Liapunov functions and error bounds for approximate solutions of ordinary differential equations

Summary It is shown that Liapunov functions may be used to obtain error bounds for approximate solutions of systems of ordinary differential equations. These error bounds may reflect the behaviour of the error more accurately than other bounds.     

Regularity theorems and optimal error estimates for linear parabolic Cauchy problems

Summary We prove some regularity results for the solution of a linear abstract Cauchy problem of parabolic type. As an application, we study the approximation of the solution by means of an implicit-Euler discretization in time, which is stable with respect to a wide class of Galerkin approximation methods in space. The error is evaluated in norms of typeL ²(0, ,L ²) andL ²(0, ,V)(H ₀₀ ^1/2 (0, ,H)+H ¹(0, ,V)), whereVHV are Hilbert spaces (the embeddings are supposed to be dense and continuous). We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation.The author was supported by G.N.A.F.A. and I.A.N. of C.N.R. and by M.P.I.     

Product integration with the Clenshaw-Curtis points: Implementation and error estimates

Summary This paper is concerned with the practical implementation of a product-integration rule for approximating , wherek is integrable andf is continuous. The approximation is , where the weightsw _ni are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.     

Sharp error bounds for Newton's process

Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].     

The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions

Summary This paper gives a method for finding sharpa posteriori error bounds for Newton's method under the assumptions of Kantorovich's theorem. On the basis of this method, new error bounds are derived, and comparison is made among the known bounds of Dennis [2], Döring [4], Gragg-Tapia [5], Kantorovich [6, 7], Kornstaedt [9], Lancaster [10], Miel [11–13], Moret [14], Ostrowski [17, 18], Potra [19], and Potra-Pták [20].This paper was written while the author was visiting the Mathematics Research Center, University of Wisconsin-Madison, U.S.A. from March 29, 1985 to October 21, 1985Sponsored by the Ministry of Education in Japan and the United States Army under Contract No. DAAG 29-80-C-0041     

Thirteen ways to estimate global error

Summary Various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations, particularly ordinary differential equations, are classified, described, and compared. For most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated.This research is partially supported by NSF Grant No. MCS-8107046     

Error bounds and estimates for eigenvalues of integral equations. II

Summary In a previous paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of non-symmetric integral equations. In this note an alternative analysis is presented leading to equivalent dominant error terms with error bounds which are quicker to calculate than those derived previously.     

On the local error and the local truncation error of linear multistep methods

Two different measures of the local accuracy of a linear multistep method — the local error and the local trunction error — appear in the literature. It is shown that the principal parts of these errors are not identical for general linear multistep methods, but that they are so for a sub-class which contains all methods of Adams type. It is sometimes argued that local error is the more natural measure; this view is challenged.     

Truncation error bounds for limit-periodic continued fractionsK(a n /1) with lima n =0

Summary Truncation error bounds are developed for continued fractionsK(a _n /1) where |a _n |1/4 for alln sufficiently large. The bounds are particularly suited (some are shown to be best) for the limit-periodic case when lima _n =0. Among the principal results is the following: If |a _n |/n ^p for alln sufficiently large (with constants >0,p>0), then |f–f _m |C[D/(m+2)] ^p(m+2) for allm sufficiently large (for some constantsC>0,D>0). Heref denotes the limit (assumed finite) ofK(a _n /1) andf _m denotes itsmth approximant. Applications are given for continued fraction expansions of ratios of Kummer functions₁ F ₁ and of ratios of hypergeometric functions₀ F ₁. It is shown thatp=1 for₁ F ₁ andp=2 for₀ F ₁, wherep is the parameter determining the rate of convergence. Numerical examples indicate that the error bounds are indeed sharp.Research supported in part by the National Science Foundation under Grant MCS-8202230 and DMS-8401717     

Realistic error bounds for a simple eigenvalue and its associated eigenvector

Summary An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.Dedicated to A.S. Householder on his 75th birthday     

Direct and inverse error estimates for finite elements with mesh refinements

The finite element method is used to solve a second order elliptic boundary value problem on a polygonal domain. Mesh refinements and weighted Besov spaces are used to obtain optimal error estimates and inverse theorems.Research performed while at the University of Maryland under a Fulbright fellowshipResearch supported in part by the Department of Energy under the contract E(40-1)3443Research supported in part by the National Institutes of Health under the grant 5R01-AM-20373     

Error estimates for interpolatory quadrature formulae

Summary In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pólya type formulae.Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainderR[f] of a formula of orderm can be represented asc·f ^(m)(). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases.Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.     

Pointwise error estimates for a streamline diffusion finite element scheme

Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5].     

Error estimates for spline interpolants on equidistant grids

Summary For oddm, the error of them-th-degree spline interpolant of power growth on an equidistant grid is estimated. The method is based on a decomposition formula for the spline function, which locally can be represented as an interpolation polynomial of degreem which is corrected by an (m+1)-st.-order difference term.Dedicated to Prof. Dr. Karl Zeller on the occasion of his 60th birthday     

Asymptotic expansions of the global error of fixed-stepsize methods

Summary In his fundamental paper on general fixed-stepsize methods, Skeel [6] studied convergence properties, but left the existence of asymptotic expansions as an open problem. In this paper we give a complete answer to this question. For the special cases of one-step and linear multistep methods our proof is shorter than the published ones.Asymptotic expansions are the theoretical base for extrapolation methods.     

On the solutions to polynomial systems obtained by homotopy methods

Summary A natural class of homotopy methods for solving polynomial systems is considered. It is shown that at least one solution from each connected component of the solution set is obtained. This generalizes the results of previous papers which concentrated on isolated solutions, i.e. connected components with one single point. The number of solution paths ending in a connected component is independent of the particular homotopy in use and defines in a natural way the multiplicity of the connected component. A few numerical experiments illustrate the obtained results.     

Lower norm error estimates for approximate solutions of differential equations with non-smooth coefficients

Summary In this paper, we derive error estimates inL _p-norm, 1p, for the ₂-Finite Element approximation to solutions of boundary value problems, where the coefficients are functions of bounded variation. The ₂-Finite Element Method was introduced in [3] and was shown to be effective for problems with non-smooth coefficient.The results of this paper form a part of a Ph.D. thesis written at the University of Maryland under the direction of Professor J.E. Osborn     

Numerical solution of retarded initial value problems: Local and global error and stepsize control

Summary Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme. Hence one can control the global error of the modified problem but not directly the actual global error of the original problem. In this paper we give an estimate for the actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems. Along with the new definition we are led to developing a reliable basis for a step selection scheme.