benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

some applications of the hermite matrix polynomials series expansions this paper deals with hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems properties of hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well known methods results are applied to compute accurate approximations of certain differential systems in terms of hermite matrix polynomials

efficient computation of the matrix cosine trigonometric matrix functions play a fundamental role in second order differential equation systems this work presents an algorithm for computing the cosine matrix function based on taylor series and the cosine double angle formula it uses a forward absolute error analysis providing sharper bounds than existing methods the proposed algorithm had lower cost than state of the art algorithms based on hermite matrix polynomial series and pade approximants with higher accuracy in the majority of test matrices

on using a priori knowledge in space time adaptive processing in space time adaptive processing stap the clutter covariance matrix is routinely estimated from secondary ldquotarget freerdquo data because this type of data is more often than not rather scarce the so obtained estimates of the clutter covariance matrix are typically rather poor in knowledge aided ka stap an a priori guess of the clutter covariance matrix e g derived from knowledge of the terrain probed by the radar is available in this note we describe a computationally simple and fully automatic method for combining this prior guess with secondary data to obtain a theoretically optimal in the mean squared error sense estimate of the clutter covariance matrix the authors apply the proposed method to the kassper data set to illustrate the type of achievable performance

efficient and accurate algorithms for computing matrix trigonometric functions trigonometric matrix functions play a fundamental role in second order differential equations this work presents an algorithm based on taylor series for computing the matrix cosine it uses a backward error analysis with improved bounds numerical experiments show that matlab implementations of this algorithm has higher accuracy than other matlab implementations of the state of the art in the majority of tests furthermore we have implemented the designed algorithm in language c for general purpose processors and in cuda for one and two nvidia gpus we obtained a very good performance from these implementations thanks to the high computational power of these hardware accelerators and our effort driven to avoid as much communications as possible all the implemented programs are accessible through the matlab environment

shrinkage to tapering estimation of large covariance matrices in this paper we introduce a shrinkage to tapering approach for estimating large covariance matrices when the number of samples is substantially fewer than the number of variables i e n p and p n the proposed estimator improves upon both shrinkage and tapering estimators by shrinking the sample covariance matrix to its tapered version we first show that under both normalized frobenius and spectral risks the minimum mean squared error mmse shrinkage to identity estimator is inconsistent and outperformed by a minimax tapering estimator for a class of high dimensional and diagonally dominant covariance matrices motivated by this observation we propose a shrinkage to tapering oracle sto estimator for efficient estimation of general large covariance matrices a closed form formula of the optimal coefficient of the proposed sto estimator is derived under the minimum frobenius risk since the true covariance matrix is to be estimated we further propose a sto approximating stoa algorithm with a data driven bandwidth selection procedure to iteratively estimate the coefficient and the covariance matrix we study the finite sample performances of different estimators and our simulation results clearly show the improved performances of the proposed sto estimators finally the proposed stoa method is applied to a real breast cancer gene expression data set

a new efficient and accurate spline algorithm for the matrix exponential computation abstract in this work an accurate and efficient method based on matrix splines for computing matrix exponential is given an algorithm and a matlab implementation have been developed and compared with the state of the art algorithms for computing the matrix exponential we also developed a parallel implementation for large scale problems this implementation allowed us to get a much better performance when working with this kind of problems

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

an efficient and accurate algorithm for computing the matrix cosine based on new hermite approximations abstract in this work we introduce new rational polynomial hermite matrix expansions which allow us to obtain a new accurate and efficient method for computing the matrix cosine this method is compared with other state of the art methods for computing the matrix cosine including a method based on pade approximants showing a far superior efficiency and higher accuracy the algorithm implemented on the basis of this method can also be executed either in one or two nvidia gpus which demonstrates its great computational capacity

shrinkage algorithms for mmse covariance estimation we address covariance estimation in the sense of minimum mean squared error mmse when the samples are gaussian distributed specifically we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples large p small n first we improve on the ledoit wolf lw method by conditioning on a sufficient statistic by the rao blackwell theorem this yields a new estimator called rblw whose mean squared error dominates that of lw for gaussian variables second to further reduce the estimation error we propose an iterative approach which approximates the clairvoyant shrinkage estimator convergence of this iterative method is established and a closed form expression for the limit is determined which is referred to as the oracle approximating shrinkage oas estimator both rblw and oas estimators have simple expressions and are easily implemented although the two methods are developed from different perspectives their structure is identical up to specified constants the rblw estimator provably dominates the lw method for gaussian samples numerical simulations demonstrate that the oas approach can perform even better than rblw especially when n is much less than p we also demonstrate the performance of these techniques in the context of adaptive beamforming

regression coefficient and autoregressive order shrinkage and selection via the lasso summary the least absolute shrinkage and selection operator lasso has been widely used in regression shrinkage and selection we extend its application to the regression model with autoregressive errors two types of lasso estimators are carefully studied the first is similar to the traditional lasso estimator with only two tuning parameters one for regression coefficients and the other for autoregression coefficients these tuning parameters can be easily calculated via a data driven method but the resulting lasso estimator may not be fully efficient to overcome this limitation we propose a second lasso estimator which uses different tuning parameters for each coefficient we show that this modified lasso can produce the estimator as efficiently as the oracle moreover we propose an algorithm for tuning parameter estimates to obtain the modified lasso estimator simulation studies demonstrate that the modified estimator is superior to the traditional estimator one empirical example is also presented to illustrate the usefulness of lasso estimators the extension of the lasso to the autoregression with exogenous variables model is briefly discussed

pade approximation for the exponential of a block triangular matrix abstract in this work we obtain improved error bounds for pade approximations to e a when a is block triangular as a result improved scaling strategies ensue which avoid some common overscaling difficulties

regularized estimation of linear functionals of precision matrices for high dimensional time series this paper studies a dantzig selector type regularized estimator for linear functionals of high dimensional linear processes explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from independent identically distributed samples to long range dependent time series and from sub gaussian innovations to those with mild polynomial moments it is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes the dantzig selector estimator is applied to the sparse markowitz portfolio allocation and the optimal linear prediction for time series in which the ratio consistency when compared with an oracle estimator is established the effect of dependence and innovation moment conditions is further illustrated in the simulation study finally the regularized estimator is applied to classify the cognitive states on a real functional magnetic resonance imaging dataset and to portfolio optimization on a financial dataset

covariance matrix estimation and classification with limited training data a new covariance matrix estimator useful for designing classifiers with limited training data is developed in experiments this estimator achieved higher classification accuracy than the sample covariance matrix and common covariance matrix estimates in about half of the experiments it achieved higher accuracy than regularized discriminant analysis but required much less computation

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

a direct approach to sparse discriminant analysis in ultra high dimensions summary sparse discriminant methods based on independence rules such as the nearest shrunken centroids classifier tibshirani et al 2002 and features annealed independence rules fan fan 2008 have been proposed as computationally attractive tools for feature selection and classification with high dimensional data a fundamental drawback of these rules is that they ignore correlations among features and thus could produce misleading feature selection and inferior classification we propose a new procedure for sparse discriminant analysis motivated by the least squares formulation of linear discriminant analysis to demonstrate our proposal we study the numerical and theoretical properties of discriminant analysis constructed via lasso penalized least squares our theory shows that the method proposed can consistently identify the subset of discriminative features contributing to the bayes rule and at the same time consistently estimate the bayes classification direction even when the dimension can grow faster than any polynomial order of the sample size the theory allows for general dependence among features simulated and real data examples show that lassoed discriminant analysis compares favourably with other popular sparse discriminant proposals

efficient orthogonal matrix polynomial based method for computing matrix exponential the matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields this paper presents an efficient method for computing matrix exponentials based on hermite matrix polynomial expansions hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods a backward error analysis of the approximation in exact arithmetic is given this analysis is used to provide a theoretical estimate for the optimal scaling of matrices two algorithms based on this method have been implemented as matlab functions they have been compared with matlab functions funm and expm obtaining greater accuracy in the majority of tests a careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs reaching in numerical tests relative higher average costs than expm of only 4 43 for the final hermite selected order and obtaining better accuracy results in the 77 36 of the test matrices the matlab implementation of the best hermite matrix polynomial based algorithm has been made available online

on bernoulli matrix polynomials and matrix exponential approximation abstract we present in this paper a new method based on bernoulli matrix polynomials to approximate the exponential of a matrix the developed method has given rise to two new algorithms whose efficiency and precision are compared to the most efficient implementations that currently exist for that a state of the art test matrix battery that allows deeply exploring the highlights and downsides of each method has been used since the new algorithms proposed here do make an intensive use of matrix products we also provide a gpus based implementation that allows to achieve a high performance thanks to the optimal implementation of matrix multiplication available on these devices

l_ 0 sparse inverse covariance estimation recently there has been focus on penalized log likelihood covariance estimation for sparse inverse covariance precision matrices the penalty is responsible for inducing sparsity and a very common choice is the convex l 1 norm however the best estimator performance is not always achieved with this penalty the most natural sparsity promoting norm is the nonconvex l 0 penalty but its lack of convexity has deterred its use in sparse maximum likelihood estimation in this paper we consider nonconvex l 0 penalized log likelihood inverse covariance estimation and present a novel cyclic descent algorithm for its optimization convergence to a local minimizer is proved which is highly nontrivial and we demonstrate via simulations the reduced bias and superior quality of the l 0 penalty as compared to the l 1 penalty

affine cellularity of bln algebras abstract we show that the bln algebra which was introduced by mcgerty is affine cellular in the sense of koenig and xi in fact we establish a stronger property namely that the affine cell ideals are generated by idempotents this particularly implies that the global dimension of the bln algebra is finite for affine type a we obtain that the affine q schur algebra u d n n when d n is affine cellular and has finite global dimension

efficient minimax estimation of a class of high dimensional sparse precision matrices estimation of the covariance matrix and its inverse the precision matrix in high dimensional situations is of great interest in many applications in this paper we focus on the estimation of a class of sparse precision matrices which are assumed to be approximately inversely closed for the case that the dimensionality p can be much larger than the sample size n which is fundamentally different from the classical case that p n different in nature from state of the art methods that are based on penalized likelihood maximization or constrained error minimization based on the truncated neumann series representation we propose a computationally efficient precision matrix estimator that has a computational complexity of o p3 we prove that the proposed estimator is consistent in probability and in l2 under the spectral norm moreover its convergence is shown to be rate optimal in the sense of minimax risk we further prove that the proposed estimator is model selection consistent by establishing a convergence result under the entry wise norm simulations demonstrate the encouraging finite sample size performance and computational advantage of the proposed estimator the proposed estimator is also applied to a real breast cancer data and shown to outperform existing precision matrix estimators

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

theory and application of covariance matrix tapers for robust adaptive beamforming we unify several seemingly disparate approaches to robust adaptive beamforming through the introduction of the concept of a covariance matrix taper cmt this is accomplished by recognizing that an important class of adapted pattern modification techniques are realized by the application of a conformal matrix taper to the original sample covariance matrix from the schur product theorem for positive semi definite matrices and kolmogorov s existence theorem we further establish that cmts are in fact the solution to a minimum variance optimum beamformer associated with an auxiliary stochastic process that is related to the original by a hadamard schur product this allows us to gain deeper insight into the design of both existing pattern modification techniques and new cmts that can for example simultaneously address several different design constraints such as pattern distortion due to insufficient sample support and weights mismatch due to nonstationary interference a new two dimensional 2 d cmt for space time adaptive radar applications designed to provide more robust clutter cancellation is also introduced since the cmt approach only involves a single matrix haddamard product it is also inherently low complexity the practical utility of the cmt approach is illustrated through its application to both spatial and spatio temporal adaptive beamforming examples

optimal feature selection in high dimensional discriminant analysis we consider the high dimensional discriminant analysis problem for this problem different methods have been proposed and justified by establishing exact convergence rates for the classification risk as well as the ell 2 convergence results to the discriminative rule however sharp theoretical analysis for the variable selection performance of these procedures have not been established even though model interpretation is of fundamental importance in scientific data analysis this paper bridges the gap by providing sharp sufficient conditions for consistent variable selection using the sparse discriminant analysis through careful analysis we establish rates of convergence that are significantly faster than the best known results and admit an optimal scaling of the sample size n dimensionality p and sparsity level s in the high dimensional setting sufficient conditions are complemented by the necessary information theoretic limits on the variable selection problem in the context of high dimensional discriminant analysis exploiting a numerical equivalence result our method also establish the optimal results for the road estimator and the sparse optimal scoring estimator furthermore we analyze an exhaustive search procedure whose performance serves as a benchmark and show that it is variable selection consistent under weaker conditions extensive simulations demonstrating the sharpness of the bounds are also provided

on the fly backward error estimate for matrix exponential approximation by taylor algorithm abstract in this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on the fly without the need to precompute in high precision quantities related to specific accuracies in this way the scaling parameter s and the degree m of the truncated taylor series the underlying method are adapted to the input matrix and not to a class of matrices sharing some spectral properties such as with the classical backward error analysis the result is a very flexible method which can approximate the matrix exponential at any desired accuracy moreover the risk of overscaling that is the possibility to select a value s larger than necessary is mitigated by a new choice as the sum of two powers of two finally several numerical experiments in matlab with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives

efficient mixed rational and polynomial approximation of matrix functions abstract this paper presents an efficient method for computing approximations for general matrix functions based on mixed rational and polynomial approximations a method to obtain this kind of approximation from rational approximations is given reaching the highest efficiency when transforming nondiagonal rational approximations with a higher numerator degree than the denominator degree then the proposed mixed rational and polynomial approximation can be successfully applied for matrix functions which have any type of rational approximation such as pade chebyshev etc with maximum efficiency for higher numerator degrees than the denominator degrees the efficiency of the mixed rational and polynomial approximation is compared with the best existing evaluating schemes for general polynomial and rational approximations providing greater theoretical accuracy with the same cost in terms of matrix multiplications it is well known that diagonal rational approximants are generally more accurate than the corresponding nondiagonal rational approximants which have the same computational cost using the proposed mixed approximation we show that the above statement is no longer true and nondiagonal rational approximants are in fact generally more accurate than the corresponding diagonal rational approximants with the same cost

comparison and anti concentration bounds for maxima of gaussian random vectors slepian and sudakov fernique type inequalities which compare expectations of maxima of gaussian random vectors under certain restrictions on the covariance matrices play an important role in probability theory especially in empirical process and extreme value theories here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of gaussian random vectors without any restriction on the covariance matrices we also establish an anti concentration inequality for the maximum of a gaussian random vector which derives a useful upper bound on the levy concentration function for the gaussian maximum the bound is dimension free and applies to vectors with arbitrary covariance matrices this anti concentration inequality plays a crucial role in establishing bounds on the kolmogorov distance between maxima of gaussian random vectors these results have immediate applications in mathematical statistics as an example of application we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size

a direct estimation approach to sparse linear discriminant analysis this article considers sparse linear discriminant analysis of high dimensional data in contrast to the existing methods which are based on separate estimation of the precision matrix and the difference of the mean vectors we introduce a simple and effective classifier by estimating the product directly through constrained l1 minimization the estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant lpd rule the lpd rule is shown to have desirable theoretical and numerical properties it exploits the approximate sparsity of and as a consequence allows cases where it can still perform well even when and or cannot be estimated consistently asymptotic properties of the lpd rule are investigated and consistency and rate of convergence results are given the lpd classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation

on the marcenko pastur law for linear time series institute of mathematical statistics 2015 this paper is concerned with extensions of the classical ma enko pastur law to time series specifically p dimensional linear processes are considered which are built from innovation vectors with independent identically distributed real or complex valued entries possessing zero mean unit variance and finite fourth moments the coefficient matrices of the linear process are assumed to be simultaneously diagonalizable in this setting the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high dimensional setting p n c 0 for which dimension p and sample size n diverge to infinity at the same rate the results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case

computing a matrix function for exponential integrators an efficient numerical method is developed for evaluating a where a is a symmetric matrix and is the function defined by x ex 1 x 1 x 2 x2 6 this matrix function is useful in the so called exponential integrators for differential equations in particular it is related to the exact solution of the ode system dy dt ay b where a and b are t independent our method avoids the eigenvalue decomposition of the matrix a and it requires about 10n3 3 operations for a general symmetric n n matrix when the matrix is tridiagonal the required number of operations is only o n2 and it can be further reduced to o n if only a column of the matrix function is needed these efficient schemes for tridiagonal matrices are particularly useful when the lanczos method is used to calculate the product of this matrix function for a large symmetric matrix with a given vector

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions