a central limit theorem for temporally non homogenous markov chains with applications to dynamic programming we prove a central limit theorem for a class of additive processes that arise naturally in the theory of nite horizon markov decision problems the main theorem generalizes a classic result of dobrushin 1956 for tem porally non homogeneous markov chains and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain we show through several examples that this added exibility gives one a direct path to asymptotic nor mality of the optimal total reward of nite horizon markov decision problems the same examples also explain why such results are not easily obtained by alternative markovian techniques such as enlargement of the state space

tensor notation and cumulants of polynomials a modified and extended tensor notation is introduced that is sufficient to cover multivariate moments and cumulants as special cases using this notation two basic identities are given the first of these expresses generalized cumulants in terms of ordinary cumulants the second gives the joint cumulant generating function of any polynomial transformation in terms of the cumulants of the original variables three applications of the basic identities are given the first application is concerned with sample cumulants or k statistics the second to edgeworth series and the third to exponential family models

estimating quadratic variation consistently in the presence of endogenous and diurnal measurement error abstract we propose an econometric model that captures the effects of market microstructure on a latent price process in particular we allow for correlation between the measurement error and the return process and we allow the measurement error process to have a diurnal heteroskedasticity we propose a modification of the tsrv estimator of quadratic variation we show that this estimator is consistent with a rate of convergence that depends on the size of the measurement error but is no worse than n 1 6 we investigate in simulation experiments the finite sample performance of various proposed implementations

accurate solution of dense linear systems part i algorithms in rounding to nearest we investigate how extra precise accumulation of dot products can be used to solve ill conditioned linear systems accurately for a given p bit working precision extra precise evaluation of a dot product means that the products and summation are executed in 2p bit precision and that the final result is rounded into the p bit working precision denote by u 2 p the relative rounding error unit in a given working precision we treat two types of matrices first up to condition number u 1 and second up to condition number u 2 for both types of matrices we present two types of methods first for calculating an approximate solution and second for calculating rigorous error bounds for the solution together with the proof of non singularity of the matrix of the linear system in the first part of this paper we present algorithms using only rounding to nearest in part ii we use directed rounding to obtain better results all algorithms are given in executable matlab code and are available from my homepage

efficient implementation of markov chain monte carlo when using an unbiased likelihood estimator summary when an unbiased estimator of the likelihood is used within a metropolis hastings chain it is necessary to trade off the number of monte carlo samples used to construct this estimator against the asymptotic variances of the averages computed under this chain using many monte carlo samples will typically result in metropolis hastings averages with lower asymptotic variances than the corresponding averages that use fewer samples however the computing time required to construct the likelihood estimator increases with the number of samples under the assumption that the distribution of the additive noise introduced by the loglikelihood estimator is gaussian with variance inversely proportional to the number of samples and independent of the parameter value at which it is evaluated we provide guidelines on the number of samples to select we illustrate our results by considering a stochastic volatility model applied to stock index returns

the algebra of two scales estimation and the s tsrv high frequency estimation that is robust to sampling times abstract in this paper we derive a new algebraic property of two scales estimation in high frequency data under which the effect of sampling times is canceled to high order this is a particular robustness property of the two scales construction in general irregular asynchronous or endogenous times can cause problems in estimators based on equidistant observation of trade or quote times the new algebraic property can be combined with pre averaging giving rise to the smoothed two scales realized volatility s tsrv we derive a finite sample solution to controlling edge effects and for handling irregular and endogenous observation times and asynchronously observed multivariate data in connection with this development we use the algebraic approach to define a version of the s tsrv which has particularly small edge effect in microstructure noise the main result of the paper is a representation of the statistical error of the estimator in terms of simple components as an application of this representation the paper develops a central limit theory for multivariate volatility estimators the approach can also handle leads and lags in the signal process

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

the first sign change of fourier coefficients of cusp forms let f sum no a n q n a n real be an arbitrary nonzero cusp form of even integral weight k geq 2 on gamma 0 n square free n we obtain an effective bound that the first sign change of a n occurs

a hausman test for the presence of market microstructure noise in high frequency data abstract we develop tests that help assess whether a high frequency data sample can be treated as reasonably free of market microstructure noise at a given sampling frequency for the purpose of implementing high frequency volatility and other estimators the tests are based on the hausman principle of comparing two estimators one that is efficient but not robust to the deviation being tested and one that is robust but not as efficient we investigate the asymptotic properties of the test statistic in a general nonparametric setting and compare it with several alternatives that are also developed in the paper empirically we find that improvements in stock market liquidity over the past decade have increased the frequency at which simple uncorrected volatility estimators can be safely employed

composition constants for raising the orders of unconventional schemes for ordinary differential equations many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general purpose software designed to solve numerically a wide range of differential equations if those properties are to be exploited fully for the sake of better numerical stability accuracy and or speed the differential equations may have to be solved by unconventional methods this short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods called reflexive

computation of the monodromy matrix in floating point arithmetic with the wilkinson model in this study results have been obtained that compute the monodromy matrix in floating point arithmetic using the wilkinson model these results have been applied to the asymptotic stability of systems of linear difference equations with periodic coefficients also the effect of floating point arithmetic has been investigated on numerical examples

rosenthal type inequalities for the maximum of partial sums of stationary processes and examples the aim of this paper is to propose new rosenthal type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations as in the recent results by peligrad et al proc amer math soc 135 2007 541 550 and rio j theoret probab 22 2009 146 163 the estimates of the moments are expressed in terms of the norms of projections of partial sums the proofs of the results are essentially based on a new maximal inequality generalizing the doob maximal inequality for martingales and dyadic induction various applications are also provided

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

the stability of barycentric interpolation at the chebyshev points of the second kind we present a new analysis of the stability of the first and second barycentric formulae for interpolation at the chebyshev points of the second kind our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice we also extend our current understanding regarding the accuracy problems of the first barycentric formula

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

on the bahadur representation of sample quantiles for dependent sequences we establish the bahadur representation of sample quantiles for linear and some widely used nonlinear processes local fluctuations of empirical processes are discussed applications to the trimmed and winsorized means are given our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables

invariance principles under the maxwell woodroofe condition in banach spaces we prove that for adapted stationary processes the so called maxwell woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense that result actually holds in the context of banach valued stationary processes including the case of lplp valued random variables with 1 p 1 p in this setting we also prove the weak invariance principle hence generalizing a result of peligrad and utev ann probab 33 2005 798 815 the proofs make use of a new maximal inequality and of approximation by martingales for which some of our results are also new

minimizing roundoff errors of prefix sums via dynamic construction of huffman trees the prefix sum operation which returns all prefix sums on a sequence of numbers plays an important role in many applications we study how to efficiently evaluate prefix sums on positive floating point numbers such that the worst case roundoff error of each sum is minimized a direct approach to this problem builds a huffman tree for each prefix subsequence from scratch requiring exactly quadratic time for every input x we can do better by taking advantage of the current huffman tree to build the next huffman tree using dynamic insertions and deletions on huffman trees consequently subquadratic time suffices for various input patterns we also provide experimental comparisons of all the algorithms studied in this paper on inputs that are randomly and uniformly generated

central limit theorems for additive functionals of markov chains central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic markov chain say s n g x 1 g x n where e g x 1 0 and e g x 1 2 the conditions imposed restrict the moments of g and the growth of the conditional means e s n x 1 no other restrictions on the dependence structure of the chain are required when specialized to shift processes the conditions are implied by simple integral tests involving g

accurate sum and dot product algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time we show that the computed results are as accurate as if computed in twice or k fold working precision k ge 3 for twice the working precision our algorithms for summation and dot product are some 40 faster than the corresponding xblas routines while sharing similar error estimates our algorithms are widely applicable because they require only addition subtraction and multiplication of floating point numbers in the same working precision as the given data higher precision is unnecessary algorithms are straight loops without branch and no access to mantissa or exponent is necessary

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

periodicity of signs of fourier coefficients of eta quotients we study the periodicity of signs of fourier coefficients of the function d f qd rd where is a positive integer f q n 1 1 qn and rd z as an application we will prove the periodicity of signs for the crank differences conjectured by g e andrews and r lewis and for the weighted counts of certain types of partitions

the observed asymptotic variance hard edges and a regression approach abstract high frequency financial data has become an essential component of the digital economy yielding an increasing number of estimators however it is hard to reliably assess the uncertainty of such estimators the observed asymptotic variance observed avar is a non parametric estimator for squared standard error in high frequency data the device is related to observed information in likelihood theory but in this case it is non parametric and uses the high frequency data structure an earlier paper has developed the estimator in the case where edge effects are small to moderate in practical data it is often more realistic to assume that edge effects can be large and this is the problem that we tackle in the current paper we here find a regression approach to observed avar which is highly robust to large edges this approach covers most high frequency estimators

on the first sign change of hecke eigenvalues of newforms let f be a non zero cusp form of even integral weight k 2 and level n with real fourier coefficients a n n n using a classical theorem of landau together with the facts that the hecke l function of f is entire and the rankin selberg zeta function of f has a pole at its abscissa of convergence with residue essentially equal to the square of the petersson norm of f one easily shows that there are infinitely many n n such that a n 0 as well as infinitely many n with a n 1 siegel s argument does not apply and the result above is not known now suppose that n is squarefree and f is a newform which is a normalized hecke eigenform so a 1 1 and a n is the n th hecke eigenvalue of f we shall prove

combinatorial characterization of the null spaces of symmetric h matrices abstract we characterize the structure of null spaces of symmetric diagonally dominant sdd matrices and symmetric h matrices with non negative diagonal entries we show that the structure of the null space of a symmetric sdd matrix or h matrix a depends on the structure of the connected components of its underlying graph each connected component contributes at most one vector to the null space this paper provides a combinatorial characterization of the rank of each connected component and a combinatorial characterization of a null vector if one exists for sdd matrices we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space the paper also shows a close connection between gain graphs and h matrices which extends known results regarding the connection between undirected graphs and laplacian matrices and between signed graphs and sdd matrices we show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite precision arithmetic

analysis of adaptive mesh refinement for imex discontinuous galerkin solutions of the compressible euler equations with application to atmospheric simulations the resolutions of interests in atmospheric simulations require prohibitively large computational resources adaptive mesh refinement amr tries to mitigate this problem by putting high resolution in crucial areas of the domain we investigate the performance of a tree based amr algorithm for the high order discontinuous galerkin method on quadrilateral grids with non conforming elements we perform a detailed analysis of the cost of amr by comparing this to uniform reference simulations of two standard atmospheric test cases density current and rising thermal bubble the analysis shows up to 15 times speed up of the amr simulations with the cost of mesh adaptation below 1 of the total runtime we pay particular attention to the implicit explicit imex time integration methods and show that the ark2 method is more robust with respect to dynamically adapting meshes than bdf2 preliminary analysis of preconditioning reveals that it can be an important factor in the amr overhead the compiler optimizations provide significant runtime reduction and positively affect the effectiveness of amr allowing for speed ups greater than it would follow from the simple performance model

assessment of uncertainty in high frequency data the observed asymptotic variance the availability of high frequency financial data has generated a series of estimators based on intra day data improving the quality of large areas of financial econometrics however estimating the standard error of these estimators is often challenging the root of the problem is that traditionally standard errors rely on estimating a theoretically derived asymptotic variance and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated standard errors are important they are used to assess the precision of estimators in the form of confidence intervals to create feasible statistics for testing to build forecasting models based on say daily estimates and also to optimize the tuning parameters the contribution of this paper is to provide an alternative and general solution to this problem which we call observed asymptotic variance it is a general nonparametric method for assessing asymptotic variance avar it provides consistent estimators of avar for a broad class of integrated parameters t dt where the spot parameter process can be a general semimartingale with continuous and jump components the observed avar is implemented with the help of a two scales method its construction works well in the presence of microstructure noise and when the observation times are irregular or asynchronous in the multivariate case the methodology is valid for a wide variety of estimators including the standard ones for variance and covariance and also for more complex estimators such as of leverage effects high frequency betas and semivariance

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

martingale approximations for sums of stationary processes approximations to sums of stationary and ergodic sequences by martingales are investigated necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained it is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a lindeberg feller condition using the explicit construction of the approximating martingales a central limit theorem is derived for the sample means of linear processes the conditions are not sufficient for the functional version of the central limit theorem this is shown by an example and a slightly stronger sufficient condition is given

reducing floating point error in dot product using the superblock family of algorithms this paper discusses both the theoretical and statistical errors obtained by various well known dot products from the canonical to pairwise algorithms and introduces a new and more general framework that we have named superblock which subsumes them and permits a practitioner to make trade offs between computational performance memory usage and error behavior we show that algorithms with lower error bounds tend to behave noticeably better in practice unlike many such error reducing algorithms superblock requires no additional floating point operations and should be implementable with little to no performance loss making it suitable for use as a performance critical building block of a linear algebra kernel

a distillation algorithm for floating point summation the addition of two or more floating point numbers is fundamental to numerical computations this paper describes an efficient distillation style algorithm which produces a precise sum by exploiting the natural accuracy of compensated cancellation the algorithm is applicable to all sets of data but is particularly appropriate for ill conditioned data where standard methods fail due to the accumulation of rounding error and its subsequent exposure by cancellation the method uses only standard floating point arithmetic and does not rely on the radix used by the arithmetic model the architecture of specific machines or the use of accumulators

an application of the discrete time toda lattice to the progressive algorithm by lanczos and related problems the finite or the semi infinite discrete time toda lattice has many applications to various areas in applied mathematics the purpose of this paper is to review how the toda lattice appears in the lanczos algorithm through the quotient difference algorithm and its progressive form pqd then a multistep progressive algorithm mpa for solving linear systems is presented the extended lanczos parameters can be given not by computing inner products of the extended lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost the asymptotic behavior of the pqd algorithm brings us some applications of mpa related to eigenvectors

on floating point summation in this paper we focus on some general error analysis results in floating point summation we emphasize analysis that is useful from both a scientific and a teaching point of view

computation and storage of the bianisotropic scalar green s function and its derivatives a set of algorithms is proposed for the accurate and efficient computation and storage of the bianisotropic scalar green s function the computation is based on an expansion of the green s function into chebyshev polynomials the analytical properties of these polynomials are exploited to allow the accurate computation of the derivatives of the green s function as well as the green s function itself for lossy materials the proposed computation strategy is provably robust in addition a multilevel storage scheme with a favorable complexity based on the chebyshev polynomial expansion is proposed for the storage of the expansion coefficients numerical results showcase the accuracy and computational complexity of the proposed algorithms

parallel algorithms to evaluate orthogonal polynomial series in this paper four parallel algorithms for the evaluation of finite series of orthogonal polynomials are introduced the algorithms are based on the forsythe and clenshaw sequential algorithms several tests carried out on a cray t3d are presented

a new distillation algorithm for floating point summation the summation of n floating point numbers is ubiquitous in numerical computations we present a new distillation algorithm for floating point summation which is stable efficient and accurate the algorithm iteratively distills the summands without discarding any significant digit until the partial sums cannot change the whole sum it uses standard floating point arithmetic and does not rely on the choice of radix or any other specific assumption furthermore the error bound of our algorithm is independent of n and less than 1 ulp

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

inequalities for ranks of partitions and the first moment of ranks and cranks of partitions abstract we prove two monotonicity properties of n m n the number of partitions of n with rank m they are i for any nonnegative integers m and n n m n n m 2 n and ii for any nonnegative integers m and n such that n 12 n m 2 n m n n m n 1 g e andrews b kim and the first author introduced ospt n a function counting the difference between the first positive rank and crank moments they proved that ospt n 0 in another article k bringmann and k mahlburg gave an asymptotic estimate for ospt n the two monotonicity properties for n m n lead to stronger inequalities for ospt n that imply the asymptotic estimate

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

long term prediction intervals of time series we consider the problem of predicting aggregates or sums of future values of a process based on its past values in contrast with the conventional prediction problem in which one predicts a future value given past values of the process in our setting the number of aggregates can go to infinity with respect to the number of available observations consistency and bahadur representations of the prediction estimators are established a simulation study is carried out to assess the performance of different prediction estimators

exact computation of the sign of a finite sum many geometric algorithms are dependent on the sign of a simple expression like a finite sum examples of such algorithms are left turn test orientation questions boolean algorithms point in circle versus not in circle etc implementing such algorithms in fixed length floating point arithmetic can lead to inaccurate or wrong geometric configurations due to falsification of the computation by rounding errors in this paper we present an algorithm for determining the sign of a sum of a finite set of real numbers the result is exact i e the computation is free of rounding errors when the real numbers are machine numbers the algorithm is extremely simple only computation with fixed word length is required for example single precision floating point arithmetic no splitting or other mantissa manipulations are necessary one only needs to know the exponential parts of the represented summands and it is almost never necessary to compute the value of the whole sum

mass conservation of the unified continuous and discontinuous element based galerkin methods on dynamically adaptive grids with application to atmospheric simulations we perform a comparison of mass conservation properties of the continuous cg and discontinuous dg galerkin methods on non conforming dynamically adaptive meshes for two atmospheric test cases the two methods are implemented in a unified way which allows for a direct comparison of the non conforming edge treatment we outline the implementation details of the non conforming direct stiffness summation algorithm for the cg method and show that the mass conservation error is similar to the dg method both methods conserve to machine precision regardless of the presence of the non conforming edges for lower order polynomials the cg method requires additional stabilization to run for very long simulation times we addressed this issue by using filters and or additional artificial viscosity the mathematical proof of mass conservation for cg with non conforming meshes is presented in appendix b

jump robust volatility estimation using nearest neighbor truncation abstract we propose two new jump robust estimators of integrated variance that allow for an asymptotic limit theory in the presence of jumps specifically our medrv estimator has better efficiency properties than the tripower variation measure and displays better finite sample robustness to jumps and small zero returns we stress the benefits of local volatility measures using short return blocks as this greatly alleviates the downward biases stemming from rapid fluctuations in volatility including diurnal intraday u shape patterns an empirical investigation of the dow jones 30 stocks and extensive simulations corroborate the robustness and efficiency properties of our nearest neighbor truncation estimators

estimating the number of stable configurations for the generalized thomson problem given a natural number n one may ask what configuration of n points on the two sphere minimizes the discrete generalized coulomb energy if one applies a gradient based numerical optimization to this problem one encounters many configurations that are stable but not globally minimal this led the authors of this manuscript to the question how many stable configurations are there in this manuscript we report methods for identifying and counting observed stable configurations and estimating the actual number of stable configurations these estimates indicate that for n approaching two hundred there are at least tens of thousands of stable configurations

accurate and efficient floating point summation we present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator let f and f be the number of significant bits in the summands and the accumulator respectively then assuming gradual underflow no overflow and round to nearest arithmetic up to approximately 2f f numbers can be added accurately by simply summing the terms in decreasing order of exponents yielding a sum correct to within about 1 5 units in the last place ulps we apply this result to the floating point formats in the ieee floating point standard for example a dot product of single precision vectors of length at most 33 computed using double precision and sorting is guaranteed correct to nearly 1 5 ulps if double extended precision is used the vector length can be as large as 65 537 we also investigate how the cost of sorting can be reduced or eliminated while retaining accuracy

accurate floating point summation a new approach the aim of this paper is to find an accurate and efficient algorithm for evaluating the summation of large sets of floating point numbers we present a new representation of the floating point number system in which a number is represented as a linear combination of integers and the coefficients are powers of the base of the floating point system the approach allows to build up an accurate floating point summation algorithm based on the fact that no rounding error occurs whenever two integer numbers are summed or a floating point number is multiplied by powers of the base of the floating point system the proposed algorithm seems to be competitive in terms of computational effort and under some assumptions the computed sum is greatly accurate with such assumptions less conservative in the practical applications we prove that the relative error of the computed sum is bounded by the unit roundoff

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time assumption on the underlying returns under our framework it becomes clear why and where the usual volatility estimator fails when the returns are sampled at the highest frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

limit theorems for functionals of moving averages let x n i 1 a i e n 1 where the e i are i i d with mean 0 and finite second moment and the a i are either summable or regularly varying with index 1 1 2 the sequence x n has short memory in the former case and long memory in the latter for a large class of functions k a new approach is proposed to develop both central n rate and noncentral non n rate limit theorems for s n n 1 n k x n ek x n specifically we show that in the short memory case the central limit theorem holds for s n and in the long memory case s n can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non central limit theorem respectively further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems connections with the usual hermite expansion in the gaussian setting are also explored

an lll algorithm with quadratic complexity the lenstra lenstra lovasz lattice basis reduction algorithm called lll or rm l 3 is a fundamental tool in computational number theory and theoretical computer science which can be viewed as an efficient algorithmic version of hermite s inequality on hermite s constant given an integer d dimensional lattice basis with vectors of euclidean norm less than b in an n dimensional space the rm l 3 algorithm outputs a reduced basis in o d 3n rm log b cdot mathcal m d rm log b bit operations where mathcal m k denotes the time required to multiply k bit integers this worst case complexity is problematic for applications where d or and rm log b are often large as a result the original rm l 3 algorithm is almost never used in practice except in tiny dimension instead one applies floating point variants where the long integer arithmetic required by gram schmidt orthogonalization is replaced by floating point arithmetic unfortunately this is known to be unstable in the worst case the usual floating point rm l 3 algorithm is not even guaranteed to terminate and the output basis may not be rm l 3 reduced at all in this article we introduce the rm l 2 algorithm a new and natural floating point variant of the rm l 3 algorithm which provably outputs rm l 3 reduced bases in polynomial time o d 2n d rm log b rm log b cdot mathcal m d this is the first rm l 3 algorithm whose running time without fast integer arithmetic provably grows only quadratically with respect to rm log b like euclid s gcd algorithm and lagrange s two dimensional algorithm

simulating low precision floating point arithmetic the half precision fp16 floating point format defined in the 2008 revision of the ieee standard for floating point arithmetic and a more recently proposed half precision format bfloat16 are in

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability

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

linear time approximation algorithms for computing numerical summation with provably small errors given a multiset x x1 xn of real numbers the floating point set summation problem asks for sn x1 xn let e n denote the minimum worst case error over all possible orderings of evaluating sn we prove that if x has both positive and negative numbers it is np hard to compute sn with the worst case error equal to e n we then give the first known polynomial time approximation algorithm that has a provably small error for arbitrary x our algorithm incurs a worst case error at most 2 lceil log n 1 rceil 1 e n all logarithms log in this paper are base 2 after x is sorted it runs in o n time for the case where x is either all positive or all negative we give another approximation algorithm with a worst case error at most lceil log log n rceil e n even for unsorted x this algorithm runs in o n time previously the best linear time approximation algorithm had a worst case error at most lceil log n rceil e n while e n was known to be attainable in o n log n time using huffman coding

reducing round off errors in rigid body dynamics in several recent publications numerical integrators based on jacobi elliptic functions are proposed for solving the equations of motion of the rigid body although this approach yields theoretically the exact solution a standard implementation shows an unexpected linear propagation of round off errors we explain how deterministic error contribution can be avoided so that round off behaves like a random walk