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

accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

the richardson extrapolation process with a harmonic sequence of collocation points let a y sim a sum infty k 1 alpha ky sigma k as y rightarrow 0 where y is a discrete or continuous variable assume that sigma k are known numbers that may be complex in general and that a y is known for y in 0 b for some b 0 the aim is to find or approximate a the limit or antilimit of a y as y rightarrow0 one very effective way of approximating a is by the richardson extrapolation process that is defined via the linear systems a y l a j n sum n k 1 overline alpha ky sigma k l j leq l leq j n here a j n are approximations to a and overline alpha k are additional unknowns the yl are picked such that y0 y1 y2 0 and lim l rightarrow infty y l 0 in this paper we give a detailed analysis of the convergence and stability of the column sequences a j n infty j 0 with n fixed when y l c l eta q for some positive c eta and q specifically we prove that convergence takes place as j to infty and give the precise rate at which it does we also prove that the process is unstable and quantify its instability asymptotically this instability may be dealt with numerically by using high precision floating point arithmetic

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

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

confidence limits to the risk ratio for the problem of obtaining confidence limits to the risk ratio or the ratio of two binomial probabilities katz et al 1978 biometrics 34 469 474 proposed a method based on the logarithm of the observed ratio and using fieller s result we propose a method based on a power of the observed ratio the power being chosen to minimize the skewness of the pivotal random variable the method is simple to use and more stable than that of katz et al a continuity correction is suggested if a conservative interval is desired

eight interval estimators of a common rate ratio under stratified poisson sampling under the assumption that the rate ratio rr is constant across strata we consider eight interval estimators of rr under stratified poisson sampling the weighted least squares wls interval estimator with the logarithmic transformation the interval estimator using the principle analogous to that of fieller s theorem the interval estimators using wald s statistic with and without the logarithmic transformation the interval estimators using the mantel haenszel statistic with and without the logarithmic transformation the score test based interval estimator and the asymptotic likelihood ratio test based interval estimator we apply monte carlo simulation to evaluate and compare the performance of these estimators with respect to the coverage probability and the average length in a variety of situations we find that the coverage probability of the commonly used wls interval estimator tends to be smaller than the desired confidence level especially when we have a large number of strata with a small expected total number of cases etnc per stratum and the underlying rr is far away from 1 i e rr18 or rr8 we further find that the two estimators with the logarithmic transformation as well as the two test based estimators can consistently perform well in a variety of situations when rr1 with a given reasonable size of etnc per stratum we note that the interval estimators without the logarithmic transformation can be preferable to the corresponding ones with the logarithmic transformation in the situations considered here however when evaluating the non coverage probability in the two tails we find that the former tends to shift the left while the latter is generally not subject to this concern we also note that interval estimator using the mantel haenszel mh statistic with the logarithmic transformation is likely less efficient than the two test based interval estimators using the score and the likelihood ratio tests finally we use the data taken from a study of the postmenopausal hormone use on the risk of breast cancer in women as an example to illustrate the use of these interval estimators considered here

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

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

confidence intervals for the risk ratio under cluster sampling based on the beta binomial model in cohort studies the risk ratio rr is one of the most commonly used epidemiologic indices to quantify the effect of a suspected risk factor on the probability of developing a disease when we employ cluster sampling to collect data an interval estimator that does not account for the intraclass correlation between subjects within clusters is likely inappropriate in application of the beta binomial model to account for the intraclass correlation we develop four asymptotic interval estimators of the rr which are direct extensions of some recently developed estimators for independent binomial sampling we then use monte carlo simulation to evaluate the finite sample performance of these four interval estimators in a variety of situations we find that the estimator using the logarithmic transformation generally performs well and is preferable to the other three estimators in most of the situations considered here finally we include an example from a study of an educational intervention with emphasis on behaviour change to illustrate the use of the estimators developed in this paper

sample sizes for repeated measurements in dichotomous data when the measurement of outcome varies within studied subjects and the cost of additional subjects is high taking more than one measurement for each subject constitutes a useful alternative to increase the power or to reduce the total cost of the study in this paper i present sample size formulae for repeated measurements in dichotomous data under different situations i also discuss optimal sample allocation for repeated measurements

procedures for testing the homogeneity of relative difference in sparse data to quantify the excess effect of an experimental treatment over a placebo group in clinical trials we often consider use of the relative difference defined as the proportion of patients who would respond to the experimental treatment among those who would not otherwise if they were assigned to the placebo group to control the effects due to confounders on the response of interest we frequently employ stratified analysis in practice before obtaining a summary estimate of the relative difference it is desirable to assess whether this measure is constant across strata based on the beta binomial model we develop simple procedures for testing the homogeneity of relative difference for sparse data in which we have many strata but few patients per stratum using monte carlo simulations we demonstrate that the proposed test procedures can generally perform well with respect to type i error in a variety of situations we further evaluate and study the power of these test procedures finally we note some robustness in using the test procedures proposed here

accelerated non linear finite volume method for diffusion a non linear finite volume method with monotone matrix for the diffusion equation is presented it does not extrapolate the primary variable to neumann boundaries as this was previously done in similar methods this change results in faster convergence computation time is significantly shortened further using the reduced rank extrapolation method rre and imposing an upper limit on the number of linear iterations per non linear step second order accuracy and performance improvement are demonstrated by numerical examples

krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices abstract in this paper we propose a general approach by which eigenvalues with a special property of a given matrix a can be obtained in this approach we first determine a scalar function c c whose modulus is maximized by the eigenvalues that have the special property next we compute the generalized power iterations uinj 1 a u j j 0 1 where u 0 is an arbitrary initial vector finally we apply known krylov subspace methods such as the arnoldi and lanczos methods to the vector u n for some sufficiently large n we can also apply the simultaneous iteration method to the subspace span x n 1 x n k with some sufficiently large n where x j 1 m a x j m j 0 1 m 1 k in all cases the resulting ritz pairs are approximations to the eigenpairs of a with the special property we provide a rather thorough convergence analysis of the approach involving all three methods as n for the case in which a is a normal matrix we also discuss the connections and similarities of our approach with the existing methods and approaches in the literature

prediction intervals for overdispersed binomial data with application to historical controls bioassays are highly standardized trials for assessing the impact of a chemical compound on a model organism in that context it is standard to compare several treatment groups with an untreated control if the same type of bioassay is carried out several times the amount of information about the historical controls rises with every new study this information can be applied to predict the outcome of one future control using a prediction interval since the observations are counts of success out of a given sample size like mortality or histopathological findings the data can be assumed to be binomial but may exhibit overdispersion caused by the variability between historical studies we describe two approaches that account for overdispersion asymptotic prediction intervals using the quasi binomial assumption and prediction intervals based on the quantiles of the beta binomial distribution both interval types were calibrated using bootstrap methods for an assessment of the intervals coverage probabilities a simulation study based on various numbers of historical studies and sample sizes as well as different binomial proportions and varying levels of overdispersion was run it could be shown that calibration can improve the coverage probabilities of both interval types the coverage probability of the calibrated intervals calculated based on at least 10 historical studies was satisfactory close to the nominal 95 in a last step the intervals were computed based on a real data set from the ntp homepage using historical controls from bioassays with the mice strain b6c3f1

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

convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms the minimal polynomial and reduced rank extrapolation algorithms are two acceleration of convergence methods far sequences of vectors in a recent survey these methods were tested and compared with the scalar vector and topological epsilon algorithms and were observed to be more efficient than the latter it was also observed that the two methods have similar convergence properties the purpose of the present work is to analyze the convergence and stability properties of these methods and to show that they are bona fide acceleration methods when applied to a class of vector sequences that includes those sequences obtained from systems of linear equations by using matrix iterative methods

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

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

an interval estimate for the interclass correlation in beta binomial sampling the beta binomial distribution has been frequently applied to model intraclass correlation among dichotomous responses in cluster sampling to avoid using sophisticated numerical procedures to calculate maximum likelihood estimates many authors have proposed different moment estimators most of them focused their discussion however on point estimation only in this paper we have developed a closed form interval estimate for the intraclass correlation and have shown it has asymptotically correct coverage we have further used a monte carlo simulation to evaluate the performance of the proposed interval estimation in a variety of situations we have found that when the number of clusters is moderate 50 the proposed interval estimation performs reasonably well except for the situation where the intraclass correlation is large and the underlying mean response probability is small even when the number of clusters is as small as 20 the proposed interval estimation may still be useful in the situation where the intraclass correlation is small we have also extended the proposed interval estimate to accommodate stratified sampling in which each stratum has a different mean response probability but has a common intraclass correlation

a new extrapolation method for pagerank computations abstract the pagerank algorithm is widely considered these years because of its great significance in search engine technology and other scientific domains though the power method is the initial measure to settle the pagerank problem it gives poor convergence when the damping factor is sufficiently close to 1 this difficulty encourages researchers to present improved iterative methods for accelerating pagerank computations in this paper a cheap and practical extrapolation approach which is determined by the trace of the google matrix is proposed and it is more attractive when combined with the well known arnoldi type algorithm convergence analysis of our algorithms is given numerical examples illustrate the efficiency of these accelerated schemes

fom accelerated by an extrapolation method for solving pagerank problems this paper formulates the pagerank problem a x x into a consistent singular linear system i a x 0 and applies the full orthogonalization method fom to solve it this singular system is characterized by index one namely i n d e x i a 1 we analyze the breakdown performance of fom on a general singular linear system and conclude that fom can determine a solution if it converges without any unfortunate breakdowns for our target problem then we propose to use a vector extrapolation method to speed up the convergence performance of fom this extrapolation procedure is based on ritz values which directly stems from the arnoldi extrapolation algorithm wu and wei 2010 eventually numerical experiments are presented to illustrate the effectiveness of our approaches

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

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

convergence theorems for least change secant update methods the purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means the theorems and proofs given here can be viewed as generalizations of those given by broyden dennis more j inst math appl 12 1973 pp 223 246 and by dennis more math comp 28 1974 pp 549 560 the analysis is done in the orthogonal projection setting of dennis schnabel siam rev 21 1980 pp 443 459 and many readers might feel that it is easier to understand the theorems here readily imply local and q superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of marwil and toint and the nonlinear least squares method of dennis gay welsch

approximate interval estimation of the ratio of binomial parameters a review and corrections for skewness various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically it is found that the method based on likelihood scores koopman 1984 biometrics 40 513 517 miettinen and nurminen 1985 statistics in medicine 4 213 226 performs best in achieving the nominal confidence coefficient but it may distribute the tail probabilities quite disparately using general theory of bartlett 1953 biometrika 40 306 317 1955 biometrika 42 201 203 we correct this method for asymptotic skewness following gart 1985 biometrika 72 673 677 we extend this correction to the case of estimating the common ratio in a series of two by two tables computing algorithms are given and applied to numerical examples parallel methods for the odds ratio and the ratio of poisson parameters are noted

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

confidence intervals for the ratio of two binomial proportions sometimes the ratio of two binomial proportions is the parameter of major interest for instance as the risk ratio in a two group cohort study fleiss 1973 or as the likelihood ratio for a diagnostic test procedure mcneill keeler and adelstein 1975 let x and y be independent binomial variates based on sample sizes m and n and parameters p and p2 respectively let 0 pm p2 we consider the construction of confidence intervals for 0 santner and snell 1980 have proposed a nonanalytical procedure which guarantees a lower bound on the confidence coefficient of the 1 a interval for 0 their algorithm is usable for small m and n but may produce conservative intervals in 2 an analytical method for obtaining approximate confidence intervals for 0 is proposed and compared to the one recommended by katz et al 1978

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

a sparse quasi newton update derived variationally with a nondiagonally weighted frobenius norm the problem of symmetric sparse updating is considered from a variational point of view and a new class of sparse symmetric quasi newton updating formulae is derived this class results from the use of a nondiagonally weighted frobenius norm the computa tion of the update involves only one positive definite and symmetric linear system that has the same sparsity pattern as the problem itself

a note about sparsity exploiting quasi newton updates this paper is concerned with two questions relating to quasi newton updates for unconstrained optimization that exploit any sparsity present in the second derivative matrix of the objective function first a family of such updates is derived that reduces to any a priori known dense update formula when no sparsity is imposed this family uses the frobenius projection of the desired update on the subspace of matrices that satisfy all the needed conditions in the second part we prove that under mild assumptions a positive definite sparse quasi newton update always exists the proof of this result includes the explicit determination of such an update

quantitative and constructive aspects of the generalized koenig s and de montessus s theorems for pade approximants abstract the generalized koenig s theorem and de montessus s theorem are two classical results concerning the convergence of the rows of the pade table for meromorphic functions employing a technique that was recently developed for the analysis of vector extrapolation methods refined versions of these theorems are proved in the present work specifically complete expansions for the numerators and denominators of pade approximants are derived these expansions are then used to obtain 1 precise asymptotic rates of convergence of the poles of the pade approximants to the corresponding poles simple or multiple of the meromorphic function in question and 2 the precise asymptotic behavior of the error in the relevant pade approximants one important feature of the asymptotic results derived in this work is that these are expressed in terms of a very small number of parameters approximations of optimal accuracy to multiple poles and the principal parts of the corresponding laurent expansions are also constructed in addition the convergence problem for the case in which the only singularities on the circle of meromorphy are poles is solved completely through the solution of a nonlinear integer programming problem

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

on extensions of the power method for normal operators abstract in a recent work by sidi and bridger some old and some new extensions of the power method have been considered and some of these extensions have been shown to produce estimates of several dominant eigenvalues of an arbitrary square matrix in the present work we continue the analysis of two versions of one of these extensions called the mpe extensions as they are applied to normal matrices we show that the convergence rate of these methods for normal matrices is twice that for nonnormal matrices we also give precise asymptotic bounds on the errors of the estimates obtained for the eigenvalues further deflation type extensions of the power method for normal matrices are suggested and analyzed for their convergence all the results are stated and proved in the general setting of inner product spaces