a re evaluation of the quantile approximation method for random effects meta analysis the quantile approximation method has recently been proposed as a simple method for deriving confidence intervals for the treatment effect in a random effects meta analysis although easily implemented the quantiles used to construct intervals are derived from a single simulation study here it is shown that altering the study parameters and in particular introducing changes to the distribution of the within study variances can have a dramatic impact on the resulting quantiles this is further illustrated analytically by examining the scenario where all trials are assumed to be the same size a more cautious approach is therefore suggested where the conventional standard normal quantile is used in the primary analysis but where the use of alternative quantiles is also considered in a sensitivity analysis copyright 2008 john wiley sons ltd

a non parametric procedure for evaluating treatment effect in the meta analysis of survival data this paper addresses the problem of combining information from independent clinical trials which compare survival distributions of two treatment groups current meta analytic methods which take censoring into account are often not feasible for meta analyses which synthesize summarized results in published or unpublished references as these methods require information usually not reported the paper presents methodology which uses the log log survival function difference i e log logs2 t log logs1 t as the contrast index to represent the multiplicative treatment effect on survival in independent trials this article shows by the second mean value theorem for integrals that this contrast index denoted as theta is interpretable as a weighted average on a natural logarithmic scale of hazard ratios within the interval 0 t in a trial when the within trial proportional hazards assumption is true theta is the logarithm of the proportionality constant for the common hazard ratio for the interval considered within the trial in this situation an important advantage of using theta as a contrast index in the proposed methodology is that the estimation of theta is not affected by length of follow up time other commonly used indices such as the odds ratio risk ratio and risk differences do not have this invariance property under the proportional hazard model since their estimation may be affected by length of follow up time as a technical artefact thus the proposed methodology obviates problems which often occur in survival meta analysis because trials do not report survival at the same length of follow up time even when the within trial proportional hazards assumption is not realistic the proposed methodology has the capability of testing a global null hypothesis of no multiplicative treatment effect on the survival distributions of two groups for all studies a discussion of weighting schemes for meta analysis is provided in particular a weighting scheme based on effective sample sizes is suggested for the meta analysis of time to event data which involves censoring a medical example illustrating the methodology is given a simulation investigation suggested that the methodology performs well in the presence of moderate censoring

monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem this paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic reaction diffusion problem firstly the monotone method known as the method of lower and upper solutions is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem secondly a monotone domain decomposition algorithm based on a modification of the schwarz alternating method is constructed this monotone algorithm solves only linear discrete systems at each iterative step of the iterative process the rate of convergence of the monotone domain decomposition algorithm is estimated numerical experiments are presented

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

simple robust procedures for combining risk differences in sets of 2 2 tables meta analyses often use a random effects model to incorporate unexplained heterogeneity of study results trimmed versions of meta analytic estimators for the risk difference adapted from procedures designed for a random effects analysis can resist the impact of a few anomalous studies a simulation study compared untrimmed and trimmed versions of four meta analytic procedures that give weighted averages of risk differences an adaptation of winsorized estimates of components of variance gains some resistance to anomalous studies when estimating variability the simulations found that a modified version of the dersimonian laird estimator is attractive when risk differences reveal the added variability described by a random effects model and that a 20 per cent trimmed weighted version of this procedure offers resistance against the impact of highly anomalous results among four trimmed procedures considered the trimmed version of the modified dersimonian laird estimator offers the best performance over a wide range of simulation designs and sample sizes none of the methods whether trimmed or untrimmed is uniformly preferable a published meta analysis of a vaccination against tb provides data that serve to illustrate differences among the eight procedures

the fluid dynamic limit of the broadwell model of the nonlinear boltzmann equation in the presence of shocks this paper studies the asymptotic equivalence of the broadwell model of the nonlinear boltzmann equation to its corresponding euler equation of compressible gas dynamics in the limit of small mean free path e it is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow although there are thin shock layers in which the convergence does not hold more precisely by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations we can show that there exist solutions to the broadwell equations such that the broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order e as the mean free path e goes to zero for the proof we first construct a formal solution for the broadwell equation by matching the truncated hilbert expansion and shock layer expansion then the existence of broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution we use an energy method which makes full use of the inner structure of time dependent shock profiles for the broadwell equations

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

ssor and assor preconditioners for block broyden method solving nonlinear equations is a problem that needs to be dealt with in the practical engineering application this paper uses block broyden method for solving large scale nonlinear systems and two preconditioners are applied for solving the underlying linear systems including ssor preconditioner as well as assor method which is based on ssor it discusses their implementation processes and compares the two algorithms from different aspects finally it solves the nonlinear systems arising from the bratu problem experimental results show that the preconditioning technique is effective for the block broyden method and that the preconditioner assor has better performance as a whole therefore it can be used in the large scale problems arising from scientific and engineering computing

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

an evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed problems in this paper we try to achieve h independent convergence with preconditioned gmres y saad and m h schultz siam j sci comput 7 1986 pp 856 869 and bicgstab h a van der vorst siam j sci comput 13 1992 pp 63 644 for two dimensional 2d singularly perturbed equations three recently developed multigrid methods are adopted as a preconditioner they are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners two of the multigrid methods differ only in the transfer operators one uses standard matrix dependent prolongation operators from j e dendy jr j comput phys 48 1982 pp 366 386 w hackbusch multi grid methods and applications springer berlin 1985 the second uses upwind prolongation operators developed in p m de zeeuw j comput appl math 33 1990 pp 1 27 both employ the galerkin coarse grid approximation and an alternating zebra line gauss seidel smoother the third method is based on the block lu decomposition of a matrix and on an approximate schur complement this multigrid variant is presented in a reusken a multigrid method based on incomplete gaussian elimination university of eindhoven the netherlands 1995 all three multigrid algorithms are algebraic methods the eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of the three algorithms furthermore the construction of the search directions for the multigrid preconditioned gmres solvers is investigated by the calculation and solution of the minimal residual polynomials for poisson and convection diffusion problems all solution methods are investigated and evaluated for finite volume discretizations on fine grids the methods have been parallelized with a grid partitioning technique and are compared on an mimd machine

error minimizing krylov subspace methods iterative methods for the solution of linear systems are usually controlled by the observation of the norm of the residual in reality the error should be controlled but the error is not available the residuals and the errors are connected by the condition number of the system matrix if the system is well conditioned the decrease of the errors is closely connected to the decrease of the residuals for these cases krylov subspace methods that minimize the residuals in the euclidean norm or in the energy norm are powerful solution techniques if the system is ill conditioned the residuals can decrease while the errors increase for these systems arising even from very simple and commonly used differential equations iterative methods that minimize the residuals may require a large number of iterations to reduce the errors the user may be misled to stop the iteration too early by small residuals two families of error minimizing krylov subspace methods are proposed to overcome these difficulties ea

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

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

a robust ilu with pivoting based on monitoring the growth of the inverse factors abstract an incomplete lu decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of l u the information on the growth of the inverse factors is used as feedback for dropping entries in l and u this method often yields a robust and effective preconditioner especially when the system is highly indefinite numerical examples demonstrate the effectiveness of this approach

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

accelerating with rank one updates abstract consider the iteration xk 1 xk h b axk for solving ax b a is n x n nonsingular we discuss rank one updates to improve h as an approximation to a 1 during the iteration the update kills and reduces singular values of i ah and thus speeds up the convergence the algorithm terminates after at most n sweeps and if all n sweeps are needed then a 1 has been computed

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

misunderstandings about q and cochran s q test in meta analysis many meta analyses report using cochran s q test to assess heterogeneity of effect size estimates from the individual studies some authors cite work by w g cochran without realizing that cochran deliberately did not use q itself to test for heterogeneity further when heterogeneity is absent the actual null distribution of q is not the chi squared distribution assumed for cochran s q test this paper reviews work by cochran related to q it then discusses derivations of the asymptotic approximation for the null distribution of q as well as work that has derived finite sample moments and corresponding approximations for the cases of specific measures of effect size those results complicate implementation and interpretation of the popular heterogeneity index i 2 also it turns out that the test based confidence intervals used with i 2 are based on a fallacious approach software that outputs q and i 2 should use the appropriate reference value of q for the particular measure of effect size and the current meta analysis q is a key element of the popular dersimonian laird procedure for random effects meta analysis but the assumptions of that procedure and related procedures do not reflect the actual behavior of q and may introduce bias the dersimonian laird procedure should be regarded as unreliable

shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

variance stabilization and the bootstrap summary we investigate the use of a variance stabilizing transformation for the computation of a bootstrap t confidence interval the transformation is estimated in an automatic manner through an initial bootstrap step a bootstrap t interval is then computed for the variance stabilized parameter and the interval is mapped back to the original scale the resultant procedure is second order correct in some settings invariant and in a number of examples it performs better than the usual untransformed bootstrap t interval it also requires far less computation the new interval is compared with efron s bca procedure and the two methods are seen to produce similar results

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

student s t approximations for three simple robust estimators abstract suppose that tn is an estimator of a location parameter sn is an estimator of its standard error and both are based on a random sample of size n then approximations of tn sn in terms of student s t are convenient and useful in confidence interval estimation and tests of hypotheses of in this article some moment based t approximations of studentized trimmed means the trimean and gastwirth s estimator are constructed at the normal and several near normal parents and compared with some earlier t approximations closely associated with tukey s jackknife and an approach of huber 1970 the latter two approximations are based on conjectures involving the large sample distributional behavior of tn and sn whereas for the new approximations the moments are computed using monte carlo techniques the trimmed means trimean and gastwirth s estimator are studentized by using the winsorized standard deviations the interquartile range and the intertertile range respectively the moment

ordering strategies for modified block incomplete factorizations in this study the aim is to first investigate how the rate of convergence of unperturbed modified block incomplete factorization methods depends on the discrete partial differential equation pde to solve and next on this basis to provide simple practical rules for easily selecting orderings that result in rapid convergence it emerges from the analysis here that for discrete pdes ordering schemes that optimize the rate of convergence strongly depend on the variation of both the pde coefficients and the mesh sizes the arguments made which bring to light the appreciable potentialities of modified methods also display insight into why block versions are more robust than pointwise ones

an analysis of the composite step biconjugate gradient method the composite step biconjugate gradient method csbcg is a simple modification of the standard biconjugate gradient algorithm bcg which smooths the sometimes erratic convergence of bcg by computing only a subset of the iterates we show that 2 2 composite steps can cure breakdowns in the biconjugate gradient method caused by near singularity of principal submatrices of the tridiagonal matrix generated by the underlying lanczos process we also prove a best approximation result for the method some numerical illustrations showing the effect of roundoff error are given

meta analysis of ratios of sample variances when conducting a meta analysis of standardized mean differences smds it is common to use cohen s d or its variants that require equal variances in the two arms of each study while interpretation of these smds is simple this alone should not be used as a justification for assuming equal variances until now researchers have either used an f test for each individual study or perhaps even conveniently ignored such tools altogether in this paper we propose a meta analysis of ratios of sample variances to assess whether the equality of variances assumptions is justified prior to a meta analysis of smds quantile quantile plots an omnibus test for equal variances or an overall meta estimate of the ratio of variances can all be used to formally justify the use of less common methods when evidence of unequal variances is found the methods in this paper are simple to implement and the validity of the approaches are reinforced by simulation studies and an application to a real data set

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

stability relaxation and oscillation of biodegradation fronts we study the stability and oscillation of travelingfronts in a three component advection reaction biodegradation model the three components are pollutant nutrient and bac teria concentrations under an explicit condition on the biomass growth and decay coefficients we derive reduced two component semilinear hyperbolic models through a relaxation procedure duringwhich biomass is slaved to pollutant and nutrient concentration variables the reduced two component models resemble the broadwell model of the discrete velocity gas the traveling fronts of the reduced system are explicit and are expressed in terms of hyperbolic tangent function in the nutrient deficient regime we perform energy estimates to prove the asymptotic stability of these fronts under explicit conditions on the coefficients in the system in the small dampinglimit we carry out wentzel kramers brillouin wkb analysis on front perturbations and show that fronts are always stable in the two component models we extend the wkb analysis to derive amplitude equations for front perturbations in the original three component model because of the bacteria kinetics we find two asymptotic regimes where perturbation amplitudes grow or oscillate in time we perform numerical simulations to illustrate the predictions of the wkb theory

on the real convergence rate of the conjugate gradient method abstract we present a parametrized class of matrices for which the rate of convergence of the conjugate gradient method varies greatly with the parameter and does not appreciably depend on the algorithm implementation a small change in the eigenvalue distribution can lead to a large change in the sensitivity of cg to rounding errors a theorem is proved which gives a necessary and sufficient condition for ordering exact arithmetic cg processes for systems with different spectra according to the energy norm of the error theorems 4 1 and 4 2 continue paige s and greenbaum s work

beta binomial model for meta analysis of odds ratios in meta analysis of odds ratios ors heterogeneity between the studies is usually modelled via the additive random effects model rem an alternative multiplicative rem for ors uses overdispersion the multiplicative factor in this overdispersion model odm can be interpreted as an intra class correlation icc parameter this model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta distributed resulting in beta binomial distributions we propose two new estimators of the icc for meta analysis in this setting one is based on the inverted breslow day test and the other on the improved gamma approximation by kulinskaya and dollinger 2015 p 26 to the distribution of cochran s q the performance of these and several other estimators of icc on bias and coverage is studied by simulation additionally the mantel haenszel approach to estimation of ors is extended to the beta binomial model and we study performance of various icc estimators when used in the mantel haenszel or the inverse variance method to combine ors in meta analysis the results of the simulations show that the improved gamma based estimator of icc is superior for small sample sizes and the breslow day based estimator is the best for n 100 the mantel haenszel based estimator of or is very biased and is not recommended the inverse variance approach is also somewhat biased for ors 1 but this bias is not very large in practical settings developed methods and r programs provided in the web appendix make the beta binomial model a feasible alternative to the standard rem for meta analysis of ors 2017 the authors statistics in medicine published by john wiley sons ltd

transforming the model t random effects meta analysis with stable weights standard meta analytic theory assumes that study outcomes are normally distributed with known variances however methods derived from this theory are often applied to effect sizes having skewed distributions with estimated variances both shortcomings can be largely overcome by first applying a variance stabilizing transformation here we concentrate on study outcomes with student t distributions and show that we can better estimate parameters of fixed or random effects models with confidence intervals using stable weights or with profile approximate likelihood intervals following stabilization we achieve even better coverage with a finite sample bias correction further a simple t interval provides very good coverage of an overall effect size without estimation of the inter study variance we illustrate the methodology on two meta analytic studies from the medical literature the effect of salt reduction on systolic blood pressure and the effect of opioids for the relief of breathlessness substantial simulation studies compare traditional methods with those newly proposed we can apply the theoretical results to other study outcomes for which an effective variance stabilizer is available copyright c 2012 john wiley sons ltd

orderings for incomplete factorization preconditioning of nonsymmetric problems numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned krylov subspace methods for the solution of nonsymmetric linear systems is shown the preconditioners used in this study are different variants of incomplete factorizations it is shown that certain reorderings for direct methods such as reverse cuthill mckee can be very beneficial the benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity

reply to discussion of empirical vs natural weighting in random effects meta analysis three discussion papers raise interesting points but at the end of the day we continue to recommend that past meta analyses which have influenced public policy or clinical paradigms be reanalyzed by unweighted methods for all future meta analyses that employ either a fixed effects where concomitant treatment and or eligibility are diverse or b classical random effects weighted methods unweighted methods should serve as the primary analysis other analyses would be reasonable as secondary approaches two commentaries suggest that weights are only random to the extent of estimation errors in the between study and within study variance components we shall demonstrate that even if these components are known there is still considerable random variability in the weights in fact methods that try to weight the estimates inversely proportional to the variance have a number of undesirable properties including bias incorrect standard errors inconsistency including coverage of confidence intervals and counter intuitive properties that the expectation of the estimator changes both with the number of studies sampled and with constant multiples of sample size across all studies these adverse properties do not exist for the unweighted approach from the numerical example of phenylephrine 10 mg despite the arguments of waksman the proper conclusion is that the collection of studies does not constitute evidence based support for efficacy in terms of lowering nasal airway resistance in the final section we present two compelling examples where questionable inferences were made with potential major public implications copyright 2010 john wiley sons ltd

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

estimation in meta analyses of mean difference and standardized mean difference methods for random effects meta analysis require an estimate of the between study variance 2 the performance of estimators of 2 measured by bias and coverage affects their usefulness in assessing heterogeneity of study level effects and also the performance of related estimators of the overall effect however as we show the performance of the methods varies widely among effect measures for the effect measures mean difference md and standardized md smd we use improved effect measure specific approximations to the expected value of q for both md and smd to introduce two new methods of point estimation of 2 for md welch type and corrected dersimonian laird and one wt interval method we also introduce one point estimator and one interval estimator for 2 in smd extensive simulations compare our methods with four point estimators of 2 the popular methods of dersimonian laird restricted maximum likelihood and mandel and paule and the less familiar method of jackson and four interval estimators for 2 profile likelihood q profile biggerstaff and jackson and jackson we also study related point and interval estimators of the overall effect including an estimator whose weights use only study level sample sizes we provide measure specific recommendations from our comprehensive simulation study and discuss an example

a comparison of statistical methods for meta analysis meta analysis may be used to estimate an overall effect across a number of similar studies a number of statistical techniques are currently used to combine individual study results the simplest of these is based on a fixed effects model which assumes the true effect is the same for all studies a random effects model however allows the true effect to vary across studies with the mean true effect the parameter of interest we consider three methods currently used for estimation within the framework of a random effects model and illustrate them by applying each method to a collection of six studies on the effect of aspirin after myocardial infarction these methods are compared using estimated coverage probabilities of confidence intervals for the overall effect the techniques considered all generally have coverages below the nominal level and in particular it is shown that the commonly used dersimonian and laird method does not adequately reflect the error associated with parameter estimation especially when the number of studies is small

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

inexact block jacobi broyden methods for solving nonlinear systems of equations we describe a parallelizable jacobi type block broyden algorithm for solving nonlinear systems of equations we study the conditions under which the algorithm is locally convergent basically the jacobian matrix in a newton algorithm is replaced by a block broyden like matrix which can be easily calculated by a recurrence relation a family of nonlinear solvers could be generated combining it with some iterative or direct linear solvers different initializations for the block broyden matrix are proposed as well as several partitioning schemes producing an effective block partitioning of the jacobian matrix coupled with the restarted version of the linear gmres m algorithm the method parallelizes very well and gives good cpu time savings which are shown by several numerical tests

analysis of binary data from a multicentre clinical trial summary we develop several methods for estimating the treatment effect difference defined as the overall log odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response a simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions multicentre randomized clinical trials are frequently used to test the efficacy of new medical regimens these trials are attractive due to the ease of getting the desired sample size in a short period of time and the broad coverage of the patient population and medical practitioners the analysis of data from such trials requires combining information from the centres in a way that properly accounts for the variation due to the centres and the differential efficacy of treatments between the centres in this paper we investigate several approaches to pooling the data across centres and for handling the effects associated with the centres chakravarthi grizzle 1975 and beitler landis 1985 argue for treating these effects as random and fleiss 1986 argues against because the centres are seldom chosen at random from a well defined population of clinics and therefore the effects cannot be associated with any population we prefer the random effect model approach for the following reasons the goal of a trial is to extend the results of the study to the general population duly accounting for variation in the patient population and the medical skills of practitioners hence if we believe that the variability between centres as estimated by the data is typical of the variability in the population of centres then the inference based on the random effects model is more appropriate than the one based on the fixed effects model furthermore the variation among the centres may be more than that predicted from samples from a binomial distribution treating the centre effects as random introduces this intra class correlation into our model assumptions let the outcome variable from each centre be a pair of binomial random variables xaj

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

use of iterative refinement in the solution of sparse linear systems it is well known that if gaussian elimination with iterative refinement ir is used in the solution of systems of linear algebraic equations ax b whose matrices are dense then the accuracy of the results will usually be greater than the accuracy obtained by the use of gaussian elimination without iterative refinement ds however both more storage about 100 because a copy of matrix a is needed and more computing time some extra time is needed to perform the iterative process must be used with ir normally when the matrix is sparse the accuracy of the solution computed by some sparse matrix technique and ir will still be greater in this paper it is verified by many numerical experiments that the use of sparse matrix techniques with ir may also result in a reduction of both the computing time and the storage requirements this will never happen when ir is applied for dense matrices two parameters a drop tolerance t geqq 0 and a stability factor u 1 are introduced in the effo

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

reaction advection random motions in inhomogeneous media abstract a probabilistic approach to reaction advection systems with a finite propagation speed is developed the convergence to traveling wave solutions in inhomogeneous media is studied the results are heavily based on the author s previous paper branching random motions nonlinear hyperbolic systems and travelling waves preprint of simon bolivar university caracas venezuela 2003 32 pp submitted for publication

gpbi cg generalized product type methods based on bi cg for solving nonsymmetric linear systems recently bi cgstab as a variant of bi cg has been proposed for solving nonsymmetric linear systems and its attractive convergence behavior has been confirmed in many numerical experiments bi cgstab can be characterized by its residual polynomial which consists of the product of the residual polynomial of bi cg with other polynomials generated from two term recurrence relations in this paper we propose a unified way to generalize a class of product type methods whose residual polynomials can be factored by the residual polynomial of bi cg and other polynomials with standard three term recurrence relations such product type methods which are based on bi cg can be regarded as generalizations of bi cgstab from the unified way the well known variants of the product type methods like cgs bi cgstab bi cgstab2 are reacquired again

valid inference in random effects meta analysis the standard approach to inference for random effects meta analysis relies on approximating the null distribution of a test statistic by a standard normal distribution this approximation is asymptotic on k the number of studies and can be substantially in error in medical meta analyses which often have only a few studies this paper proposes permutation and ad hoc methods for testing with the random effects model under the group permutation method we randomly switch the treatment and control group labels in each trial this idea is similar to using a permutation distribution for a community intervention trial where communities are randomized in pairs the permutation method theoretically controls the type i error rate for typical meta analyses scenarios we also suggest two ad hoc procedures our first suggestion is to use a t reference distribution with k 1 degrees of freedom rather than a standard normal distribution for the usual random effects test statistic we also investigate the use of a simple t statistic on the reported treatment effects

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

empirical vs natural weighting in random effects meta analysis this paper brings into serious question the validity of empirically based weighting in random effects meta analysis these methods treat sample sizes as non random whereas they need to be part of the random effects analysis it will be demonstrated that empirical weighting risks substantial bias two alternate methods are proposed the first estimates the arithmetic mean of the population of study effect sizes per the classical model for random effects meta analysis we show that anything other than an unweighted mean of study effect sizes will risk serious bias for this targeted parameter the second method estimates a patient level effect size something quite different from the first to prevent inconsistent estimation for this population parameter the study effect sizes must be weighted in proportion to their total sample sizes for the trial the two approaches will be presented for a meta analysis of a nasal decongestant while at the same time will produce counter intuitive results for the dersimonian laird approach the most popular empirically based weighted method it is concluded that all past publications based on empirically weighted random effects meta analysis should be revisited to see if the qualitative conclusions hold up under the methods proposed herein it is also recommended that empirically based weighted random effects meta analysis not be used in the future unless strong cautions about the assumptions underlying these analyses are stated and at a minimum some form of secondary analysis based on the principles set forth in this article be provided to supplement the primary analysis

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

a random effects regression model for meta analysis many meta analyses use a random effects model to account for heterogeneity among study results beyond the variation associated with fixed effects a random effects regression approach for the synthesis of 2 x 2 tables allows the inclusion of covariates that may explain heterogeneity a simulation study found that the random effects regression method performs well in the context of a meta analysis of the efficacy of a vaccine for the prevention of tuberculosis where certain factors are thought to modify vaccine efficacy a smoothed estimator of the within study variances produced less bias in the estimated regression coefficients the method provided very good power for detecting a non zero intercept term representing overall treatment efficacy but low power for detecting a weak covariate in a meta analysis of 10 studies we illustrate the model by exploring the relationship between vaccine efficacy and one factor thought to modify efficacy the model also applies to the meta analysis of continuous outcomes when covariates are present

explaining heterogeneity in meta analysis a comparison of methods exploring the possible reasons for heterogeneity between studies is an important aspect of conducting a meta analysis this paper compares a number of methods which can be used to investigate whether a particular covariate with a value defined for each study in the meta analysis explains any heterogeneity the main example is from a meta analysis of randomized trials of serum cholesterol reduction in which the log odds ratio for coronary events is related to the average extent of cholesterol reduction achieved in each trial different forms of weighted normal errors regression and random effects logistic regression are compared these analyses quantify the extent to which heterogeneity is explained as well as the effect of cholesterol reduction on the risk of coronary events in a second example the relationship between treatment effect estimates and their precision is examined in order to assess the evidence for publication bias we conclude that methods which allow for an additive component of residual heterogeneity should be used in weighted regression a restricted maximum likelihood estimator is appropriate although a number of other estimators are also available methods which use the original form of the data explicitly for example the binomial model for observed proportions rather than assuming normality of the log odds ratios are now computationally feasible although such methods are preferable in principle they often give similar results in practice

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

quadratic inference function approach to merging longitudinal studies validation and joint estimation summary merging data from multiple studies has been widely adopted in biomedical research in this paper we consider two major issues related to merging longitudinal datasets we first develop a rigorous hypothesis testing procedure to assess the validity of data merging and then propose a flexible joint estimation procedure that enables us to analyse merged data and to account for different within subject correlations and follow up schedules in different studies we establish large sample properties for the proposed procedures we compare our method with meta analysis and generalized estimating equations and show that our test provides robust control of type i error against both misspecification of working correlation structures and heterogeneous dispersion parameters our joint estimating procedure leads to an improvement in estimation efficiency on all regression coefficients after data merging is validated

a comparison of heterogeneity variance estimators in combining results of studies for random effects meta analysis seven different estimators of the heterogeneity variance are compared and assessed using a simulation study the seven estimators are the variance component type estimator vc the method of moments estimator mm the maximum likelihood estimator ml the restricted maximum likelihood estimator reml the empirical bayes estimator eb the model error variance type estimator mv and a variation of the mv estimator mvvc the performance of the estimators is compared in terms of both bias and mean squared error using monte carlo simulation the results show that the reml and especially the ml and mm estimators are not accurate having large biases unless the true heterogeneity variance is small the vc estimator tends to overestimate the heterogeneity variance in general but is quite accurate when the number of studies is large the mv estimator is not a good estimator when the heterogeneity variance is small to moderate but it is reasonably accurate when the heterogeneity variance is large the mvvc estimator is an improved estimator compared to the mv estimator especially for small to moderate values of the heterogeneity variance the two estimators mvvc and eb are found to be the most accurate in general particularly when the heterogeneity variance is moderate to large

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems we propose an algorithm for computing a class of least squares polynomials on polygonal regions of the complex plane an important application of this technique to solving large sparse linear systems is considered the advantage of using general polygonal regions instead of ellipses as was done in previous work is that elliptic regions may fail to accurately represent the convex hull of the spectrum of the matrix a an attractive feature of the algorithm is that it does not explicitly require any numerical integration numerical experiments show that the least squares based methods for solving linear systems are competitive with the chebyshev based methods and are more reliable

a quasi minimal residual variant of the bi cgstab algorithm for nonsymmetric systems motivated by a recent method of freund siam j sci comput 14 1993 pp 470 482 who introduced a quasi minimal residual qmr version of the conjugate gradients squared cgs algorithm a qmr variant of the biconjugate gradient stabilized bi cgstab algorithm of van der vorst that is called qmrcgstab is proposed for solving nonsymmetric linear systems the motivation for both qmr variants is to obtain smoother convergence behavior of the underlying method the authors illustrate this by numerical experiments that also show that for problems on which bi cgstab performs better than cgs the same advantage carries over to qmrcgstab

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

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

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

exact inference on the random effects model for meta analyses with few studies we describe an exact unconditional non randomized procedure for producing confidence intervals for the grand mean in a normal normal random effects meta analysis the procedure targets meta analyses based on too few primary studies 7 say to allow for the conventional asymptotic estimators e g dersimonian and laird 1986 or non parametric resampling based procedures e g liu et al 2017 meta analyses with such few studies are common with one recent sample of 22 453 heath related meta analyses finding a median of 3 primary studies per meta analysis davey et al 2011 reliable and efficient inference procedures are therefore needed to address this setting the coverage level of the resulting ci is guaranteed to be above the nominal level up to monte carlo error provided the meta analysis contains more than 1 study and the model assumptions are met after employing several techniques to accelerate computation the new ci can be easily constructed on a personal computer simulations suggest that the proposed ci typically is not overly conservative we illustrate the approach on several contrasting examples of meta analyses investigating the effect of calcium intake on bone mineral density

incorporating variability in estimates of heterogeneity in the random effects model in meta analysis when combining results from separate investigations in a meta analysis random effects methods enable the modelling of differences between studies by incorporating a heterogeneity parameter tau 2 that accounts explicitly for across study variation we develop a simple form for the variance of cochran s homogeneity statistic q leading to interval estimation of tau 2 utilizing an approximating distribution for q this enables us to extend the point estimation of dersimonian and laird we also develop asymptotic likelihood methods and compared them with this method we then use these approximating distributions to give a new method of calculating the weight given to the individual studies results when estimating the overall mean which takes into account variation in these point estimates of tau 2 two examples illustrate the methods presented where we show that the new weighting scheme is between the standard fixed and random effects models in down weighting the results of large studies and up weighting those of small studies

a stable richardson iteration method for complex linear systems the richardson iteration method is conceptually simple as well as easy to program and parallelize this makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues we change the ordering of the relaxation parameters of a richardson iteration method proposed by eiermann niethammer and varga for the solution of such problems the new method obtained is shown to be stable and to have better convergence properties

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

physics based preconditioner for iterative algorithms in mom problems an efficient method to solve electromagnetic scattering problems involving several metallic scatterers or bodies composed of dielectric and metallic regions is proposed for this the block jacobi preconditioner is combined with the iterative transpose free quasi minimum residual tfqmr algorithm for best performance the blocks for the preconditioner are chosen according to individual scatterers or groups of scatterers or according to the electric and magnetic current basis functions for dielectric metallic scatterers

high performance finite element approximate inverse preconditioning abstract a new parallel normalized optimized approximate inverse algorithm based on the concept of the fish bone computational approach satisfying an antidiagonal data dependency for computing classes of explicit approximate inverses is introduced for symmetric multiprocessor systems the parallel normalized explicit approximate inverses are used in conjunction with parallel normalized explicit preconditioned conjugate gradient square schemes for the efficient solution of finite element sparse linear systems the parallel design and implementation issues of the new proposed algorithms are discussed and the parallel performance is presented using openmp

a method of moments estimator for random effect multivariate meta analysis meta analysis is a powerful approach to combine evidence from multiple studies to make inference about one or more parameters of interest such as regression coefficients the validity of the fixed effect model meta analysis depends on the underlying assumption that all studies in the meta analysis share the same effect size in the presence of heterogeneity the fixed effect model incorrectly ignores the between study variance and may yield false positive results the random effect model takes into account both within study and between study variances it is more conservative than the fixed effect model and should be favored in the presence of heterogeneity in this paper we develop a noniterative method of moments estimator for the between study covariance matrix in the random effect model multivariate meta analysis to our knowledge it is the first such method of moments estimator in the matrix form we show that our estimator is a multivariate extension of dersimonian and laird s univariate method of moments estimator and it is invariant to linear transformations in the simulation study our method performs well when compared to existing random effect model multivariate meta analysis approaches we also apply our method in the analysis of a real data example

performance analysis of preconditioners based on broyden method solving nonlinear equations is a problem often needed to be dealt with in the practical engineering application this paper discusses preconditioning methods based on block broyden method it first introduces the block broyden method and preconditioning technique then it presents four different preconditioners for the block broyden method and discusses the implementation process it also analyzes the time complexity of the four algorithms finally it solves the nonlinear systems arising from the bratu problem by the four preconditioning algorithms experimental results show that the preconditioning technique is effective for the block broyden method and that the preconditioner ssor has the fastest solving speed and the best performance among the four preconditioners therefore it can be used in the large scale problems arising from scientific and engineering computing

nested grids ilu decomposition ngilu a preconditioning technique is described which shows in many cases grid independent convergence this technique only requires an ordering of the unknowns based on the different levels of multigrid and an incomplete lu decomposition based on a drop tolerance the method is demonstrated on a variety of well known elliptic test problems including strongly varying coefficients advective terms and grid refinement

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

a likelihood approach to meta analysis with random effects in a meta analysis of a set of clinical trials a crucial but problematic component is providing an estimate and confidence interval for the overall treatment effect theta since in the presence of heterogeneity a fixed effect approach yields an artificially narrow confidence interval for theta the random effects method of dersimonian and laird which incorporates a moment estimator of the between trial components of variance sigma b2 has been advocated with the additional distributional assumptions of normality a confidence interval for theta may be obtained however this method does not provide a confidence interval for sigma b2 nor a confidence interval for theta which takes account of the fact that sigma b2 has to be estimated from the data we show how a likelihood based method can be used to overcome these problems and use profile likelihoods to construct likelihood based confidence intervals this approach yields an appropriately widened confidence interval compared with the standard random effects method examples of application to a published meta analysis and a multicentre clinical trial are discussed it is concluded that likelihood based methods are preferred to the standard method in undertaking random effects meta analysis when the value of sigma b2 has an important effect on the overall estimated treatment effect

a breakdown free lanczos type algorithm for solving linear systems lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates the various lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae if the denominator in the formula for one of these coefficients is zero then a breakdown occurs in the algorithm and it must be stopped such a breakdown is in fact due to the non existence of some orthogonal polynomial in this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method the resulting algorithm called mrz is equivalent to the nongeneric biodir algorithm which is a look ahead lanczos type algorithm but our derivation is much simpler

approximate inverse techniques for block partitioned matrices this paper proposes some preconditioning options when the system matrix is in block partitioned form this form may arise naturally for example from the incompressible navier stokes equations or may be imposed after a domain decomposition reordering approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner the storage requirements for these preconditioners may be much less than for incomplete lu factorization ilu preconditioners for tough large scale computational fluid dynamics cfd problems the numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite

a comparison of some gmres like methods abstract gmres and cgs are well known iterative methods for the solution of certain sparse linear systems with a nonsymmetric matrix these methods have been compared experimentally in many studies and specific observations on their convergence behavior have been reported a new iterative method to solve a nonsymmetric system has been proposed by eirola and nevanlinna the purpose of this paper is to investigate this method and to compare it with gmres we have seen problems for which this method is more efficient than gmres the original method has as drawbacks that it is not scaling invariant and that it may suffer from numerical instability but it is shown that these deficiencies can be repaired a method proposed by broyden 1969 seems related to the new method and is therefore included in the comparison

modified block approximate factorization strategies two variants of modified incomplete block matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations pdes even in the case of quasi singular problems

conjugate gradient least squares algorithm for solving the generalized coupled sylvester matrix equations abstract this paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled sylvester matrix equations j 1 q a i j x j b i j f i i 1 2 p we prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum norm solution

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance