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ALGEBRAIC AND METRIC STRUCTURES IN INTERVAL ARITHMETIC AND appl.CATIONS

COMPUTING 1970年第2期5卷 144-&页

作者： MAYER, O Institut für Angewandte Mathematik und Rechenzentrum Universität Karlsruhe Englerstraße 2 D-75 Karlsruhe Bundesrepublik Deutschland

As in [1], [6] and [8] is shown, round-off errors in numerical computation can be controlled with the aid of the intrrval *** paper deals with the structure and the characteristics of the spaces occurring in interval analysis and presents some appl.cations. At first the algebraic structure of these spaces is abstractly described by the definition of the quasilinear space, a generalization of the linear space. Then metric structures in these spaces are treated. As for appl.cations only metrics are apt, which have a certain compatibility with the algebraic structure, such properties are introduced and examined. For the interval arithmetic with matrices and vectors metrics are developped, which are compatible with the algebraic structure and so are apt to estimate with. With these results for certain equations existence and uniqueness of the solution is proved as well as some iteration methods of interval analysis for the evaluation of the solution are examined and tests of convergence deduced. Finally it is shown how these iteration methods are to be carried out on a computer.

关键词： computer Science general Information Systems appl.cations (***) computer Communication Networks Software Engineering Artificial Intelligence computer appl. in administrative data processing

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SYNTAKTISCHE STRUKTUR FORMALER UND NATURLICHER SPRACHEN

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COMPUTING 1966年第2期1卷 93-&页

作者： BAUER, FL Mathematisches Institut der Technischen Hochschule Arcisstraße 21 D-8 München

A development originating from symbolic logic, programming languages and the formal investigation of natural languages leads to the abstract mathematical definition of a language Σ [A] as a free semigroup over an alphabet of generators, equipped with a binary transitive relation σ. σ represents the structure and can be expressed by a Hasse diagram. An important class contains languages for which the relation σ is isotonic under the semigroup operation (Semi-Thuë-Systems and Thuë-Systems, languages inBackus form, Chomsky-languages). An isotonic language σ can be interpreted as the transitive hull of the semigroup hull of a relation π („productions”). Essentially, we have a (partially) ordered *** central problem is the study of intervals [z, b] wherez is a fixed, distinguished minimal element: whether or not [z, b] is non-empty (word problem), what structure, i.e. relation induced by σ is carried by [z, b] (analysis problem). The analysis is particularly simple for languages which are dead-end-free with respect toz, that is for which the predomain ofb is always finite and coincides with [z, b], ifz σ b. If [z, b] contains more than one chain, then the chains are grouped, in particular for isotonic languages, in classes of phrase structures. Σ [A] is called unique with respect toz, if any non-empty [z, b] contains only one phrase structure. Languages which are dead-end-free and unique with respect toz are of particular practical importance, and in this connection also modifications leading to such a language, provided they are compatible on the postdomain ofz. Finally the importance of this abstract method for the study of formal problems in natural languages is shortly illustrated.

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CONSTRUCTION OF DIFFERENCE METHODS FOR SOLUTION OF HYPERBOLIC SYSTEMS OF 1ST ORDER PARTIAL-DIFFERENTIAL EQUATIONS BY SPECTRAL DECOMPOSITIONS

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COMPUTING 1977年第4期17卷 323-334页

作者： WOLF, G UNIV INNSBRUCK INST INFORMAT & NUMER MATH A-6010 INNSBRUCK AUSTRIA

Departing from the representation of a one-dimensional characteristic difference method by spectral decomposition, the description of a set of explicit one-step difference schemes for the solution of the initial-value problem for hyperbolic systems of first order partial differential equations inn+1 independant variables is given. Among them there are the one-dimensional methods of Friedrichs, of Courant-Isaacson-Rees, of Lax-Wendroff, and their multidimensional variants constructed by the fractional step method.

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DERIVATION-FREE METHODS WITHOUT SOLVING LINEAR EQUATIONS

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COMPUTING 1970年第1期5卷 71-&页

作者： SCHMIDT, JW LEDER, D Bereich Numerische Mathematik Technische Universität Dresden Zellescher Weg 12-14 D-x 8027 Dresden Deutsche Demokratische Republik

Iteration methods are given for the solution of nonlinear equations in normed spaces requiring functionvalues and first-order divided differences. The latter may be replaced by derivations. The linear equations of eac... 详细信息

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BREAKING DOWN OF THE INTERVAL GAUSS ALGORITHM

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COMPUTING 1979年第4期22卷 355-361页

作者： REICHMANN, K Institut für Angewandte Mathematik Albert-Ludwigs-Universität Hermann-Herder-Straße 10 D-7800 Freiburg i. Br. Bundesrepublik Deutschland

The following problem is treated: Under which conditions does the interval Gauß elimination work uncoditionally? It is shown by a counterexample that the condition 0 ∉ {detA|A∈[A]} is alwaysnecessary for the feasibility butnot sufficient for more than two *** the special case of interval tridiagonal matrices the problem of feasibility is solved completely if there is no pivoting.

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MIN-MAX-PRINCIPLES FOR EIGENVALUES OF A NONLINEAR EIGENVALUE PROBLEM

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COMPUTING 1972年第3期9卷 189-&页

作者： WERNER, B Institut für Angewandte Mathematik Rothenbaumchaussee 41 D-2000 Hamburg 13 Deutschland

Under certain conditions the eigenvalues of the nonlinear eigenvalue problemM y=N(λ)y,R1y=R2 (λ)y, for ordinary, selfadjoint differential operatorsM, N (λ) and linear boundary operatorsR1,R2 (λ) can be characterized by min-max-values of a Rayleighfunctional. For this purpose the eigenvalue problem is transformed into a Hilbert space problemA(λ)y=y, A (λ) being selfadjoint, completetly continuous operators.

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ITERATIONSVERFAHREN BEI DER KONFORMEN ABBILDUNG

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COMPUTING 1966年第2期1卷 146-+页

作者： NIETHAMMER, W Abteilung Rechenzentrum Mathematisches Institut der Universität Tübingen Wilhelmstr. 7 74 Tübingen

We consider the conformal mapping of the interior of the unit circle onto the interior of a general closed curve by the method ofTheodorsen. There is to solve a singular nonlinear integral equation. By discretisation one gets a system of nonlinear equations; we examine different iterative methods of solving this system. A theorem is proved which says that the single step iteration converges twice as fast as the total step iteration (on the latter a convergence theorem ofOpitz, Ostrowski andSaltzer is known). In the next section we appl. the SOR-method (Successive Over-Relaxation) to the systems of linear equations which appear at every step of theNewton-method. Compared with direct methods the computational work is reduced, especially on great systems. Finally we propose to appl. the SOR-method directly to the nonlinear system. Conjetures on the “limit of convergence” and the “optimal relaxation factor” are confirmed by an example.

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ARITHMETIC OF COMPLEX SETS

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COMPUTING 1980年第2-3期24卷 97-105页

作者： NICKEL, K Institut für Angewandte Mathematik Universität Freiburg i. Br. Hermann-Herder-Strasse 10 D-7800 Freiburg i. Br. Federal Republic of Germany

Let\(\mathbb{I}\)(ℝ) be the set of all real closed intervals and letΩ1:= {+, −, ×, /} be the set of arithmetic operators of ℝ. By extendingΩ1 from ℝ to\(\mathbb{I}\)(ℝ) as usual one finds that\(\mathbb{I}\)(ℝ) is closed with respect to the operations fromΩ1 (R. E. Moore [9]). In the literature several possibilities are discussed to go over from complex numbers to “complex intervals”: rectangles (Alefeld [1] et al.), discs (Henrici [4] et al.) or ellipses (Kahan [5] et al.). In all three cases the resulting sets are not closed with respect toΩ1, since the multiplication and division of such “intervals” does not lead to sets of the same kind. In what follows the question is treated whether there are classes of complex sets (“generalized intervals”) which are closed with respect toΩ1 or to subsets ofΩ1. One such class is easy to find. Also the shape of the sets involved is discussed. If it is assumed however that the sets under consideration are described by a finite number of parameters then there isno such class closed underΩ1.

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ACCELERATING CONVERGENCE OF CHEBYSHEV SERIES

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COMPUTING 1975年第3期15卷 235-246页

作者： LOCHER, F UNIV TUBINGEN MATH INSTD-7400 TUBINGENFED REP GER

The convergence rate of the Chebyshev series of an analytic function depends on the singularities. In analgy to the Euler transformation we propose a method based on conformal mapping. By this means one obtains an improvement of the convergence rate and an enlargement of the domain where the series converges. In the special case of a linear transformation we find out a connection to the Euler-Knopp method. Finally we give appl.cations to approximation problems and treat some examples.

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EXPANDED MODEL FOR PAGING DRUMS

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COMPUTING 1979年第2期23卷 99-118页

作者： BOLCH, G SCHMIDT, B Institut für Mathematische Maschinen und Datenverarbeitung IV Universität Erlangen-Nürnberg Martensstraße 3 D-8520 Erlangen Bundesrepublik Deutschland

The throughput of a computer system depends especially on the performance of the secondary storage. In order to evaluate performance it is necessary to have models of drum and disk storage units. A good summary of the existing models is given in [2].In the present paper an analytical and a simulation model for paging drums with the scheduling policies FIFO and SATF are developed. With these models it is possible to calculate variables which are important for the evaluation of system performance such as access time, system time and throughput. The model considers in addition to the usual parameters such as number of tracks, number of sectors and rotation time the probability that a page request is a read request or a write request. A write request always references the next unoccupied page of the drum. This is not considered in [1] and [2], in [4] it is considered but the results are inaccurate because of an incorrect assumption. From the results of the present paper we can see how important this differentiation *** occupancy level of the drum is considered for the first time. It is shown, that the occupancy level influences the results only if it is close to one, but then the influence is considerable.

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