TheClenshaw-Curtis quadrature formula uses the extremal points of theChebyshev polynomialTn(x) as nodes. By estimating the error of interpolation we derive rather sharp error estimates if the number of nodes is odd. T...
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TheClenshaw-Curtis quadrature formula uses the extremal points of theChebyshev polynomialTn(x) as nodes. By estimating the error of interpolation we derive rather sharp error estimates if the number of nodes is odd. Thereby it appears that a modified (open) method which uses only the inner extremal points ofTn(x) as nodes gives better error estimates. In the following we gain some derivative-free estimates using the degree of approximation by algebraic polynomials. Finally we consider holomorphic integrands; here theChebyshev series expansion of the given function offers a favourable starting point for estimates.
An algorithm for finding variously restricted maximal flows through capacitated networks with gains is given. Essentially only detecting of “shortest routes” and “negative cycles” is asked.
An algorithm for finding variously restricted maximal flows through capacitated networks with gains is given. Essentially only detecting of “shortest routes” and “negative cycles” is asked.
LetL u=f be a linear operator equation, whereu is a real-valued function of the real variablex. Letu satisfym constraints of the formS u=g, whereg is a real vector. The paper approximatesu (x) by the expression\(u_n (...
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LetL u=f be a linear operator equation, whereu is a real-valued function of the real variablex. Letu satisfym constraints of the formS u=g, whereg is a real vector. The paper approximatesu (x) by the expression\(u_n (x) = \sum\limits_{i = 1}^n {a_i v_i (x),n \geqslant m} \), withvi(x) given. The coefficientsai are determined by the requirement thatun satisfies them constraints above, and the firstn−m iterated integrals ofL un have the same value as those ofL u. The paper gives convergence criteria, error estimations, and appl.cations.
Besides the electronic digital computers electronic analog computers and their generalisations to iterativ-analog computers or analog computers controlled by digital computers and finally the connection of analog and ...
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Besides the electronic digital computers electronic analog computers and their generalisations to iterativ-analog computers or analog computers controlled by digital computers and finally the connection of analog and digital computers in hybrid computers are of increasing importance especially for those mathematical problems, which do not need the high accuracy of digital computers, whereas the typical possibilities of the analog computers allow a quick survey of the features of the solutions. From these appl.cations we choose a few problems of operations research.
The paper describes a surprisingly occuring rounding error in Gomory's all-integer algorithm for the solution of linear integer programming problems. A modification for the numerical stabilization of the algorithm...
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The paper describes a surprisingly occuring rounding error in Gomory's all-integer algorithm for the solution of linear integer programming problems. A modification for the numerical stabilization of the algorithm is quoted.
We are looking for a solution of the initial boundary value problem for the threedimensional heat equation in a compact domain with a boundary of continous curvature. We use Rothe's line method, which works by dis...
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We are looking for a solution of the initial boundary value problem for the threedimensional heat equation in a compact domain with a boundary of continous curvature. We use Rothe's line method, which works by discretisation of the time variable. For every time step there remains an elliptic boundary value problem, which is solved by means of an integral equation. The so obtained approximate solutions converge to the exact solution of the original problem. In case of a sphere we find a simple error estimate for the approximation. For two initial conditions the practical computations show, that the integral equations method yields useful results with relative small effort.
The pseudoinverseAI of a matrixA is characterized through two inAI linear equations and rank (AI)≤rank(A). A posteriori error bounds are developped for the derivation of an approximationX ofAI and the errors of the r...
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The pseudoinverseAI of a matrixA is characterized through two inAI linear equations and rank (AI)≤rank(A). A posteriori error bounds are developped for the derivation of an approximationX ofAI and the errors of the residuesAAI-AX andAIA-XA. The results are extended to the best least squares solution. A numerical example illustrates the technique.
A method, called the method of pseudolinear equations, for solving a class of quasilinear hyperbolic initial-boundary-value problems of second order in one space dimension is investigated. The finite-difference implem...
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A method, called the method of pseudolinear equations, for solving a class of quasilinear hyperbolic initial-boundary-value problems of second order in one space dimension is investigated. The finite-difference implementation of the method is improved vs. a previous version by elimination of iteration. The results of a series of numerical experiments show the improved finite-difference implementation to be between 1.8 and 13.9 times faster than solution of nonlinear systems of finite-difference approximations of the quasilinear hyperbolic problems by Newton's method. However, in cases with large nonlinearities, the method of pseudolinear equations shows instability, while Newton's method converges after a large amount of computing time.
In this paper we deal with a few definitions of stability of finite-difference methods for the approximation of differential equations, integro-differential equations etc. These definitions have been presented by Stum...
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In this paper we deal with a few definitions of stability of finite-difference methods for the approximation of differential equations, integro-differential equations etc. These definitions have been presented by Stummel [17], Spijker [14], Dahlquist [3], Ansorge [1] and Kreth [9].We present conditions on which the stability of a special finite-difference method in some sense follows from the stability in another sense.
作者：
BITZER, GFachbereich Mathematik
Universität Essen-Gesamthochschule Universitätsstraße 2 D-4300 Essen 1 Bundesrepublik Deutschland
We consider the problem posed by Erdös of minimizing the sum of the squares of the fundamental functions of the Lagrange interpolation by trigonometric polynomials at2n+1 points.
We consider the problem posed by Erdös of minimizing the sum of the squares of the fundamental functions of the Lagrange interpolation by trigonometric polynomials at2n+1 points.
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