line search acceleration of iterative methods abstract this paper describes and analyzes a simple technique that accelerates the convergence of iterative methods for solving large linear systems the proposed technique can be used with any convergent method however it is especially useful when solving unstructured systems since in this case traditional methods as sor or chebyshev semiiteration have difficulty in obtaining optimal parameters special attention is given to column relaxation and row relaxation it is demonstrated experimentally that an accelerated gauss seidel scheme runs faster than an optimally tuned sor scheme

multiparameter descent methods abstract a descent method for solving a system of linear equations ax b consists of the iterations x k 1 x k k z k where z k is a vector and k a parameter chosen to minimize some functional in this paper we will consider multiparameter generalizations of such descent methods namely iterations of the form x k 1 x k z k k where z k is a matrix and k a vector chosen to minimize some functional multiparameter generalizations of the conjugate direction method and the lanczos method will be obtained and their algebraic properties discussed multiparameter conjugate and biconjugate gradient algorithms and other lanczos type algorithms for implementing this lanczos method will also be given

two grid analysis of minimal residual smoothing as a multigrid acceleration technique we analyze the two level method accelerated by a minimal residual smoothing mrs technique the two grid analysis is sufficient for our purpose because our mrs acceleration scheme is only applied on the finest level of the multigrid method we prove that the mrs acceleration scheme is a semi iterative method with respect to the underlying two level iteration and that the mrs accelerated two level method is a polynomial acceleration of first order we explain why mrs may not effectively accelerate standard multigrid method for solving poisson like problems the iteration matrices for the mrs accelerated coarse grid correction operator and the mrs accelerated two level operator are obtained we give bounds for the residual reduction rates of the accelerated two level method numerical experiments are employed to support the analytical results

a simple test for regression specification with non nested alternatives abstract in this paper i introduce a simple test for the presence of the data generating process among several non nested alternatives the test is an extension of the classical j test for non nested regression models i also provide a bootstrap version of the test that avoids possible size distortions inherited from the j test

tests for model specification in the presence of alternative hypotheses some further results abstract in davidson and mackinnon 1981 two of the present authors proposed a novel and very simple procedure for testing the specification of a nonlinear regression model against the evidence provided by a non nested alternative in this paper we extend their results in several directions first we relax a number of the assumptions of the previous paper we admit the possibility that the nonlinear regression functions may depend on lagged dependent variables and we do not require that the error terms be normally distributed second we show how the earlier procedure may straightforwardly be generalized to the case where the two non nested models involve different transformations of the dependent variable finally we propose a simple procedure for testing non nested linear regression models which have endogenous variables on the right hand side and have therefore been estimated by two stage least squares

a statistical analysis of brain morphology using wild bootstrapping methods for the analysis of brain morphology including voxel based morphology and surface based morphometries have been used to detect associations between brain structure and covariates of interest such as diagnosis severity of disease age iq and genotype the statistical analysis of morphometric measures usually involves two statistical procedures 1 invoking a statistical model at each voxel or point on the surface of the brain or brain subregion followed by mapping test statistics e g t test or their associated p values at each of those voxels 2 correction for the multiple statistical tests conducted across all voxels on the surface of the brain region under investigation we propose the use of new statistical methods for each of these procedures we first use a heteroscedastic linear model to test the associations between the morphological measures at each voxel on the surface of the specified subregion e g cortical or subcortical surfaces and the covariates of interest moreover we develop a robust test procedure that is based on a resampling method called wild bootstrapping this procedure assesses the statistical significance of the associations between a measure of given brain structure and the covariates of interest the value of this robust test procedure lies in its computationally simplicity and in its applicability to a wide range of imaging data including data from both anatomical and functional magnetic resonance imaging fmri simulation studies demonstrate that this robust test procedure can accurately control the family wise error rate we demonstrate the application of this robust test procedure to the detection of statistically significant differences in the morphology of the hippocampus over time across gender groups in a large sample of healthy subjects

alternative nonnested specification tests of time series investment models abstract this paper develops and compares non nested hypothesis tests for linear regression models with first order serially correlated errors it extends the non nested testing procedures of pesaran fisher and mcaleer and davidson and mackinnon and compares their performance on four conventional models of aggregate investment demand using quarterly u s investment data from 1951 i to 1983 iv the data and the non nested hypothesis tests initially indicate that none of the models are correctly specified the tests are also intransitive in their assessments before rejecting these conventional models of investment demand we investigate the small sample properties of these different non nested test procedures through a series of monte carlo studies we find that these non nested tests can have significant finite sample size and power biases

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

associative subalgebras of low dimensional majorana algebras abstract a majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents called majorana axes that satisfy some of the properties of the 2a axes of the monster griess algebra the term was introduced by a a ivanov in 2009 inspired by the work of s sakuma and m miyamoto in the present paper we revisit mayer and neutsch s theorem on associative subalgebras of the griess algebra in the context of majorana theory we apply this result to determine all the maximal associative subalgebras of some low dimensional majorana algebras namely the majorana algebras generated by two majorana axes and the majorana representations of the symmetric group of degree 4 involving 3c algebras

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

monte carlo methodology and the finite sample properties of instrumental variables statistics for testing nested and non nested hypotheses using monte carlo methodology this paper investigates the effect of dynamics and simultaneity on the finite sample properties of instrumental variables statistics for testing nested and nonnested hypotheses simple numerical analytical formulae response surfaces are obtained which closely approximate the statistics unknown size and power functions for a dynamic simultaneous equations model the analysis illustrates the value and limitations of asymptotic theory in interpreting finite sample properties two practical results arise the f form and the wald statistic is favored over its chi squared form and large sigma and small effective sample size strongly affect the test of over identifying restrictions and the cox type test copyright 1991 by the econometric society

vector sequence transformations methodology and applications to linear systems in this paper a methodology for the construction of various vector sequence transformations is formulated leading to a unified presentation of the subject and to new results the connections to the general interpolation problem and to projections are discussed various particular cases are examined in more details applications to the solution of systems of linear equations will end the paper and in particular their relation with lanczos method will be studied some numerical examples will be given

regularity conditions for cox s test of non nested hypotheses abstract in this article we provide for the first time general regularity conditions and a rigorous proof of the asymptotic normality of cox s statistic for testing separate families of hypotheses the cox test for choosing between competing linear regression models is discussed as an example

statistical inference in non nested econometric models the purpose of this paper is to discuss some procedures that are available for testing non nested or separate hypotheses in the statistics and econometrics literature since many of these techniques may also be exploited in other disciplines it is hoped that an elaboration of the principal theoretical findings may make them more readily accessible to researchers in other disciplines several simple examples are used to illustrate the concepts of nested and non nested hypotheses and within the latter category global and partial non nested hypotheses two alternative methods of testing non nested hypotheses are discussed and contrasted the first of these is cox s modification of the likelihood ratio statistic and the second is atkinson s comprehensive model approach a major emphasis is placed on the role of the cox principle of hypothesis testing which enables a broad range of hypotheses to be tested within the same framework the problem associated with the application of the comprehensive model approach to composite non nested hypotheses is also highlighted roy s union intersection principle is presented as a viable method of dealing with this problem simulation results concerning the finite sample properties of various tests are discussed together with an analysis of some attempts to correct the poor size of the cox and related tests

a simulation approach to the problem of computing cox s statistic for testing nonnested models abstract this paper proposes a new procedure for computing the cox statistic for tests of nonnested hypotheses using the method of stochastic simulation the procedure is applicable to a wide class of probability distributions is relatively simple to implement and does not require an analytic derivation of the pseudo true estimators that enter the cox test statistic the paper also contains an application of the proposed method to the test of probit versus logistic formulations of the univariate binary choice models the empirical results show that the simulation method works reasonably well even for a moderate number of replications

vml a class of virtual multi level iterative methods for solving partial differential equations we introduce virtual multi level iterative methods vml which attempt to remove the low frequency errors by conducting some special smoothing residual norm minimization procedure with respect to the coarse grids however there is no coarse grid formed explicitly no inter grid transfer operator is needed and even the smoothing procedure can be done almost locally these properties are attractive to parallel computers vml with different relaxation schemes and different smoothing techniques constitute a class of vml iterative methods they may be used to accelerate general single level iterative methods or be used with the standard real multi grid method to alleviate the inherent lack of parallelism numerical experiments with some relaxation and smoothing techniques are used to show how the vml iterative methods work

on the application of robust regression based diagnostics to models of conditional means and conditional variances abstract a strategy is proposed for applying a class of robust regression based diagnostics to nonlinear models of conditional means and conditional variances for cross section or time series data the distinguishing feature of the current approach which builds on already popular residual based procedures is that no auxiliary assumptions are imposed at any testing stage consequently the statistics are guaranteed to have the correct asymptotic size under the relevant null hypothesis several new regression based conditional mean and conditional variance diagnostics are proposed the case of incompletely specified dynamics in time series models is explicitly covered

a correction of the decomposability result in a paper by meyer neutsch abstract in their paper of 1993 meyer and neutsch established the existence of a 48 dimensional associative subalgebra in the griess algebra g by exhibiting an explicit counter example the present paper shows a gap in the proof one of the key results in meyer and neutsch s paper which states that an idempotent a in the griess algebra is indecomposable if and only its peirce 1 eigenspace i e the 1 eigenspace of the linear transformation l a x a x is one dimensional the present paper fixes this gap and shows a more general result let v be a real commutative nonassociative algebra with an associative inner product and let c be a nonzero idempotent of v such that its peirce 1 eigenspace is a subalgebra then c is indecomposable if and only if its peirce 1 eigenspace is one dimensional the proof of this result is based on a general variational argument for real commutative metrised algebras with inner product

a hybrid iterative method for symmetric indefinite linear systems hybrid iterative methods that combine a conjugate direction method with a simpler iteration scheme such as chebyshev or richardson iteration were first proposed in the 1950s the ease with which chebyshev and richardson iteration can be implemented efficiently on a large variety of computer architectures has in recent years lead to renewed interest in iterative methods that use chebyshev or richardson iteration this paper presents a new hybrid iterative method for the solution of linear systems of equations with a symmetric indefinite matrix our method combines the conjugate residual method with richardson iteration special attention is paid to the determination of two real intervals one on each side of the origin that contain most of the eigenvalues of the matrix these intervals are used to compute suitable iteration parameters for richardson iteration we also discuss when to switch between the methods the hybrid scheme typically uses the richardson method for most iterations and this reduces the number of arithmetic vector operations significantly compared with the number of arithmetic vector operations required when only the conjugate residual method is used computed examples illustrate the competitiveness of the hybrid scheme

projection methods for linear systems the aim of this paper is first to give a unified framework for deriving several known projection methods for solving systems of linear equations we shall show that all these methods follow from a unique minimization problem the particular cases of the methods of steepest descent richardson and conjugate gradients will be treated in details then projection acceleration procedures for accelerating the convergence of an arbitrary iterative method will also be proposed and discussed

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

associative subalgebras of the griess algebra abstract the structure of the griess algebra g whose automorphism group is the fischer griess monster f1 is investigated we show the existence of 48 dimensional associative subalgebras in g and furthermore demonstrate that they are not contained in strictly larger ones this is an immediate consequence of a more general necessary and sufficient criterion for the maximality of associative subalgebras of g it is conjectured that the given explicit examples are of maximal possible dimension among all associative subalgebras in g this depends on the validity of a certain inequality

a robust test for non nested hypotheses we propose a robust version of cox type test statistics for the choice between two non nested hypotheses we first show that the influence of small amounts of contamination in the data on the test decision can be very large secondly we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable finally we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution unlike for the classical test statistic we apply our results to the choice between a pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

fixed point iterations for real functions abstract we give proofs of general results on the computation of fixed points of a continuous function or a lipschitz function on the real line we also show how completely these results can fail to hold in spaces of more than one dimension

multiparameter iterative schemes for the solution of systems of linear and nonlinear equations in this paper we introduce multiparameter generalizations of the linear and nonlinear iterative richardson methods for solving systems of linear and nonlinear equations the new algorithms are based on using a optimal matricial relaxation instead of the optimal scalar relaxation of the steepest descent method the optimal matrix which is defined at each iteration by minimizing the current residual is computed as the least squares solution of an associated problem whose dimension is generally much lower than that of the original problem in particular thanks to this approach we construct multiparameter versions of the delta k method introduced for solving nonlinear fixed point problems various numerical results illustrate the implementation of the new schemes they concern the solution of a linear problem and of a nonlinear one which comes out from a reaction diffusion problem which exhibits bifurcations in both cases the optimal multiparameter relaxation improves the convergence as compared to the optimal scalar one

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

the null and non null asymptotic distribution of the cox test for multivariate nonlinear regression alternatives and a new distribution free cox test abstract the asymptotic distribution of the generalized cox test for choosing between two multivariate nonlinear regression models in implicit form is derived the data is assumed to be generated by a model that need not be either the null or the non null model as the data generating model is not subjected to a pitman drift the analysis is global not local and provides a fairly complete qualitative description of the power characteristics of the generalized cox test some investigations of these characteristics are included a new test statistic is introduced that does not require an explicit specification of the error distribution of the null model the idea is to replace an analytical computation of the expectation of the cox difference with a bootstrap estimate the null distribution of this new test is derived

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

smoothing iterative block methods for linear systems with multiple right hand sides in the present paper we present smoothing procedures for iterative block methods for solving nonsymmetric linear systems of equations with multiple right hand sides these procedures generalize those known when solving one right hand linear systems we give some properties of these new methods and then using these procedures we show connections between some known iterative block methods finally we give some numerical examples

hybrid procedures for solving linear equations in this paper we introduce the notion of hybrid procedures for solving a system of linear equations a hybrid procedure consists in a combination of two arbitrary approximate solutions with coefficients summing up to one thus the combination only depends on one parameter whose value is chosen in order to minimize the euclidean norm of the residual vector obtained by the hybrid procedure properties of such procedures are studied in detail the two approximate solutions which are combined in a hybrid procedure are usually obtained by two iterative methods several strategies for combining these two methods together or with the previous iterate of the hybrid procedure itself are discussed and their properties are analyzed numerical experiments illustrate the various procedures

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

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

implicit alternatives and the local power of test statistics the local power of test statistics is analyzed by extending the notion of pitman sequences to sequences of data generating processes dgp s that approach the null hypothesis without necessarily satisfying the alternative a space of probability densities is defined and endowed with the structure of an infinite dimensional hilbert manifold which permits a geometrical interpretation of hypothesis testing the three classical test statistics lr wald and lm are shown to tend asymptotically to the same random variable under all sequences of local dgp s the power of these statistics is seen to depend on the null the alternative and the sequence of dgp s in a simple and geometrically intuitive way moreover for any test statistic that is asymptotically chi squared under the null there exists an implicit alternative hypothesis against which that statistic will have highest power and which coincides with the explicit alternative for the classical test statistics

residual smoothing techniques for iterative methods an iterative method for solving a linear system ax b produces iterates x k with associated residual norms that in general need not decrease smoothly to zero residual smoothing techniques are considered that generate a second sequence y k via a simple relation y k 1 eta k y k 1 eta k x k the authors first review and comment on a technique of this form introduced by schonauer and weiss that results in y k with monotone decreasing residual norms this is referred to as minimal residual smoothing certain relationships between the residuals and residual norms of the biconjugate gradient bcg and quasi minimal residual qmr methods are then noted from which it follows that qmr can be obtained from bcg by a technique of this form this technique is extended to generally applicable quasi minimal residual smoothing the practical performance of these techniques is illustrated in a number of numerical experiments

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

a nonnested approach to testing continuous time models against discrete alternatives abstract this paper considers a test of two competing models when one is formulated in continuous time as a system of stochastic differential equations and the other is formulated in discrete time as a system of nonlinear equations since the competing hypotheses are nonnested the appropriate formulae are derived for the implementation of the cox test this approach is illustrated with an application to the demand for food and related products in the united kingdom