数值积分的可选择误差界(英文)

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.     

Optimal Order and Stepsize Sequences

The problem is considered of choosing the most efficient orderand stepsize in an adaptive ODE code. It is shown how the stepsizeshould be selected in a given step for each of the availableorders and it is also shown how the choice should be made amongstthe alternative orders.     

Asymptotics of Landau Constants with Optimal Error Bounds

We study the asymptotic expansion for the Landau constants \(G_n\) , $$\begin{aligned} \pi G_n\sim \ln N + \gamma +4\ln 2 + \sum _{s=1}^\infty \frac{\beta _{2s}}{ N^{2s}},\quad n\rightarrow \infty , \end{aligned}$$ where \(N=n+3/4, \gamma =0.5772\ldots \) is Euler’s constant, and \((-1)^{s+1}\beta _{2s}\) are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative \(n\) . Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form $$\begin{aligned} \ln N+\gamma +4\ln 2+\sum _{s=1}^{2m}\frac{\beta _{2s}}{N^{2s}}< \pi G_n < \ln N+\gamma +4\ln 2+\sum _{s=1}^{2k-1}\frac{\beta _{2s}}{N^{2s}} \end{aligned}$$ for all \(n=0,1,2,\ldots , m=1,2,\ldots \) , and \(k=1,2,\ldots \) . The results are proved by approximating the coefficients \(\beta _{2s}\) with the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by \(G_n\) , as well as an integral representation of the constants \(\rho _k=(-1)^{k+1}\beta _{2k}/(2k-1)!\) .     

一种确定求积公式误差最优估计的简单方法

利用求积公式代数精度的概念,给出一种确定Newton-Cotes和Hermite插值型求积公式截断误差最优估计的简单方法,并通过实例验证其有效性.     

Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions,Optimal Deterministic Algorithms,and Higher-Order Convergence

We study the numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing-dimension algorithms. More precisely, the spaces of integrands we consider are weighted, reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst-case error. We study two cost models for function evaluations that depend on the number of active variables of the chosen sample points, and we study two classes of weights, namely product and order-dependent weights and the newly introduced finite projective dimension weights. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing-dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter \(\alpha \) and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing-dimension algorithms based on higher-order polynomial lattice rules.     

Asymptotic Error Formulae for Numerical Quadrature

A method, due to Fox, is used to derive asymptotic error formulaefor numerical procedures having the form (z+h, h).–(z,h)=(f,z,h).These procedures correspond to numerical quadrature for theintegrand (/h)(f,z,0)and compact expressions are given for determiningthe order of convergence as h 0, and the leading term in theerror. It is shown that a natural generalization of the Euler-Maclaurinexpansion is available. These results are applied to the particularcase where the Pt are polynomials in the differentiation operator.A related interpolation problem is also studied, and it is shownthat in certain cases higher order quadrature formulae are possiblewhen this interpolation problem is not poised.     

Explicit Error Bounds for Periodic Splines of Odd Order on a Uniform Mesh

The dependence relationships connecting equal interval splinesand their derivatives are analysed to obtain the form of theerror term when the spline is replaced by a general function.The defining equations for periodic splines of odd order ona uniform mesh are then expressed in terms of a positive definitecirculant matrix A and attainable bounds determined for thecondition number of A and for the norm of A^-1. In conjunctionwith the error term associated with the dependence relationships,this enables explicit error bounds to be established for thederivatives at the knots of the spline function. Some subsidiary results in the paper also relate to B-splineson a uniform mesh.     

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros.     

Multistep Methods for a Class of Higher Order Differential Problems: Convergence and Error Bounds

In this paper multistep methods for higher order differential systems of the type $Y^{(r)}=f(t,Y)$ are proposed. Such methods permit the numerical solutions of initial value problems for such systems, providing error bounds and avoiding the increase of the computational cost derived from the standard approach based on the consideration of an equivalent extended first order system.     

Bounds for the Third Peano Constants of Gaussian Quadrature Formulae

Denote by R _n ^G the remainder functional of the Gaussian quadrature formula involving n nodes. Using an elementary method, we derive the asymptotically best possible inequalities

Equivalent Conditions for Local Error Bounds

In this paper we present two classes of equivalent conditions for local error bounds in finite dimensional spaces. We formulate conditions of the first class by using subderivatives, subdifferentials and strong slopes for nearby points outside the referenced set, and show that these conditions actually characterize a uniform version of the local error bound property. We demonstrate this uniformity for the max function of a finite collection of smooth functions, and as a consequence we show that quasinormality constraint qualifications guarantee the existence of local error bounds. We further present the second class of equivalent conditions for local error bounds by using the various limits defined on the boundary of the referenced set. In presenting these conditions, we exploit the variational geometry of the referenced set in a systematic way and unify some existing results in the literature.     

Error Bounds for Nonnegative Dynamic Models

The extension of Markov reward models to dynamic models with nonnegative matrices is motivated by practical applications, such as economic input–output, employment, or population models. This paper studies the generalization of error bound theorems for Markov reward structures to dynamic reward structures with arbitrary nonnegative matrices. Both irreducible and reducible matrices are covered. In addition, results for the stochastic case are unified and extended. First, generalized expressions are derived for average reward functions. The special normalization case is distinguished and is shown to be transformable into the stochastic case. Its interpretation is of interest in itself. Next, error bound results are studied. Under a general normalization condition, it is shown that the results for the stochastic case can be extended. Both the average case and the transient case are included. A random walk-type example is included to illustrate the results.     

Improved Error Bounds for Lattice Rules

We establish results on the worst-case errors that can be achieved by well-chosen lattice rules for standard classes of multivariate periodic functions. These theorems improve or generalize earlier results of this type.     

On Error Bounds for Quasinormal Programs

The Mangasarian-Fromovitz constraint qualification is a central concept within the theory of constraint qualifications in nonlinear optimization. Nevertheless there are problems where this condition does not hold though other constraint qualifications can be fulfilled. One of such constraint qualifications is the so-called quasinormality by Hestenes. The well known error bound property (R-regularity) can also play the role of a general constraint qualification providing the existence of Lagrange multipliers. In this note we investigate the relation between some constraint qualifications and prove that quasinormality implies the error bound property, while the reciprocal is not true.     

Error Bounds for Exponential Operator Splittings

Error bounds for the Strang splitting in the presence of unbounded operators are derived in a general setting and are applied to evolutionary Schrödinger equations and their pseudo-spectral space discretization.     

Two Classes of Merit Functions for Infinite-Dimensional Second Order Complimentary Problems

In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.     

Improved Error Bounds for a Chebysheff-Halley-Type Method

Using the majorant method we find sufficient conditions for the convergence of a Chebysheff-Halley-type method in a Banach space. Our results improve all our previous results as well as those of others.     

Optimal Angle Bounds for Quadrilateral Meshes

We show that any simple planar n-gon can be meshed in linear time by O(n) quadrilaterals with all new angles bounded between 60 and 120 degrees.