarithmic potential theory the new asymptotic bounds for the rate of convergence are illustrated by discussing toeplitz systems as well as a model problem stemming from the discretization of the poisson equation

on the maslov index of multi pulse homoclinic orbits multi pulse homoclinic orbits of hamiltonian systems on can be classified by a sequence of integers in this paper we find the surprising result that this string of integers encodes the value of the maslov index of the homoclinic orbit our results include a computable formulation of the maslov index for homoclinic orbits and a robust numerical method for the evaluation of the maslov index

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

spectral analysis of random to random markov chains abstract we compute the eigenvalues and eigenspaces of random to random markov chains we use a family of maps which reveal a remarkable recursive structure of the eigenspaces yielding an explicit and effective construction of all eigenbases starting from bases of the kernels

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

convergence bounds of gmres with schwarz preconditioner for the scattering problem we consider the jacobi preconditioner of the gmres method introduced by liu and jin for the scattering problem ieee trans ante prop 2002 50 132 140 we explain why it is a particular form of the schwarz preconditioner with a complete overlap and specific transmission conditions so far a superlinear convergence has been predicted by the general theory without any additional indication on the convergence rates here we establish error bounds that provide accurate convergence rates in two and three dimensions courant weyl s min max principle applied to some kernel operators together with some polynomial approximation estimates are the milestones for the proofs

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

a note on the superlinear convergence of gmres in the paper it is shown how the rate of convergence of the gmres method for solving a linear operator equation lambda i k u f in a hilbert space is related to the degree of compactness of k measured by the products of its singular values

directed nonabelian sandpile models on trees we define two general classes of nonabelian sandpile models on directed trees or arborescences as models of nonequilibrium statistical physics unlike usual applications of the well known abelian sandpile model these models have the property that sand grains can enter only through specified reservoirs

the local convergence of broyden like methods on lipschitzian problems in hilbert spaces the classical local convergence analysis for broyden s method requires a finite number of real variables holder continuity of the jacobian and a rather accurate initial approximation b 0 here these three conditions are relaxed to allow for lipschitz continuous hilbert space operators that can be reasonably well approximated by a linear operator b that is close to b 0 up to an arbitrary compact perturbation to achieve local and r linear convergence broyden s method is modified by a very mild line search condition in case of a continuously differentiable system in euclidean space we obtain r superlinear convergence without restricting b 0 or requiring the jacobian to be holder continuous

dynamics of nearest neighbour competitions on graphs considering a collection of agents representing the vertices of a graph endowed with integer points we study the asymptotic dynamics of the rate of the increase of their points according to a very simple rule we randomly pick an an edge from the graph which unambiguously defines two agents we give a point the the agent with larger point with probability p and to the lagger with probability q such that p q 1 the model we present is the most general version of the nearest neighbour competition model introduced by ben naim vazquez and redner we show that the model combines aspects of hyperbolic partial differential equations as that of a conservation law graph colouring and hyperplane arrangements we discuss the properties of the model for general graphs but we confine in depth study to d dimensional tori we present a detailed study for the ring graph which includes a chemical potential approximation to calculate all its statistics that gives rather accurate results the two dimensional torus not studied in depth as the ring is shown to possess critical behaviour in that the asymptotic speeds arrange themselves in two coloured islands separated by borders of three other colours and the size of the islands obey power law distribution we also show that in the large d limit the d dimensional torus shows inverse sine law for the distribution of asymptotic speeds

nonequivalence deflation for the solution of matrix latent value problems the following nonlinear latent value problem is studied f x 0 where f is an n n analytic nondefective matrix function in the scalar the latent pair x has been previously found by applying newton s method to a certain equation the deflation technique is required for finding another latent pair starting from a computed latent pair several deflation strategies are examined and the nonequivalence deflation technique is developed it is demonstrated by analysis and numerical experience to be a reliable and efficient strategy for finding a few latent roots in a given region

discrete qmr and bcg in the numerical solution of linear systems of odes in this paper we consider the use of qmr and bcg in the numerical solution of linear systems of odes reformulated as second kind integral equations the implementation of the methods is performed in practice by applying them on suitably discretized equations an important issue in this contest is to compare the behaviour of the discrete and continuous iteration here we consider the case of linear systems of odes the discretization is performed using a continuous runge kutta method a numerical experiment concludes the paper

mesh independence of matrix free methods for path following in this paper we consider a matrix free path following algorithm for nonlinear parameter dependent compact fixed point problems we show that if these problems are discretized so that certain collective compactness and strong convergence properties hold then this algorithm can follow smooth folds and capture simple bifurcations in a mesh independent way

unified theory for finite markov chains abstract we provide a unified framework to compute the stationary distribution of any finite irreducible markov chain or equivalently of any irreducible random walk on a finite semigroup s our methods use geometric finite semigroup theory via the karnofsky rhodes and the mccammond expansions of finite semigroups with specified generators this does not involve any linear algebra the original tsetlin library is obtained by applying the expansions to p n the set of all subsets of an n element set our set up generalizes previous groundbreaking work involving left regular bands or r trivial bands by brown and diaconis extensions to r trivial semigroups by ayyer steinberg thiery and the second author and important recent work by chung and graham the karnofsky rhodes expansion of the right cayley graph of s in terms of generators yields again a right cayley graph the mccammond expansion provides normal forms for elements in the expanded s using our previous results with silva based on work by berstel perrin reutenauer we construct infinite semaphore codes on which we can define markov chains these semaphore codes can be lumped using geometric semigroup theory using normal forms and associated kleene expressions they yield formulas for the stationary distribution of the finite markov chain of the expanded s and the original s analyzing the normal forms also provides an estimate on the mixing time

bifurcation detection with the un preconditioned gmres m abstract when iterative solvers are used for the solution of the linearized equations sets arising from newton s method detection of bifurcation points in the solution space can be done by solving several eigenvalue problems with the jacobian matrix during parameter continuation monitoring the condition number of appropriate matrices involved in the gmres method employed here replaces or significantly reduces the costly eigenvalue computations the effectiveness of the detection method is examined in connection with the usage of unpre and preconditioned iterative solvers

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

random walks and hyperplane arrangements let be the set of chambers of a real hyperplane arrangement we study a random walk on introduced by bidigare hanlon and rockmore this includes various shuffling schemes used in computer science biology and card games it also includes random walks on zonotopes and zonotopal tilings we find the stationary distributions of these markov chains give good bounds on the rate of convergence to stationarity and prove that the transition matrices are diagonalizable the results are extended to oriented matroids

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n