similarities of the integral pade approximants ii in this paper we show further similarities of the integral pade approximant the similarity of the integral pade approximant with the functional pade approximant and the improved functional epsilon algorithms is demonstrated the purpose of demonstrating the similarity between the integral pade approximant and the functional pade approximant is to verify our conjecture which is the integral pade approximant and the hybrid functional pade approximant produce identical rational functions finally we demonstrate the improvement of the classical functional epsilon algorithms which also has a relationship with the integral pade approximant

minimal completion problem for quasi regular matrix pencils abstract in this paper we study the problem of completion of an arbitrary matrix pencil by rows and columns in order to obtain a quasi regular matrix pencil in the case when the number of added columns or rows is the minimal possible to obtain a quasi regular matrix pencil in the solution we use combinatorial methods involving lr coefficients a solution of the carlson problem and localization techniques the result is given over algebraically closed fields

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 the convergence of rows of vector pade approximants we state and prove a de montessus like theorem for vector valued meromorphic functions using vector pade approximants which are defined in the context of clifford algebras the proof is based on that used by saff for the scalar case the theorem may be regarded as an answer to a general problem posed by graves morris and saff

an accelerated poincare map method for autonomous oscillators a novel time domain method for finding the periodic steady state of a free running electrical oscillator is introduced the method is based on the extrapolation technique mpe the new method is applied to a family of artificial benchmark problems and to a real circuit colpitt s oscillator it turns out to have super linearly convergence properties in both cases full implementational details are provided

some generic invariant factor assignment results using dynamic output feedback abstract working with input output transfer functions in the frequency domain and exploiting a formulation involving generalized sylvester resultants we are able to derive necessary and sufficient conditions for generic invariant factor assignment in several cases using proper dynamic output feedback compensators

quadratic convergence of an extrapolation method abstract extrapolation methods serve to accelerate the convergence of iteration processes like nonlinear programming or root finding these methods are mostly generalizations of wellknown one dimensional schemes whose favourable properties are expected to be retained in several dimensions such an important property is quadratic acceleration which is the central topic of this paper the acceleration property is demonstrated on a simple gradient method and the result is compared to direct application of more advanced iteration methods

picard vessiot extensions of artinian simple module algebras abstract this paper pursues takeuchi s hopf algebraic approach m takeuchi a hopf algebraic approach to the picard vessiot theory j algebra 122 1989 481 509 to the picard vessiot pv theory for differential equations to involve the pv extensions of difference equations differential fields and total difference rings in the standard pv theory are unified here by artinian simple as module algebras over a cocommutative pointed smooth hopf algebra

clifford algebras and vector valued rational forms i we demonstrate how the use of clifford algebras in the theory of vector valued rational forms leads to practical recurrence relations which do not involve representations of the algebras

a chevalley theorem for difference equations by a theorem of chevalley the image of a morphism of varieties is a constructible set the algebraic version of this fact is usually stated as a result on extension of specializations or lifting of prime ideals we present a difference analog of this theorem the approach is based on the philosophy that occasionally one needs to pass to higher powers of where is the endomorphism defining the difference structure in other words we consider difference pseudo fields which are finite direct products of fields rather than difference fields we also prove a result on compatibility of pseudo fields and present some applications of the main theorem e g constrained extension and uniqueness of differential picard vessiot rings with a difference parameter

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

iterative residual based vector methods to accelerate fixed point iterations fixed point iterations are still the most common approach to dealing with a variety of numerical problems such as coupled problems multi physics domain decomposition or nonlinear problems electronic structure heat transfer nonlinear mechanics methods to accelerate fixed point iteration convergence or more generally sequence convergence have been extensively studied since the 1960 s for scalar sequences the most popular and efficient acceleration method remains the 2 of aitken various vector acceleration algorithms are available in the literature which often aim at being multi dimensional generalizations of the 2 method in this paper we propose and analyse a generic residual based formulation for accelerating vector sequences the question of the dynamic use of this residual based transformation during the fixed point iterations for obtaining a new accelerated fixed point method is then raised we show that two main classes of such iterative algorithms can be derived and that this approach is generic in that various existing acceleration algorithms for vector sequences are thereby recovered in order to illustrate the interest of such algorithms we apply them in the field of nonlinear mechanics on a simplified point wise solver used to perform mechanical behaviour unit testings the proposed test cases clearly demonstrate that accelerated fixed point iterations based on the elastic operator quasi newton method are very useful when the mechanical behaviour does not provide the so called consistent tangent operator moreover such accelerated algorithms also prove to be competitive with respect to the standard newton raphson algorithm when available

linear systems some algebraic aspects abstract in the past twenty years mathematically minded engineers working in control theory have discovered a number of results of a purely algebraic nature the present article aims at giving a survey of this area for readers interested in linear algebra but without previous knowledge of this topic

a hopf algebraic approach to the picard vessiot theory abstract this paper shows what is really happening in picard vessiot theory our hopf algebraic approach simplifies and generalizes the theory among other things the generalization gives a picard vessiot theory for positive characteristic for fields with not necessarily commuting derivations and even for fields with a set of higher derivations

the number of invariant polynomials of a matrix with prescribed off diagonal blocks abstract let a be an n n matrix over an arbitrary field f of the form a a 1 2 a 1 2 a 2 1 a 2 2 where a1 1 fp p and a2 2 fq q and p q n we characterize the possible number of nontrivial invariant polynomials of a when the submatrices a1 2 and a2 1 are prescribed

a difference version of nori s theorem we consider frobenius difference equations over mathbb f q s t phi q where phi q fixes t and acts on mathbb f q s as the frobenius endomorphism we prove that every semisimple simply connected linear algebraic group mathcal g defined over mathbb f q can be realized as a difference galois group over mathbb f q i s t phi q i for some i in mathbb n the proof uses upper and lower bounds on the galois group scheme of a frobenius difference equation that are developed in this paper the result can be seen as a difference analogue of nori s theorem which states that mathcal g mathbb f q occurs as a finite galois group over mathbb f q s

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

the change of the jordan structure of a mtrix under small perturbations abstract in this paper a complete description including multiplicity is given for the jordan structure of a matrix which is a small perturbation of a matrix of known jordan structure the problem solved here was solved independently as noted and the other solution has been published in english however the solutions are different the problem is important and it seems desirable to have this solution widely available

accelerating numerical simulations of strain adaptive bone remodeling predictions abstract bone modeling and remodeling have been extensively studied in many fields of research the process of adaptive bone remodeling can be described mathematically and simulated in a computer model integrated with the finite element method fem the fem can determine the long term behavior of bone and the impact on bone biomechanics produced by prosthetic models as whole bone remodeling simulations are time consuming the aim of this paper is to combine fe analyses with numerical extrapolation techniques to accelerate bone remodeling simulations two vector extrapolation methods reduce ranked extrapolation rre and minimal polynomial extrapolation mpe are used to reduce the simulation time these extrapolation techniques are illustrated by numerical examples of 2 d and 3 d models based on a strain adaptive bone remodeling theory

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

development of iterative techniques and extrapolation methods for drazin inverse solution of consistent or inconsistent singular linear systems abstract consider the linear system of equations bx where b is an nxn singular matrix in an earlier work by the author it was shown that iterative techniques coupled with standard vector extrapolation methods can be used to obtain or approximate a solution of this system when it is consistent in the present work we expand on that approach to treat the case in which this system is in general inconsistent starting with richardson s iterative method we develop a family of new iterative techniques and vector extrapolation methods that enable us to obtain or approximate the drazin inverse solution of this system whether the index of b is 1 or greater than 1 we show that the drazin inverse solution can be constructed from a finite number of iterations this number being at most n 2 we also provide detailed convergence analyses of the new iterative techniques and vector extrapolation methods and give their precise rates of convergence

convergence and stability analyses for some vector extrapolation methods on the presence of defective iteration matrices abstract in two previous papers 10 11 convergence and stability results for the following vector extrapolation methods were presented minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm the analyses were carried out for vector sequences that include those arising from iterative methods for linear systems of equations having diagonalizable iteration matrices in this paper the analyses of 10 11 are extended to vector sequences that include those arising from iterative methods for linear systems having defective iteration matrices the results are illustrated with numerical examples the analyses above naturally suggest some old and some new extensions of the well known power method enabling one to obtain estimates for several dominant eigenvalues of a general matrix

introduction to the determinantal formulae for the levin type algorithms abstract there are two aims of this paper firstly we shall introduce the determinantal formulae for the new levin type algorithms and secondly we shall demonstrate the similarities between the new levin type algorithms and the well established methods namely the aitken 2 algorithm the brezinski theta algorithm and the lubkin transformation actually there are two groups of similarities first of the group demonstrates the similarity between the super enhanced levin algorithm the super modified levin algorithm the brezinski theta algorithm and the lubkin transformation the second group of similarity is between the modified levin algorithm the original levin transformation and the aitken 2 algorithm

extrapolation methods for vector sequences an analogue of aitken s 2 method suitable for vector sequences is proposed aspects of the numerical performance of the vector algorithm based on using the moore penrose inverse are investigated the fact that the denominator polynomial associated with a vector pade approximant is the square of its equivalent in the scalar case is shown to be a source of approximation error in cases where the convergence of the vector sequence is dominated by real eigenvalues a hybrid form of the vector pade approximant having a denominator polynomial of minimal degree is proposed and its effectiveness is demonstrated on several standard examples

some results about vector extrapolation methods and related fixed point iterations abstract in this work we consider polynomial extrapolation methods for vector sequences we give a unified presentation of these methods and a new recursive algorithm for the implementation of the modified minimal polynomial extrapolation mmpe method properties operation counts and storage requirements of the algorithm will be given the second part of the work is devoted to the application of these methods for solving systems of nonlinear equations a sufficient condition for quadratic convergence is given

introduction to the improved functional epsilon algorithm this paper introduces the improved functional epsilon algorithm we have defined this new method in principle of the modified aitken 2 algorithm moreover we have found that the improved functional epsilon algorithm has remarkable precision of the approximation of the exact solution and there exists a relationship with the integral pade approximant the use of the improved functional epsilon algorithm for accelerating the convergence of sequence of functions is demonstrated the relationship of the improved functional epsilon algorithm with the integral pade approximant is also demonstrated moreover we illustrate the similarity between the integral pade approximant and the modified aitken 2 algorithm thus we have shown that the integral pade approximant is a natural generalisation of modified aitken 2 algorithm

interlacing inequalities matrix groups abstract this paper has two parts 1 interlacing inequalities in linear algebra eigenvalues singular values invariant polynomials attempts to construct a unifying theory 2 subgroups of the full linear group defined by polynomial identities

matrices with prescribed nonprincipal blocks and the trace function abstract let a and b be fixed matrices we discuss the existence of matrices u and v such that vau 1 ubv 1 is nonscalar and simultaneously has a prescribed trace

the lasker noether theorem for commutative and noetherian module algebras over a pointed hopf algebra abstract let h be a pointed hopf algebra over a field let a be a commutative noetherian h module algebra and let i be an invariant ideal in a such that g p p for any group like element g h and any associated prime p ass i we prove that i admits an irredundant primary decomposition i q 1 q n such that each q i is invariant moreover we introduce the concept of a convolutionally hopf algebra and show that each associated prime of the ideal i is invariant provided the hopf algebra h is convolutionally reduced also it will be proved that in characteristic 0 every connected hopf algebra is convolutionally reduced

vector rational interpolation algorithms of burlisch stoer neville form we present algorithms of the bulirsch stoer neville form for calculating the value of the vector rational interpolant the derivation parallels that of the bulirsch stoer algorithm for the usual scalar rational interpolation algorithm of stoer and bulirsch using the clifford algebra foundations for vector rational interpolants of roberts

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

matrices with prescribed eigenvalues and prescribed submatrices ii let f be a field and let n p 1 p 2 p 3 be positive integers such that n p 1 p 2 p 3 let c c 1 1 c 1 2 c 1 3 c 2 1 c 2 2 c 2 3 c 3 1 c 3 2 c 3 3 f n i n where the blocks c i j f p i i p j i j 1 2 3 in this paper we describe the eigenvalues of c when c 1 2 c 1 3 c 2 2 are prescribed with c 1 3 0 and the other blocks are unknown for the same prescription of blocks for arbitrary prescription of c 1 3 we still provide a sufficient condition for the prescription of the characteristic polynomial of c

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

hopf module duality applied to x outer galois theory we work over a fixed ground field k in arbitrary characteristic let r be a prime algebra and let q denote the symmetric martindale quotient ring of r we know from 10 section 6 4 that to study group actions on r it is fundamental to consider the extended actions on q and those actions which are x outer or outer as automorphisms of q have nice properties which could be generalized for x outer in some appropriate sense hopf algebra actions to recall kharchenko s galois type correspondence theorem let g be a finite group of x outer automorphisms of r it was proved in 2 theorem 3 10 2 11 theorem b that

existence of parameterized picard vessiot extensions over fields with algebraically closed constants abstract the purpose of this short note is to establish the existence of parameterized picard vessiot extensions for systems of linear difference differential equations over difference differential fields with an algebraically closed field of constants

acceleration of convergence of vector sequences a general approach to the construction of convergence acceleration methods for vector sequences is proposed using this approach one can generate some known methods such as the minimal polynomial extrapolation the reduced rank extrapolation and the topological epsilon algorithm and also some new ones some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties the convergence analysis of these new methods is carried out and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively a stability analysis is also given and numerical examples are provided the convergence and stability properties of the topological epsilon algorithm are likewise given

differential embedding problems over laurent series fields abstract we solve the inverse differential galois problem over the fraction field of k t x and use this to solve split differential embedding problems over k t x that are induced from k x the proofs use patching as well as prior results on inverse problems and embedding problems

hybrid vector transformations we present in this paper hybrid and semi hybrid vector extrapolation methods for a vector sequence transformation we will define the generalized residual which allows us to introduce a new hybrid vector transformation when solving linear systems this new transformation reduces to the hybrid procedure defined by brezinski and redivo zaglia 1994 which is a generalization of the smoothing technique in the second part of this paper we define a new composite transformation called semi hybrid when applied to linearly generated sequences we will show that the residual polynomial of the semi hybrid transformation is the product of two residual polynomials

on the number of invariant polynomials of partially prescribed matrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 a 1 2 a 2 2 are prescribed

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

generalized q boson algebras and their integrable modules abstract we define the generalized q boson algebra b associated to a pair of nichols algebras and a skew pairing we study integrable b modules generalizing results by m kashiwara and t nakashima on integrable modules over a q boson kashiwara algebra

convergence of vectorial sequences applications a transformation of vectorial sequences is given analogous to the shank s transform in the scalar case this is possible because of the link with the vector pad e approximants defined for a power series with vectorial coefficients algorithms different from recursive computation of determinants can be defined some numerical examples are given

galois theory for iterative connections and nonreduced galois groups this article presents a theory of modules with iterative connection this theory is a generalisation of the theory of modules with connection in characteristic zero as given by n katz to modules over rings of arbitrary characteristic we show that these modules with iterative connection and also the modules with integrable iterative connection form a tannakian category assuming some nice properties for the underlying ring and we show how this generalises to modules over schemes stratifications on modules as introduced by a grothendieck generalise the notion of integrable ordinary connections in another way however over smooth rings we obtain an equivalence of stratifications and integrable iterative connections furthermore over a regular ring in positive characteristic we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by d gieseker in the second part of this article we set up a picard vessiot theory for fields of solutions for such a picard vessiot extension we obtain a galois correspondence which takes into account even nonreduced closed subgroup schemes of the galois group scheme on one hand and inseparable intermediate extensions of the picard vessiot extension on the other hand finally we compare our galois theory with the galois theory for purely inseparable field extensions given by s chase

vector extrapolation methods applications and numerical comparison the present paper is a survey of the most popular vector extrapolation methods such as the reduced rank extrapolation rre the minimal polynomial extrapolation mpe the modified minimal polynomial extrapolation mmpe the vector e algorithm vea and the topological e algorithm tea using projectors we derive a different interpretation of these methods and give some theoretical results the second aim of this work is to give a numerical comparison of the vector extrapolation methods above when they are used for practical large problems such as linear and nonlinear systems of equations

solution of integral equations using generalised inverse function valued pade approximants i abstract neumann series solutions for the linear integral equation x 1 1 0 1 x y y d y converge for small values of we use generalised inverse function valued pade approximants to accelerate the convergence of the neumann series and to estimate the singular value of these estimates are found to compare very favourably with those from the fredholm determinant method and the pade approximant method

acceleration methods for vector sequences the purpose of this paper is to propose acceleration methods for certain logarithmically convergent vector sequences these methods are obtained by extending acceleration methods for scalars lubkin s w transform the algorithm and the algorithm applied to a system of singular equations new methods are compared with the vector algorithm the minimal polynomial extrapolation and the generalized algorithm of brezinski theoretical results are also given

vector extrapolation methods with applications to solution of large systems of equations and to pagerank computations an important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors x n where x n c n with n very large such sequences arise for example in the solution of systems of linear or nonlinear equations by fixed point iterative methods and lim n x n are simply the required solutions in most cases of interest however these sequences converge to their limits extremely slowly one practical way to make the sequences x n converge more quickly is to apply to them vector extrapolation methods in this work we review two polynomial type vector extrapolation methods that have proved to be very efficient convergence accelerators namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre we discuss the derivation of these methods describe the most accurate and stable algorithms for their implementation along with the effective modes of usage in solving systems of equations nonlinear as well as linear and present their convergence and stability theory we also discuss their close connection with the method of arnoldi and with gmres two well known krylov subspace methods for linear systems we show that they can be used very effectively to obtain the dominant eigenvectors of large sparse matrices when the corresponding eigenvalues are known and provide the relevant theory as well one such problem is that of computing the pagerank of the google matrix which we discuss in detail in addition we show that a recent extrapolation method of kamvar et al that was proposed for computing the pagerank is very closely related to mpe we present a generalization of the method of kamvar et al along with a very economical algorithm for this generalization we also provide the missing convergence theory for it

vector orthogonal polynomials and matrix series abstract existence and uniqueness of the vector orthogonal polynomials associated to f z n 0cznanx0 are studied where cn c x0 c d a a square d d matrix a necessary and sufficient condition for the existence and uniqueness of the vector polynomial psd is given when a is a regular matrix the vector orthogonal polynomials associated to the resolvent of the matrix a are studied and the eigenvalues of a are computed by using the vector qd algorithm an initialization of that algorithm is given finally the normality of the pade table of the function f z ezax0 is evoked

vector extrapolated fast maximum likelihood estimation algorithms for emission tomography a new class of fast maximum likelihood estimation mle algorithms for emission computed tomography ect is developed in these cyclic iterative algorithms vector extrapolation techniques are integrated with the iterations in gradient based mle algorithms with the objective of accelerating the convergence of the base iterations this results in a substantial reduction in the effective number of base iterations required for obtaining an emission density estimate of specified quality the mathematical theory behind the minimal polynomial and reduced rank vector extrapolation techniques in the context of emission tomography is presented these extrapolation techniques are implemented in a positron emission tomography system the new algorithms are evaluated using computer experiments with measurements taken from simulated phantoms it is shown that with minimal additional computations the proposed approach results in substantial improvement in reconstruction

matrix extrapolation algorithms abstract extrapolation algorithms are used for accelerating scalar or vector sequences they are also used for solving systems of linear and nonlinear equations these algorithms are expressed in terms of a ratio of two determinants as the e algorithm and the general recursive projection algorithm grpa in this paper we define the matrix extrapolation problem and we use the schur complement and the sylvester identity for solving this problem we give two transformations equivalent to the e algorithm and the grpa for the matrix case we give also a characterization of their kernels

difference galois theory of linear differential equations abstract we develop a galois theory for linear differential equations equipped with the action of an endomorphism this theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation the galois groups here are linear difference algebraic groups i e matrix groups defined by algebraic difference equations

matrix completion problems abstract throughout the last decades several results have been published in the area of the so called matrix completion problems in this paper we survey several results in this field in particular we describe the possible eigenvalues the characteristic polynomial the invariant polynomials or the number of nontrivial invariant polynomials of a square matrix over a field when some of its entries are prescribed and the others vary finally we present our contribution generalizing some of the previous cases to an n n matrix partitioned into k k blocks with entries in a field when some of its blocks are prescribed and the others vary

acceleration schemes for the method of alternating projections abstract in this paper we develop and analyze schemes for accelerating the convergence of the alternating projections method for finding the projection of a point onto the intersection of a finite number of subspaces of a hilbert space a special method for the case of two subspaces is developed and two general schemes are determined for the case of an arbitrary number of subspaces proofs of convergence and acceleration are given for each method finally techniques for constructing test problems are outlined and the results of computational experiments are reported

efficient implementation of minimal polynomial and reduced rank extrapolation methods abstract the minimal polynomial extrapolation mpe and reduced rank extrapolation rre are two very effective techniques that have been used in accelerating the convergence of vector sequences such as those that are obtained from iterative solution of linear and nonlinear systems of equations their definitions involve some linear least squares problems and this causes difficulties in their numerical implementation in this work timewise efficient and numerically stable implementations for mpe and rre are developed a computer program written in fortran 77 is also appended and applied to some model problems among them a hypersonic flow problem involving chemical reactions

the characteristic polynomial of a matrix with prescribed off diagonal blocks abstract we present new sufficient conditions for the existence of a matrix with prescribed complementary off diagonal blocks and characteristic polynomial

quasilinear vector extrapolation methods abstract we show that almost all vector extrapolation methods posses the property of quasilinearity that is translativity and homogeneity we give a necessary and sufficient condition for the property of translativity this condition is applied to some vector extrapolation methods in particular the h algorithm the e algorithm and certain polynomial extrapolation methods an application of this condition to convergence acceleration is also given and applied to henrici s transformation and to the vector generalization of the e algorithm given by wimp

sequences of transformations and triangular recursion schemes with applications in numerical analysis abstract in many fields of numerical analysis there appear transformations of the form t k v v k i v k i v t i when v varies a sequence of transformations is obtained this approach covers for example the e and algorithms the recursion formulae for b splines bernstein polynomials and orthogonal polynomials pade approximants the divided difference scheme and projection methods in this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme the reciprocal of these results also holds furthermore we will show that t v k can be represented in terms of a complex contour integral throughout the paper we will study several examples in some detail and it will turn out that the application of our general theory leads to interesting new results in the special cases among others we will derive a new determinantal representation formula for b splines a recurrence relation for generalized bernstein polynomials a generalization of the e algorithm and we will prove that the algorithm can be represented as a quotient of determinants

the simplified topological varepsilon algorithms for accelerating sequences in a vector space when a sequence of numbers is slowly converging it can be transformed into a new sequence which under some assumptions converges faster to the same limit one of the best known sequence transformations is the shanks transformation which can be recursively implemented by the varepsilon algorithm of wynn this transformation and this algorithm have been extended in two different ways to a sequence of elements of a topological vector space e in this paper we present new algorithms for implementing these topological shanks transformations they no longer require the manipulation of elements of the algebraic dual space e of e nor do they use the duality product inside the rules of the algorithms they need the storage of fewer elements of e and the stability is improved they also allow us to prove convergence and acceleration results for some types of sequences various applications involving sequences of vectors or matrices show the interest of the new algorithms

a completion problem over the field of real numbers let f be a field almost 25 years ago g n de oliveira has proposed the following completion problems describe the possible characteristic polynomials of ai j i j 1 2 where a1 1 and a2 2 are square submatrices when some of the blocks ai j are fixed and the others vary cf linear multilinear algebra 2 1975 357 several of these problems remain unsolved this paper gives the solution over the field of real numbers of oliveira s problem where the blocks a1 2 a2 1 are fixed and the others vary

analysis of some vector extrapolation methods for solving systems of linear equations in this paper we consider some vector extrapolation methods for solving nonsymmetric systems of linear equations when applied to sequences generated linearly these methods namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre are krylov subspaces methods and are respectively equivalent to the method of arnoldi and to the gcr and gmres by considering the geometrical aspect of these methods we derive new expressions for their residual norms and give a relationship between them this allows us to compare the two methods using this new approach we will show that for nonsingular skew symmetric matrices the gmres stagnates every two iterations and the restarted version gmres m m ge 2 is always convergent finally the incomplete forms are considered and some convergence results are given