new estimates for ritz vectors the following estimate for the rayleigh ritz method is proved u u au u sin u u u 1 here a is a bounded self adjoint operator in a real hilbert euclidian space u one of its eigenpairs u a trial subspace for the rayleigh ritz method and u a ritz pair this inequality makes it possible to analyze the fine structure of the error of the rayleigh ritz method in particular it shows that u u c 2 if an eigenvector u is close to the trial subspace with accuracy and a ritz vector u is an approximation to another eigenvector with a different eigenvalue generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested and estimates of approximation of eigenspaces and invariant subspaces are proved

the concept of duality for automata over a changing alphabet and generation of a free group by such automata in the paper we deal with the notion of an automaton over a changing alphabet which generalizes the concept of a mealy type automaton we modify the methods based on the idea of a dual automaton and its action used by b steinberg et al 2011 and m vorobets and ya vorobets 2007 2010 16 18 and adapt them to automata over a changing alphabet we show that this modification provides some naturally defined automaton representations of a free nonabelian group by a 2 state automaton over a changing alphabet

equilibria of set valued maps on nonconvex domains we present new theorems on the existence of equilibria or zeros of convex as well as nonconvex set valued maps defined on compact neighborhood retracts of normed spaces the maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets among other things we show that if k is a compact neighborhood retract with nontrivial euler characteristic in a banach space e and 1 k 2e is an upper hemicontinuous set valued map with nonempty closed convex values satisfying the tangency condition x n tk x 0 for all x c k then there exists xo c k such that 0 c 4 xo here tk x denotes a new concept of retraction tangent cone to k at x suited for compact neighborhood retracts when k is locally convex at x tk x coincides with the usual tangent cone of convex analysis special attention is given to neighborhood retracts having lipschitzian behavior called l retracts below this class of sets is very broad it contains compact homeomorphically convex subsets of banach spaces epi lipschitz subsets of banach spaces as well as proximate retracts our results thus generalize classical theorems for convex domains as well as recent results for nonconvex sets

equilibria of nonconvex valued maps under constraints abstract in the paper the notion of n tangency for set valued maps defined on a subset of a banach space is considered the existence of equilibria of upper semicontinuous map being n tangent to a sleek retract with the nontrivial euler characteristic is established

a sharp version of kahan s theorem on clustered eigenvalues let n times n hermitian matrix a have eigenvalues lambda 1 lambda 2 cdots lambda n and let k times k hermitian matrix h have eigenvalues mu 1 le mu 2 le cdots le mu k and let q be an n times k matrix having full column rank so 1 le k le n it is proved that there exist k eigenvalues lambda i 1 le lambda i 2 le cdots le lambda i k of a such that max 1 le j le k mu j lambda i j le frac c sigma min q aq qh 2 always holds with c 1 where sigma min q is the smallest singular value of q and cdot 2 denotes the biggest singular value of a matrix the inequality was proved for c le sqrt 2 in 1967 by kahan who also conjectured that it should be true for c 1

the concept of self similar automata over a changing alphabet and lamplighter groups generated by such automata abstract generalizing the idea of self similar groups defined by mealy automata we introduce the notion of a self similar automaton and a self similar group over a changing alphabet we show that every finitely generated residually finite group is self similar over an arbitrary unbounded changing alphabet we construct some naturally defined self similar automaton representations over an unbounded changing alphabet for any lamplighter group k z with an arbitrary finitely generated finite or infinite abelian group k

new a priori fem error estimates for eigenvalues we analyze the ritz galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates for a simple eigenvalue we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants for a multiple eigenvalue we prove in addition what is apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace these estimates reflect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities thus resulting in different levels of errors for the approximate eigenvalues for clustered eigenvalues we derive eigenvalue error bounds that do not depend on the width of the cluster our results are readily applicable to the classical ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators

principal angles between subspaces in an a based scalar product algorithms and perturbation estimates computation of principal angles between subspaces is important in many applications e g in statistics and information retrieval in statistics the angles are closely related to measures of dependency and covariance of random variables when applied to column spaces of matrices the principal angles describe canonical correlations of a matrix pair we highlight that all popular software codes for canonical correlations compute only cosine of principal angles thus making impossible because of round off errors finding small angles accurately we review a combination of sine and cosine based algorithms that provide accurate results for all angles we generalize the method to the computation of principal angles in an a based scalar product for a symmetric and positive definite matrix a we provide a comprehensive overview of interesting properties of principal angles we prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants numerical examples and a detailed description of our code are given

piecewise automatic groups we introduce a notion of a piecewise automatic group among these groups we describe a new class of groups of intermediate growth we show that for any function f n n there exists a finitely generated torsion group of intermediate growth g for which the folner function satisfies folg s n f n for some generating set s and all sufficiently large n as a corollary we see that the asymptotic entropy of simple random walks on these groups could be arbitrarily close to being linear while the poisson boundary is trivial

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

the classification of abelian groups generated by time varying automata and by mealy automata over the binary alphabet for every natural number n we classify abelian groups generated by an n state time varying automaton over the binary alphabet as well as by an n state mealy automaton over the binary alphabet

block locally optimal preconditioned eigenvalue xolvers blopex in hypre and petsc we describe our software package block locally optimal preconditioned eigenvalue xolvers blopex recently publicly released blopex is available as a stand alone serial library as an external package to petsc portable extensible toolkit for scientific computation a general purpose suite of tools developed by argonne national laboratory for the scalable solution of partial differential equations and related problems and is also built into hypre high performance preconditioners a scalable linear solvers package developed by lawrence livermore national laboratory the present blopex release includes only one solver the locally optimal block preconditioned conjugate gradient lobpcg method for symmetric eigenvalue problems hypre provides users with advanced high quality parallel multigrid preconditioners for linear systems with blopex the same preconditioners can now be efficiently used for symmetric eigenvalue problems petsc facilitates the integration of independently developed application modules with strict attention to component interoperability and makes blopex extremely easy to compile and use with preconditioners that are available via petsc we present the lobpcg algorithm in blopex for hypre and petsc we demonstrate numerically the scalability of blopex by testing it on a number of distributed and shared memory parallel systems including a beowulf system sun fire 880 an amd dual core opteron workstation and ibm bluegene l supercomputer using petsc domain decomposition and hypre multigrid preconditioning we test blopex on a model problem the standard 7 point finite difference approximation of the 3 d laplacian with the problem size in the range of 10 5 10 8

eigenvalues of rayleigh quotient matrices support thata is ann n hermitian matrix andx 1 y 1 aren 1 matrices satisfyingx 1 h x 1 y 1 h y 1 i 1 l n such that the column space x 1 is an eigenspace ofa and y 1 is an approximation of x 1 let a 1 x 1 h ax 1 h 1 y 1 h ay 1 r ay 1 y 1 h 1 and theta arccos x 1 h y 1 y 1 h x 1 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 geqslant 0 the matricesa 1 andh 1 are called the rayleigh quotient matrices ofa with respect tox 1 andy 1 respectively this paper derives new upper bounds of the difference between h 1 and a 1 the set of the eigenvaluesh 1 anda 1 in terms of the quantities sin and or r in a simple way here denotes a unitarily invariant norm

cluster robust error estimates for the rayleigh ritz approximation i estimates for invariant subspaces this is the first part of a paper that deals with error estimates for the rayleigh ritz approximations to the spectrum and invariant subspaces of a bounded hermitian operator in a hilbert or euclidean space this part addresses estimates for the angles between the invariant subspaces and their approximations via the corresponding best approximation errors and residuals and for invariant subspaces corresponding to parts of the discrete spectrum via eigenvalue errors the paper s major concern is to ensure that the estimates in question are accurate and cluster robust i e are not adversely affected by the presence of clustered i e closely situated eigenvalues in the spectrum available estimates of such kind are reviewed and new estimates are derived the paper s main new results introduce estimates for invariant subspaces in which the operator may have clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only

optimal a priori error bounds for the rayleigh ritz method we derive error bounds for the rayleigh ritz method for the approximation to extremal eigenpairs of a symmetric matrix the bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector we also present a sharp bound

fixed points and equilibria in nonconvex sets tcentre de recherches mathtmatiques universite de montreal montreal quebec h3c 357 canada psteklov institute of mathematics moscow 117966 russia and ildepartment of mathematics and statistics concordia university montreal quebec h4b lr6 canada received 30 september 1993 received forpublication 18 august 1994 key words and phrases multivalued mapping tangent cones zeros fixed points differential inclusion weak invariance nonsmooth analysis 1 introduction

lehmann bounds and eigenvalue error estimation the paper investigates the properties of lehmann s optimal bounds for eigenvalues of hermitian problems in order to find a way to efficiently use them for eigenvalue error estimation a practical error estimation scheme is described that can be employed in the framework of a subspace iteration algorithm and is actually implemented by the hsl ea19 software package from the hsl mathematical software library of rutherford appleton laboratory

hp finite elements for elliptic eigenvalue problems error estimates which are explicit with respect to lambda h and p convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form if the mesh width h is fine enough then the eigenvalues resp eigenfunctions converge at some well defined rate in this paper we will determine the maximal mesh width h 0 more precisely the minimal dimension of a finite element space so that the asymptotic convergence estimates hold for h leq h 0 this mesh width will depend on the size and spacing of the exact eigenvalues the spatial dimension and the local polynomial degree of the finite element space for example in the one dimensional case the condition lambda 3 4 h 0 lesssim1 is sufficient for piecewise linear finite elements to compute an eigenvalue lambda with optimal convergence rates as h 0 geq h rightarrow0 it will turn out that the condition for eigenfunctions is slightly more restrictive furthermore we will analyze the dependence of the ratio of the errors of the galerkin approximation and of the best approximation of an eigenfunction on lambda and h in this paper the error estimates for the eigenvalue function are limited to the selfadjoint case however the regularity theory and approximation property cover also the nonselfadjoint case and hence pave the way towards the error analysis of nonselfadjoint eigenvalue function problems

an adaptive homotopy approach for non selfadjoint eigenvalue problems this paper presents adaptive algorithms for eigenvalue problems associated with non selfadjoint partial differential operators the basis for the developed algorithms is a homotopy method which departs from a well understood selfadjoint problem apart from the adaptive grid refinement the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources the first algorithm balances the homotopy discretization and approximation errors with respect to a fixed stepsize in the homotopy the second algorithm combines the adaptive stepsize control for the homotopy with an adaptation in space that ensures an error below a fixed tolerance e the outcome of the analysis leads to the third algorithm which allows the complete adaptivity in space homotopy stepsize as well as the iterative algebraic eigenvalue solver all three algorithms are compared in numerical examples

galerkin eigenvector approximations how close are galerkin eigenvectors to the best approximation available out of the trial subspace under a variety of conditions the galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors both orthogonal galerkin and petrov galerkin methods are considered here with a special emphasis on nonselfadjoint problems thus extending earlier studies by chatelin babuska and osborn and knyazev consequences for the numerical treatment of elliptic pdes discretized either with finite element methods or with spectral methods are discussed new lower bounds to the sep of a pair of operators are developed as well

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

existence of equilibria of set valued maps on bounded epi lipschitz domains in hilbert spaces without invariance conditions abstract in this paper we provide a new result of the existence of equilibria for set valued maps on bounded closed subsets k of hilbert spaces we do not impose either convexity or compactness assumptions on k but we assume that k has epi lipschitz sections i e its intersection with suitable finite dimensional spaces is locally the epigraph of lipschitz functions in finite dimensional spaces the famous brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set k to itself our result could be viewed as a kind of generalization of this classical result in the context of hilbert spaces and when the function or the set valued map does not necessarily map k into itself k is not invariant under the map our approach is based firstly on degree theory for compact and for condensing set valued maps and secondly on flows generated by trajectories of differential inclusions

fixed point index for weakly inward mappings abstract the fixed point index for weakly inward mappings which are not cone mappings is defined in this paper its properties are also investigated consequently we study the derivatives of weakly inward mappings and nonzero fixed points of such mappings

constrained topological degree and positive solutions of fully nonlinear boundary value problems abstract in the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form a f where a d a e is a densely defined m accretive operator in a banach space e and f u e is a tangent field defined on an open subset u of a neighborhood retract m being invariant with respect to the resolvents of a the construction is performed under the assumption that resolvents of a are completely continuous in the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction diffusion equations and equations with one dimensional p laplacian with possibly non positive perturbations as well as some controlled neumann like problems

positive stationary solutions for p laplacian problems with nonpositive perturbation the paper is devoted to the existence of positive solutions of nonlinear elliptic equations with p laplacian we provide a general topological degree that detects solutions of the problem a u f u u m where a x d a x is a maximal monotone operator in a banach space x and f m x is a continuous mapping defined on a closed convex cone m x next we apply this general framework to a class of partial differential equations with p laplacian under dirichlet boundary conditions in the paper we employ general ideas from cwiszewski and kryszewski 2009 5 where a setting suitable for the one dimensional p laplacian was introduced

on relative residual bounds for the eigenvalues of a hermitian matrix abstract let h be a hermitian matrix x an orthonormal matrix and m x hx then the eigenvalues of m approximate some eigenvalues of h with an absolute error bounded by hx xm 2 the main interest in this work is the relative distance between the eigenvalues of m and some part of the spectrum of h it is shown that distance depends on the angle between the ranges of x and hx

homotopy invariants for tangent vector fields on closed sets abstract in the paper we construct a topological degree theory for single and convex valued tangent vector fields defined on locally compact closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds the degree allows to study vector fields on sets which are neither smooth nor convex and may be applied to study continuation and bifurcation of equilibria in parameterized families of closed sets

fixed points of weakly inward multis soit x un espace de banach reel k x convexe ferme borne f k 2 x semicontinue superieurement et condensant f x convexe ferme x k et f faiblement interieur sur k alors f a un point fixe

a generalization of saad s theorem on rayleigh ritz approximations abstract let x be an eigenpair of the hermitian matrix a of order n and let u be a ritz pair from a subspace k of c 2 y saad numerical methods for large eigenvalue problems theory and algorithm wiley new york 1992 has given a simple inequality bounding sin x u in terms of sin x k in this paper we show that this inequality can be extended to an equally simple inequality for eigenspaces of non hermitian matrices

cluster robust accuracy bounds for ritz subspaces abstract given an approximating subspace for a hermitian matrix a the rayleigh ritz procedure is commonly used to compute a few approximate eigenvalues called ritz values and corresponding approximate eigenvectors called ritz vectors in this paper new bounds on the canonical angles between the invariant subspace of a associated with its few extreme smallest or largest eigenvalues and its approximating ritz subspace in terms of the differences between ritz values and the targeted eigenvalues are obtained from this result various bounds are readily available to estimate how accurate the ritz vectors computed from the approximating subspace may be based on information on approximation accuracies in the ritz values the result is helpful in understanding how ritz vectors move towards eigenvectors while ritz values are made to move towards eigenvalues

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

an oscillation free adaptive fem for symmetric eigenvalue problems a refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first order adaptive finite element method afem is presented the h 1 stability of the l 2 projection provides reliability and efficiency of the edge contribution of standard residual based error estimators for p 1 finite element methods in fact the volume contributions and even oscillations can be omitted for courant finite element methods this allows for a refined averaging scheme and so improves mao et al in adv comput math 25 1 3 135 160 2006 the proposed afem monitors the edge contributions in a bulk criterion and so enables a contraction property up to higher order terms and global convergence numerical experiments exploit the remaining l 2 error contributions and confirm our theoretical findings the averaging schemes show a high accuracy and the afem leads to optimal empirical convergence rates

cluster robust error estimates for the rayleigh ritz approximation ii estimates for eigenvalues this is the second part of a paper that deals with error estimates for the rayleigh ritz approximations of the spectrum and invariant subspaces of a bounded hermitian operator in a hilbert or euclidean space this part addresses the approximation of eigenvalues two kinds of estimates are considered i estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces and ii estimates for the same via the corresponding residuals estimates of these two kinds are needed for respectively the a priori and a posteriory error analysis of numerical methods for computing eigenvalues the paper s major concern is to ensure that the estimates in question are accurate and cluster robust i e are not adversely affected by the presence of clustered i e closely situated eigenvalues among those of interest the paper s main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only