On componentwise error estimates for inverse matrices

LetA be an×n-matrix with the property I–A<1. LetY be an approximation of the inverse ofA. This paper shows how to get a componentwise error estimate forY, that does not require too much numerical effort but generally presents better results than global error estimates do. Although proved by means of interval mathematics, the given error estimate can also be calculated in absence of any implementation of interval arithmetic.

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Über komponentenweise Fehlerabschätzungen für inverse Matrizen

Zusammenfassung SeiA einen×n-Matrix mit der Eigenschaft I–A<1. SeiY eine Approximation der Inversen vonA. In dieser Arbeit wird gezeigt, wie man eine komponentenweise Fehlerabschätzung fürY erhalten kann, deren Berechnung nicht sehr aufwendig ist, die aber im allgemeinen schärfer ist als globale Fehlerabschätzungen. Obwohl mit intervallmathematischen Mitteln bewiesen, kann die angegebene Fehlerabschätzung auch berechnet werden, wenn keine Intervallarithmetik implementiert ist.

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This research was supported in part by Sonderforschungsbereich 72-Approximation und Optimierung, University of Bonn.     

Explicit,optimal stability functionals and their application to cyclic discretization methods

The present paper contains a stability concept for discretization methods of a certain, very general classM, which is optimal (in the sense of yielding the best general, two-sided error bounds) without being more restrictive than any of the classical stability definitions. The optimal stability functional Ψ_h related to it depends on the linear part of the discretization operator, and has the important property that Ψ_h [δ] may be of orderq+1, i.e. Ψ_h [δ] = O(h ^q+1), even if the local error δ only has orderq, δ = O(h ^q). This result may be used for the construction of methods with maximum order. Its application to linear cyclic methods, for example, furnishes a new approach to the theory of linearM-cyclick-step methods of maximum order.     

Konvergenz von quadratischen Interpolations- und Flächenabgleichssplines

The purpose of this paper is to show that for continuous functions the related quadratic splines converge without any assumption on the spline grid. The points of the interpolatory grid can be chosen between the corresponding points of the spline grid with a division ratio from \(\frac{{\sqrt 2 }}{2}\) to \(1 - \frac{{\sqrt 2 }}{2}\) . In the case of continuously differentiable functions the division ratio can even be taken between 0 and 1; in addition, the order of convergence is increased. For twice differentiable functions the full order of convergence is obtained. Analogous results about the convergence of histo splines are proved.     

Optimal interval enclosing of certain sets of matrices,with application to monotone enclosing of square roots

In this paper, two kinds of partial ordering for symmetric matrices are related to each other, namely, the natural partial ordering ≤ generated by the coneK of elementwise nonnegative matrices, and the definite partial ordering \( \leqslant \cdot\) generated by the coneK _D of nonnegative definite matrices. The main result of this paper shows how a matrix interval in the sense of the definite partial ordering can be enclosed between optimal bounds with respect to the natural partial ordering. By means of this result, it is possible to compute a numerically practicable inclusion based on the natural partial ordering from a given inclusion of some matrix with the definite partial ordering. In this way, an always and moreover quadratically convergent method of elementwise enclosing the square root of a positive definite, symmetric matrix can be constructed.     

Unsteady mixed convection from an isothermal circular cylinder

Summary Combined unsteady convection from an isothermal horizontal cylinder in a stream flowing vertically upwards has been investigated. Numerical solutions of the unsteady boundary-layer equations have been obtained at any station along the cylinder using the series truncation method. Solutions which are valid near the front and near stagnation points have been obtained using standard finite-difference methods. A series solution in powers of time has been obtained with which the numerical solutions has been checked.

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Unstetige, gemischte Konvektion um einen isothermen Kreiszylinder

Zusammenfassung Es wird eine kombinierte, unstetige Konvektion eines isothermen, horizontalen Zylinders in einer vertikal nach oben gerichteten Strömung untersucht. Numerische Lösungen der unstetigen Grenzschichtgleichungen werden an jeder Stelle längs des Zylinders durch die Verwendung der Reihenabbruchsmethode erhalten.Nahe der Vorderseite und nahe bei den Staupunkten gültige Lösungen werden durch Verwendung üblicher Methoden der finiten Differenzen erhalten. Mit Hilfe einer Reihenlösung in Potenzen der Zeit wird das numerische Ergebnis überprüft.

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Notation a radius of the cylinder - g acceleration of gravity - G _r Grashof number =g|T|a ³/v ² - Q heat transfer - R _e Reynolds number =U ₀ a/v - T temperature of fluid in the boundary layer - T ₀ temperature of the ambient fluid - T ₁ temperature of the cylinder - T temperature difference=T ₁–T ₀ - t time - U ₀ free stream - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r /R _e ² - coefficient of expansion - coefficient of kinematic viscosity - _w skin friction With 7 Figures     

MARKAL — Ein LP-Modell für die Internationale Energieagentur

Zusammenfassung MARKAL ist ein LP-Modell der Energieversorgung, mit dem für die Internationale Energieagentur Rechnungen zur langfristigen Technologiepolitik zur Öleinsparung durchgeführt werden. Als Mehrperiodenmodell deckt es einen Planungszeitraum von 40 Jahren ab und umfaßt alle wesentlichen Sektoren und Umwandlungsprozesse der Energiewirtschaft. Optimierungsrechnungen, die die Energieversorgung der wesentlichen westlichen Industriestaaten umfassen, zeigen unter anderem auf, daß die Chancen einer substantiellen Öleinsparung skeptisch zu beurteilen sind.

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MARKAL is a linear programming model of the energy supply used by the International Energy Agency for a long term technology evaluation to save oil. As a multi-period model, it covers a planning period of 40 years for all important sectors and technical processes within the energy economy. The model applications to the energy supplies of most industrialized western countries justify a certain scepticism about the chances for a substantial oil saving in the future.

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Diese Arbeit wurde als Vortrag auf der Sitzung der DGOR-Arbeitsgruppe Praxis der linearen Optimierung (PRALINE) unter dem Generalthema Energieplanung am 18. 1. 1980 in Jülich präsentiert.     

Estimating errors of numerical approximation for periodic analytic functions

We give a simple method based on Cauchy's integral formula for estimating the errors of numerical approximation for periodic analytic functions. We then obtain error estimates for the quadrature formulas of Chawla and Ramakrishnan [1] for the numerical evaluation of the Cauchy principal value integral $$I\left( {f;a} \right) = \int\limits_0^{2\pi } {f(x)\cot \left( {\left( {x - a} \right)/2} \right)dx,} $$ and for the quadrature formula of Garrick [2] for the evaluation ofI (f;o), Based on these error estimates, we are led to conclude that for the evaluation ofI (f;o), Garrick's formula has a better error estimate than the formula of Chawla and Ramakrishnan with the same number of function evaluations. Finally, we extend Garrick's formula for the evaluation ofI (f;a) for arbitrarya∈[0,2π); the extended formula has, for alla, the same error estimate as Garrick's formula. While, for a=x_j, the extended formula is identical with the quadrature formula of Wittich [3], fora≠x _j, the extended formula is much better in that it uses only half the number of function evaluations of Wittich's formula for the same accuracy.     

Approximation der optimalen Vorhersage der Ausgabe stochastischer Quellen

This paper discusses the prediction of the output of a stochastic sourceQ having positive transition function. If the sourceQ has two output alphabets,X andY, then there exists such a stochastic automaton \(\mathfrak{A}(V_{opt} )\) with input alphabetX, output alphabetY, and a countable set of states, that realizes the optimal predictionV _opt fromX toY. Here it will be shown thatV _opt can be approximated with any precision by a predictionV which is realized by a stochastic automaton with finite set of states.     

Experimente zum Strömungswiderstand in gekrümmten,rotierenden Kanälen mit quadratischem Querschnitt

Zusammenfassung Bei ausgebildeter Strömung wurde der Druckverlust in rotierenden, gekrümmten Kanalstrecken experimentell ermittelt. Speziell interssierte der Einfluß der Krümmung auf den Reibungswiderstand und im besonderen der der Rotation, die sowohl in Richtung der Durchflußgeschwindigkeit als auch entgegengesetzt dazu erfolgte. Die strömungsspezifischen Parameter wie Reynolds-und Taylor-Zahl erfuhren eine Änderung im Bereich 3·10²<Re<2·10⁵ und 0Ta<10⁵. Das Krümmungsverhältnis variierte zwischen 0,0496a/r _m 0,337. Aufgezeigt wird die Möglichkeit einer universellen Darstellung des Widerstandsbeiwertes für den nicht rotierenden Kanal bei einerseits laminarer und anderseits bei turbulenter Bewegung. Im rotierenden Kanal beobachtet man beträchtliche Druckverlusterhöhungen, deren Größe von der Rotationsrichtung abhängt. Bei großen Taylor-Zahlen wurde eine Unabhängigkeit des Strömungswiderstandes vom Krümmungsverhältnis festgestellt.

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Experimental investigations on the friction resistance in rotating curved channels with square cross-section

Summary For fully developed flow the pressure loss in rotating curved channels was investigated experimentally. The influence of the curvature on the friction coefficient and particularly on the rotation, which was realized in the direction of the through flow as well as in the opposite direction was of special interest. The specific parameters of the flow field, like the Reynolds-number and the Taylor-number were varied within the range of 3·10²<Re<2·10⁵ and 0Ta<10⁵. The curvature ratio were varied within 0,0496a/r _m 0,337. The possibility of an universal presentation of the friction coefficient for the nonrotating channel is shown for laminar and turbulent through flow. In the rotating channel a considerable pressure loss increase can be observed. The magnitude depends on the direction of rotation. In case of great Taylor-numbers the curvature ratio was found to be independent of the friction resistance.

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Bezeichnungen a Kantenlänge - mittlere Geschwindigkeit - l Kanallänge - r _m Krümmungsradius - Volumenstrom - _p Druckdifferenz - Widerstandsbeiwert - =57/Re Widerstandsbeiwert des geraden Kanals - dynamische Viskosität - Dichte - Winkelgeschwindigkeit - Dean-Zahl - Reynolds-Zahl - Taylor-Zahl Mit 6 Abbildungen     

On grammar forms with terminal context

Summary A context-free grammar form G defines a family of context-free grammars {ie397-01}(G). In this paper {ie397-02}(G) is extended to give the family of terminal context grammars {ie397-03}(G), which were introduced by Book [4]. It is shown that the families of languages defined by {ie397-04}(G) and {ie397-05}(G) are equal when G is an -free context-free grammar form. This generalizes the result of Book [4] for context-free grammars and underlines the usefulness of the notion of grammar forms in formal language theory.The work of the second author was carried out partially under a National Research Council of Canada Grant. No. A-7700 and partially under the auspices of the University of Karlsruhe