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901.
We solve the functional equation
  相似文献   
902.
LetC m be a compound quadrature formula, i.e.C m is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ n to every subinterval. LetR m be the corresponding error functional. Iff (r) > 0 impliesR m [f] > 0 (orR m [f] < 0),=" then=" we=" say=">C m is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf (r) > 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC m be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR m [f], where , denotes the modulus of continuity of orderr:
  相似文献   
903.
Summary Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wonicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.This work was supported by the National Science Foundation grants DMS-8807338 and INT-8918502  相似文献   
904.
Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n –2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw , (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw , is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday  相似文献   
905.
Summary Utilizing kernel structure properties a unified construction of Hankel matrix inversion algorithms is presented. Three types of algorithms are obtained: 1)O(n 2) complexity Levinson type, 2)O (n) parallel complexity Schur-type, and 3)O(n log2 n) complexity asymptotically fast ones. All algorithms work without additional assumption (like strong nonsingularity).  相似文献   
906.
Summary The good Boussinesq equationu tt =–u xxxx +u xx +(u 2) xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation. Numerical experimentas are also reported.  相似文献   
907.
Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL p[O,T;L 2()] norms for boundary conditions inL p[O,T;L 2()], 1p.  相似文献   
908.
Summary For the numerical integration of boundary value problems for first order ordinary differential systems, collocation on Gaussian points is known to provide a powerful method. In this paper we introduce a defect correction method for the iterative solution of such high order collocation equations. The method uses the trapezoidal scheme as the basic discretization and an adapted form of the collocation equations for defect evaluation. The error analysis is based on estimates of the contractive power of the defect correction iteration. It is shown that the iteration producesO(h 2), convergence rates for smooth starting vectors. A new result is that the iteration damps all kind of errors, so that it can also handle non-smooth starting vectors successfully.  相似文献   
909.
Theorem.Let the sequences {e i (n) },i=1, 2, 3,n=0, 1, 2, ...be defined by where the e (0) s satisfy and where all square roots are taken positive. Then where the convergence is quadratic and monotone and where The discussions of convergence are entirely elementary. However, although the determination of the limits can be made in an elementary way, an acquaintance with elliptic objects is desirable for real understanding.  相似文献   
910.
Summary The problem of the construction of an equilibrium surface taking the surface tension into account leads to Laplace-Young equation which is a nonlinear elliptic free-boundary problem. In contrast to Orr et al. where an iterative technique is used for direct solution of the equation for problems with simple geometry, we propose here an alternative approach based on shape optimization techniques. The shape of the domain of the liquid is varied to attain the optimality condition. Using optimal control theory to derive expressions for the gradient, a numerical scheme is proposed and simple model problems are solved to validate the scheme.  相似文献   
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