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

damping oscillatory integrals let s be a smooth compact convex hypersurface of finite type in r n l with surface measure and let u be a nonnegative real valued function on s we shall study the behaviour of the fourier transform of the measure u in particular we are interested in obtaining inequalities for the decay of u 4 as 4 goes to inifinty this problem has been considered by various authors including hlawka 5 herz 4 littman 10 randol 12 and svensson 15 for an overview of such results and their significance in harmonic analysis we refer our readers to stein 14 by taking a partition of unity and using suitable coordinate systems the study of the fourier transform integral may be reduced to that of an oscillatory integral i of the form 1 2 v x exp i2 a x dx v2 r r

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

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

variational conditions under the constant rank constraint qualification this paper studies solution properties of a parametric variational condition under the constant rank constraint qualification crcq and properties of its underlying set we start by showing that if the crcq holds at a point in a fixed set then there exists a one to one correspondence between the collection of nonempty faces of the normal cone to the set at that point and the collection of active index sets at points nearby we then study the behavior of the euclidean projector and prove under the crcq that the set of multipliers associated with the euclidean projection is locally a continuous multifunction following that we apply the degree theory to a localized normal map to show that the combination of the crcq and the so called strong coherent orientation condition suffices for the parametric variational condition to have a locally unique continuous solution which is selected from finitely many c1 functions applications of this result under additional assumptions extend or recover some earlier results on parametric variational inequalities

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

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

sensitivity analysis in variational inequalities in this paper we derive conditions for the local uniqueness continuity and differentiability of a solution of parametric variational inequalities our approach is new and has a strong geometric flavor as an illustration we discuss the sensitivity of the classical complementarity problem

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

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

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

second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs in this paper we study second order differential properties of an optimal value function x it is shown that under certain conditions x possesses second order directional derivatives which can be calculated by solving corresponding quadratic programs also upper and lower bounds on these derivatives are introduced under weaker assumptions in particular we show that the second order directional derivative is infinite if the corresponding quadratic program is unbounded finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs

lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint it is proved that the metric projection from a point onto a moving polyhedron is lipschitz continuous with respect to the perturbations on the right hand sides of the linear inequalities defining the polyhedron the property leads to a simple sufficient condition for lipschitz continuity of a locally unique solution of parametric variational inequalities with a moving polyhedral constraint set applications of these results to traffic network equilibrium problems are given in detail

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

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

on uniqueness of kuhn tucker multipliers in nonlinear programming recently fujiwara han and mangasarian introduced a new constraint qualification which is a slight tightening of the well known mangasarian fromovitz constraint qualification we show that this new qualification is a necessary and sufficient condition for the uniqueness of kuhn tucker multipliers we also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum

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

linear stability of generalized equations part ii applications to nonlinear programming dedicated to anthony v fiacco on the occasion of his 65th birthday we present an approach by which the quantitative stability of solutions of a general nonlinear programming problem is obtained in the spirit of part i of the paper under assumptions of the constraint regularity and the general second order sufficient condition we show that the solution set is linearly stable under small allowed perturbations in the sense of part i of the paper and derive bounds for the linear stability number which characterizes the quantitative stability of solutions of the problem in question for standard nonlinear programs that are most commonly encountered in practical situations we develop a method to compute these bounds the results obtained here complement those of robinson 1982

solution differentiability for variational inequalities in this paper we study solution differentiability properties for variational inequalities we characterize frechet differentiability of perturbed solutions to parametric variational inequality problems defined on polyhedral sets our result extends the recent result of pang and it directly specializes to nonlinear complementarity problems variational inequality problems defined on perturbed sets and to nonlinear programming problems

the strong positivity conditions in this paper we consider a class of generalized equations which are a unified way of formulating problems of linear and nonlinear programming complementarity variational inequalities and mathematical economics among others we introduce a set of conditions which we call the strong positivity conditions prove that they are sufficient for a solution of a generalized equation to be locally unique and discuss their stability when the functions involved in the problem are subjected to small perturbations applications to linear generalized equations and nonlinear programming problems are discussed

holderian continuity of solutions of a dynamic walrasian price equilibrium problem with application this paper considers a qualitative analysis of the solution of a pure exchange general economic equilibrium problem regulated by walras law in the dynamic parametric case the equilibrium conditions that describe this pure exchange economic model are expressed in terms of a time dependent parametric variational inequality for which a holderian type stability of the solutions have been proved for this model we have also focused our attention on a numerical application with the parameter n m l where m represents the variation of the average annual inflation rate and l represents the variation of the vat share both considered in italy in the year 2008 for this numerical example it has been proved the existence of the solutions of the evolutionary parametric variational inequality that describes the equilibrium conditions by means of the direct method and by using matlab computation after a linear interpolation the curves of equilibria have been visualized

sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers we extend the principal sensitivity analysis results for parametric nonlinear programs and variational inequalities by considering problems with nonunique optimal lagrange multipliers specifically we derive sufficient conditions for the existence continuity and directional differentiability of a locally unique perturbed solution of a nonlinear program or a variational inequality

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

generalized sensitivity analysis of nonlinear programs this paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set this is accomplished using lexicographic directional derivatives a recently developed tool in nonsmooth analysis based on nesterov s lexicographic differentiation a nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric kkt system it is shown that the sufficient conditions for this implicit function theorem variant are implied by a kkt point satisfying the linear independence constraint qualification and strong second order sufficiency mirroring the classical theory the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution practically implementable algorithms are provided for calculating the nonsmooth sensitivity system s unique solution which is then used to furnish b subdifferenti

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

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

directional derivatives of the solution of a parametric nonlinear program consider a parametric nonlinear optimization problem subject to equality and inequality constraints conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known we show under the additional assumption of constant rank of the active constraint gradients that the optimal solution is actually piecewise smooth hence b differentiable we show for the first time to our knowledge a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique

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

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