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

semigroup characterization of besov type morrey spaces and well posedness of generalized navier stokes equations abstract the well posedness of generalized navier stokes equations with initial data in some critical homogeneous besov spaces and in some critical q spaces was known in this paper we establish a wavelet characterization of besov type morrey spaces under the action of semigroup as an application we obtain the well posedness of smooth solution for the generalized navier stokes equations with initial data in some critical homogeneous besov type morrey spaces b p p 1 2 n 1 2 1 1 2 1 2 1 p 2 and n p 2 2 2 n p or 2 p and max n p 2 2 1 2 n p with divergence free these critical homogeneous besov type morrey spaces are larger than corresponding classical besov spaces and cover q spaces

well posedness of quasi geostrophic equations with data in besov q spaces abstract by wavelets and multi resolution analysis we convert the bilinear estimate on tent spaces associated with besov q spaces b p p 1 2 r n to the estimates of the numerical quantities as an application we get the well posedness of the quasi geostrophic equations with initial data in b p p 1 2 r 2

generalized matrix diagonal stability and linear dynamical systems let a aij be a real square matrix and 1 p we present two analogous developments one for schur stability and the discrete time dynamical system x t 1 ax t and the other for hurwitz stability and the continuous time dynamical system x t ax t here is a description of the latter development for a we define and study hurwitz diagonal stability with respect to p norms abbreviated as hdsp hds2 is the usual concept of diagonal stability a is hdsp implies re 0 for every eigenvalue of a which means a is hurwitz stable abbreviated as hs when the off diagonal elements of a are nonnegative a is hs iff a is hdsp for all p for the dynamical system x t ax t we define diagonally invariant exponential stability relative to the p norm abbreviated as diesp meaning there exist time dependent sets which decrease exponentially and are invariant with respect to the system we show that diesp is a special type of exponential stability and the dynamical system has this property iff a is hdsp

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

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

rounding errors in solving block hessenberg systems a rounding error analysis is presented for a divide and conquer algorithm to solve linear systems with block hessenberg matrices conditions are derived under which the algorithm computes a stable solution the algorithm is shown to be stable for block diagonally dominant matrices and for m matrices

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 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

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

k version of finite element method in gas dynamics higher order global differentiability numerical solutions in this paper we consider and examine alternate finite element computational strategies for time dependent navier stokes equations describing high speed compressible flows with shocks in a viscous and conducting medium with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems ivp in which a the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations gdes i e the ivp b the solutions are admissible in the non discretized form of the gdes in the pointwise sense i e anywhere and everywhere in the entire space time domain and hence in the integrated sense as well c the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions d it is possible to time march the solutions this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time i e to minimize the error build up in the time marching process e the computational process is unconditionally stable and non degenerate regardless of the choice of discretization nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process f there are no issues of stability cfl number limitations and g the mathematical and computational methodology is independent of the nature of the space time differential operators we consider one dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all ivp the riemann shock tube with a single diaphragm serves as a model problem the specific details presented in the paper discuss 1 choice of the form of the gdes i e strong form or weak form 2 various choices of variables the paper establishes and considers density velocity and temperature as variables of choice 3 details of the space time least squares ls integral forms meritorious over all others in all aspects are presented and choice of approximation spaces are discussed 4 in all numerical studies we consider a viscous and conducting medium with ideal gas law however results are also presented for non conducting medium extension of this work to real gas models will be presented in a separate paper it is worth noting that when the medium is viscous and conducting the solutions of gas dynamics equations are analytic 5 it is also significant to note that upwinding methods based on addition of artificial diffusion such as supg supg dc supg dc ls and their many variations are neither needed nor used in this present work 6 numerical studies are aimed at resolving the localized details of the shock structure i e shock relations shock width shock speed etc as well as the over all global behaviour of the solution in the entire space time domain 7 numerical studies are presented for riemann shock tube for high mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non discretized form of the gdes in the pointwise sense and hence in the integrated sense as well 8 comparisons are made with published results as well as theoretical solutions when possible it is established that space time least squares processes are the only processes that yield variationally consistent space time integral forms and hence unconditionally non degenerate space time computational processes which when considered in higher order scalar product spaces provide the desired mathematical framework in which progressively higher order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the ivp in all aspects copyright 2006 john wiley sons ltd

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

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

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

well posedness of the cauchy problem for the fractional power dissipative equations abstract this paper studies the cauchy problem for the nonlinear fractional power dissipative equation u t u f u for initial data in the lebesgue space l r r n with either r r d n b 2 d or the homogeneous besov space b p r n with 2 d b n p and 1 p where 0 f u f u or q d f u with q d being a homogeneous pseudo differential operator of order d 0 2 and f u is a function of u which behaves like u b u with b 0

analysis of heat conduction in solids by space time finite element method a functional in the form of a convolution of the temperature variation in the space time domain has been derived it has been used as the basis for a finite element formulation of heat conduction in solids numerical illustrations have indicated that the space time finite element algorithm provides a more rapid convergence to the exact solutions than the usual finite element analysis with discretization in space only

a streamline upwind full galerkin method for space time convection dominated transport problems an original space time finite element approach for the solution of the diffusion convection equation is proposed in this paper a slight manipulation of the differential equation suggests that transient transport problems may in fact be seen as steady state space time transport problems accurately and easily soluble by the standard galerkin technique however concerning convective transport involving sharp fronts or coarse discretization it is shown that implementation of dissipation along space time trajectories significantly improves the solutions classical comparative test problems are run to establish the performances of this method and to show the limits of the more sophisticated petrov and taylor galerkin schemes evocation of a possible space time anisotropy generated by usual finite difference time stepping procedures as well as comparative analysis of amplification matrices help to understand the accuracy and the robustness of the proposed approach

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

a new class of function spaces connecting triebel lizorkin spaces and q spaces abstract let s r 0 p 1 and q 1 in this paper we introduce a new class of function spaces f p q s r n which unify and generalize the triebel lizorkin spaces with both p 1 and p and q spaces by establishing the carleson measure characterization of q space we then determine the relationship between triebel lizorkin spaces and q spaces which answers a question posed by dafni and xiao in g dafni j xiao some new tent spaces and duality theorem for fractional carleson measures and q r n j funct anal 208 2004 377 422 moreover via the hausdorff capacity we introduce a new class of tent spaces f t p q s r n 1 and determine their dual spaces f w p q s q r n where s r p q 1 max p q 1 0 q max p q and t denotes the conjugate index of t 1 as an application of this we further introduce certain hardy hausdorff spaces f h p q s r n and prove that the dual space of f h p q s r n is just f p q s q r n when p q 1

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

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

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 space time coupled p version least squares finite element formulation for unsteady fluid dynamics problems this paper presents a p version least squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled the dimensionless form of the differential equations describing the problem are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c element approximation the element properties are derived by utilizing p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach for non linear problems the non linear algebraic equations resulting from the least squares process are solved using newton s method with a line search this procedure results in a symmetric hessian matrix equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the generality success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection diffusion and burgers equations the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements the work presented in this paper provides the basis for the extension of the space time coupled least squares minimization concept to two and three dimensional unsteady fluid flow

the optimal temporal decay estimates for the fractional power dissipative equation in negative besov spaces in this paper we first generalize a new energy approach developed by guo and wang commun partial differ equations 37 2165 2208 2012 in the framework of homogeneous besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation then we apply this approach to the critical and supercritical surface quasi geostrophic equation and the critical keller segel system we show that certain weighted negative besov norm of solutions is preserved along time evolution and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the fourier splitting approach and the interpolation techniques

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

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

new besov type spaces and triebel lizorkin type spaces including q spaces let s tau in mathbb r and q in 0 infty we introduce besov type spaces dot b s tau p q mathbb r n for p in 0 infty and triebel lizorkin type spaces dot f s tau p q mathbb r n rm for p in 0 infty which unify and generalize the besov spaces triebel lizorkin spaces and q spaces we then establish the varphi transform characterization of these new spaces in the sense of frazier and jawerth using the varphi transform characterization of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n we obtain their embedding and lifting properties moreover for appropriate we also establish the smooth atomic and molecular decomposition characterizations of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n for s in mathbb r p in 1 infty q in 1 infty and tau in 0 frac 1 max p q via the hausdorff capacity we introduce certain hardy hausdorff spaces b dot h s tau p q mathbb r n and prove that the dual space of b dot h s tau p q mathbb r n is just dot b s tau p q mathbb r n where t denotes the conjugate index of t in 1 infty

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

well posedness and regularity of generalized navier stokes equations in some critical q spaces we study the well posedness and regularity of the generalized navier stokes equations with initial data in a new critical space q 1 rn q rn n 12 1 which is larger than some known critical homogeneous besov spaces here q rn is a space defined as the set of all measurable functions with sup l i 2 1 n i i f x f y 2 x y n 2 1 dxdy where the supremum is taken over all cubes i with edge length l i and edges parallel to the coordinate axes in rn in order to study the well posedness and regularity we give a carleson measure characterization of q rn by investigating a new type of tent spaces and an atomic decomposition of the predual for q rn in addition our regularity results apply to the incompressible navier stokes equations with initial data in q 1 1 rn

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

componentwise asymptotic stability of linear constant dynamical systems this note deals with a special type of asymptotic stability namely componentwise asymptotic stability with respect to the vector gamma t cwas of system s dot x ax bu t geq 0 where gamma t 0 componentwise inequality and gamma t rightarrow 0 as t rightarrow infty s is cwas if for each t 0 geq 0 and for each x t 0 leq gamma t 0 x t 0 with the components x i t 0 the free response of s satisfies x t leq gamma t for each t geq t 0 for gamma t underline underline delta alphae beta t t geq 0 with alpha 0 and beta 0 scalar the cweas e exponential may be defined s is cwas cweas if and only if dot gamma t geq bar a gamma t t geq 0 bar a alpha and bar a has the elements a ij and a ij i neq j these results may be used in order to evaluate in a more detailed manner the dynamical behavior of s as well as to stabilize s componentwise by a suitable linear state feedback

on the solution of diffusion convection equations by the space time finite element method a functional has been developed for the finite element solution of diffusion convection problems this functional is suitable for the application of the variational principle on discretization schemes in the space time domain this algorithm has shown to be computationally efficient over the conventional finite element discretization in the space domain alone numerical examples on one dimensional energy transport have been included to illustrate the merit of this technique

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

some new tent spaces and duality theorems for fractional carleson measures and qα rn several duality questions for fractional carleson measures and the spaces q rn are resolved using a new type of tent spaces these tent spaces are defined in terms of choquet integrals with respect to hausdorff capacity a predual for q rn is then defined as a space of distributions containing the hardy space h1 and an atomic decomposition is proved

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

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

componentwise stability of 1d and 2d linear discrete time systems this paper deals with a special type of asymptotic exponential stability namely componentwise asymptotic exponential stability for 1 d and 2 d linear discrete time systems the main motivation for these results is the need particularly felt in the evaluation in a more detailed manner of the dynamical behaviour of 1 d and 2 d linear discrete time systems necessary and sufficient conditions for componentwise asymptotic exponential stability are given

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

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

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

vector norms as lyapunov functions for linear systems a unified theory of quadratic and piecewise linear lyapunov functions for continuous and discrete time linear systems is presented the key to this work is the description of these lyapunov functions by vector norms the main results are sufficient and necessary conditions for a vector norm to be a lyapunov function as well as a method based on these conditions of constructing such lyapunov functions

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