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  • Articles  (6,945)
  • 1995-1999
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  • 1995-1999
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  • 1
    Electronic Resource
    Electronic Resource
    On the number of solutions to a class of complementarity problems (1981)
    Springer
    Mathematical programming 21 (1981), S. 190-203 
    Details
    ISSN: 1436-4646
    Keywords: Complementarity Problem ; Cones ; Topological Degree
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we consider the problem of establishing the number of solutions to the complementarity problem. For the case when the Jacobian of the mapping has all principal minors negative, and satisfies a condition at infinity, we prove that the problem has either 0, 1, 2 or 3 solutions. We also show that when the Jacobian has all principal minors positive, and satisfies a condition at infinity, the problem has a unique solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584240
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  • 2
    Electronic Resource
    Electronic Resource
    Finding simplicial subdivisions of polytopes (1981)
    Springer
    Mathematical programming 21 (1981), S. 233-234 
    Details
    ISSN: 1436-4646
    Keywords: Polytope ; Subdivision ; Simplex ; Volume ; Centroid
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Subdivisions of polytopes are described which place any chosen vertex in a (near-) minimal number of simplices. Ancillary procedures to find volumes and centroids of polytopes are indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584244
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  • 3
    Electronic Resource
    Electronic Resource
    Direct theorems in semi-infinite convex programming (1981)
    Springer
    Mathematical programming 21 (1981), S. 301-318 
    Details
    ISSN: 1436-4646
    Keywords: Semi-infinite Programs ; Open Helly-type Theorems ; Finite Subprograms ; Convex Programs ; Multi-criteria Programs ; Quasi-convex Programs ; Quasi-differentiable Programs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We show that a semi-infinite quasi-convex program with certain regularity conditions possesses finitely constrained subprograms with the same optimal value. This result is applied to various problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584251
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  • 4
    Electronic Resource
    Electronic Resource
    A statistical estimate of the structure of multi-extremal problems (1981)
    Springer
    Mathematical programming 21 (1981), S. 348-356 
    Details
    ISSN: 1436-4646
    Keywords: Multi-extremal Optimization ; Multinomial Distribution ; Bayes Estimate ; Monte Carlo
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The solution of ak-extremal problem is defined as the set of pairs (x i * , θi),i = 1, ⋯ ,k, where x t * isi th local minimum andθ i is the volume of the set of attraction of this minimum. A Bayesian estimate ofk and (θ 1 , ⋯ , θ k ) is constructed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584254
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  • 5
    Electronic Resource
    Electronic Resource
    Symposium on network flows (1981)
    Springer
    Mathematical programming 21 (1981), S. 120-120 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584234
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  • 6
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    On solving a primal geometric program by partial dual optimization (1981)
    Springer
    Mathematical programming 21 (1981), S. 319-330 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Geometric Programming ; Duality Theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The several published methods for mapping a dual solution estimate to a primal solution estimate in posynomial geometric programming provide no criteria for deciding how much deviation from primal feasibility, or discrepancy between the primal and dual objective function values, should be permitted before the primal solution estimate is accepted by the designer. This paper presents a new and simple dual-to-primal conversion method that uses the cost coefficients to provide a sound economic criterion for determining when to accept a primal solution estimate. The primal solution estimate generated is the exact solution to a modified primal obtained from the given primal by modifying the cost coefficients, with the exponent matrix left unchanged. The method is shown to have desirable properties when coupled with a convergent dual algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584252
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  • 7
    Electronic Resource
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    Minimum cost spanning tree games (1981)
    Springer
    Mathematical programming 21 (1981), S. 1-18 
    Details
    ISSN: 1436-4646
    Keywords: Game Theory ; Cost Allocation ; Spanning Tree
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584227
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  • 8
    Electronic Resource
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    Institute on deterministic and stochastic scheduling (1981)
    Springer
    Mathematical programming 21 (1981), S. 119-119 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584233
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  • 9
    Electronic Resource
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    Polynomial algorithms for a class of linear programs (1981)
    Springer
    Mathematical programming 21 (1981), S. 121-136 
    Details
    ISSN: 1436-4646
    Keywords: Linear Programming ; Polynomial Algorithms ; Total Unimodularity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes several algorithms for solution of linear programs. The algorithms are polynomial when the problem data satisfy certain conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584235
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  • 10
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    An algorithm for computing equilibria in a linear monetary economy (1981)
    Springer
    Mathematical programming 21 (1981), S. 182-189 
    Details
    ISSN: 1436-4646
    Keywords: Equilibrium ; Linear Monetary Economy ; Linear Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm is presented for computing equilibria in a linear monetary economy, that is, an exchange economy in which all individuals have linear utility functions and in which goods are bought and sold only in exchange for money. The algorithm computes the equilibrium prices by solving a finite sequence of linear programming problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584239
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  • 11
    Electronic Resource
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    Comment on a paper by M.C. Cheng (1981)
    Springer
    Mathematical programming 21 (1981), S. 229-232 
    Details
    ISSN: 1436-4646
    Keywords: Primal Feasible Basis ; Dual Feasible Basis ; Basic Variable ; Nonbasic Variable ; Optimal Basic Solution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In a recent paper M.C. Cheng proposed new criteria for the simplex algorithm which guarantee that (i) a nonbasic variable of a basic feasible solution will remain nonbasic in an optimal basic solution, (ii) a basic variable of a basic solution will remain basic in an optimal basic solution. This comment gives (i) a slight generalization of the first result and (ii) a counterexample to the second proposition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584243
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  • 12
    Electronic Resource
    Electronic Resource
    Euro V — tims XXV (1981)
    Springer
    Mathematical programming 21 (1981), S. 240-240 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584247
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  • 13
    Electronic Resource
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    An elementary survey of general duality theory in mathematical programming (1981)
    Springer
    Mathematical programming 21 (1981), S. 241-261 
    Details
    ISSN: 1436-4646
    Keywords: Dual problem ; Duality Theory ; Optimality Conditions ; Price Functions ; Nonlinear Programming ; Integer Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming. The idea of replacing dual variables (prices) by price functions, suggested by Everett and developed by Gould, is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming. The dual problem adopted has a (traditional) economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality. In addition we attempt to make precise the relationship between primal, dual and saddlepoint results in both the traditional Lagrangean and the more general duality theories and to see the implications of passing from prices to price functions. Finally, and perhaps surprisingly, it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584248
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  • 14
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    International symposium on the occasion of the 25th anniversary of the econometric institute, rotterdam (1981)
    Springer
    Mathematical programming 21 (1981), S. 358-358 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584256
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  • 15
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    Monotone Einschließung durch Verfahren vom Regula-falsi-Typ unter Verwendung eines verallgemeinerten Steigungsbegriffes (1981)
    Springer
    Computing 26 (1981), S. 33-44 
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    ISSN: 1436-5057
    Keywords: 65 J ; 65 H
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Summary A generalization of the concept of a divided difference operator introduced by J. W. Schmidt is derived. The inclusion of solutions by certain Regula-falsi-methods realized by a well known approximation of the Jacobian matrix, which is no divided difference operator in the sense of Schmidt but a realization of this generalization, is proved.
    Notes: Zusammenfassung Durch Verallgemeinerung des von J. W. Schmidt eingeführten Steigungsbegriffes gelingt es, bei Durchführung gewisser Verfahren vom Regula-falsi-Typ mit einer naheliegenden und häufig verwendeten Approximation der Funktionalmatrix, die keine Steigung im herkömmlichen Sinne darstellt, jedoch eine Realisierung der oben genannten Verallgemeinerung liefert, Einschließungsaussagen für Lösungen von Operatorgleichungen nachzuweisen.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243421
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  • 16
    Electronic Resource
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    Schrankentreue Berechnung der Exponentialfunktion wesentlich-nichtnegativer Matrizen (1981)
    Springer
    Computing 26 (1981), S. 57-66 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract An algorithm is proposed for computation of the exponential of a matrixX which uses the well known continued fraction expansion of tanhX. ForX essentially-nonnegative the following is proved: In interval arithmetic the algorithm is feasible, numerically convergent and bound conserving; after possibly a few initial steps it gives alternatively lower and upper bounds to the exact result.
    Notes: Zusammenfassung Es wird ein Algorithmus zur Berechnung der Exponentialfunktion von MatrizenX vorgeschlagen, der die Kettenbruch-Entwicklung von tanhX benutzt. IstX wesentlich-nichtnegativ, dann gilt: Der Algorithmus ist in Intervall-Arithmetik durchführbar, numerisch konvergent und schrankentreu; von einem Index an liefert er alternierend untere und obere Schranken für das exakte Ergebnis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243423
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  • 17
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    On the convergence of some methods for determining zeros of order-convex operators (1981)
    Springer
    Computing 26 (1981), S. 45-56 
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    ISSN: 1436-5057
    Keywords: Order-convex operator ; Banach space ; Existence ; Convergence ; Operator equation ; partially ordered topological linear space ; 65J15 ; 47H17 ; 47H07
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung SeiF:X→Y ein ordnungskonvexer Operator, woX, Y halbgeordnete Banachräume sind. Es werden zwei verwandte Verfahren zur Lösung der GleichungF(x)=0 diskutiert, von denen eines schon von Pasquali beschrieben worden ist (s. [2]), das andere von Wolfe [12]. Existenz- und Konvergenzsätze für diese Verfahren sind dargestellt und mit Hilfe von Beispielen illustriert. Ferner liegen einige Bemerkungen über ein Verfahren von Traub vor, das auch schon von Wolfe diskutiert worden ist [12].
    Notes: Abstract LetF:X→Y be an order-convex operator, whereX, Y are partially ordered Banach spaces. Two related methods for the solution ofF(x)=0 are discussed, one of which has been studied by Pasquali (see [2]) and the other by Wolfe [12]. Existence-convergence theorems for the methods are given, and these are illustrated with the aid of example. Some remarks are also made on a method due to Traub [7] which has also been discussed by Wolfe [12].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243422
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  • 18
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    Computer methods for efficient sampling from largely arbitrary statistical distributions (1981)
    Springer
    Computing 26 (1981), S. 19-31 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Gegeben sei ein Pseudo-Zufallszahlen-Generator, der gleichverteilte StichprobenU aus dem Intervall (0, 1) liefert, und eine statistische Verteilung, beschrieben durch ihre VerteilungsfunktionF(x). Dann erzeugt die InversionsmethodeX←F −1(U) Stichproben vonF(x). Ein Verfahren wird entwickelt, das “Guide”-Tafeln erstellt, mit dem Ziel, diese Inversion zu ermöglichen, so daß das Verfahren für beliebigeF(x) effizient wird. Für diskrete Verteilungen sind diese Tafeln klein und leicht zu erstellen, und der entstehende Stichproben-Algorithmus kann mit bekannten allgemeinen Verfahren gut konkurrieren. Stetige Verteilungen erfordern längere Vorbereitungszeiten und mehr Speicherplatz für die Tafeln. Diese werden mit Hilfe gegebener Wahrscheinlichkeitsdichtenf(x) vorbereitet. Die Methode ist anwendbar auf “vernünftige”f(x), einschließlich der normalerweise in der Statistik vorkommenden Fälle. Aufgrund der angeführten Rechenerfahrung mit Poisson-, Normal-, Gamma- und Cauchy-Verteilungen zeigt sich, daß unser allgemeines Verfahren fast so schnell ist, wie die besten bekannten Methoden, die speziell auf diese Verteilungen zugeschnitten wurden.
    Notes: Abstract Given a basic pseudo-random number generator which returns uniformly distributed samplesU from the interval (0, 1) and a statistical distribution as defined by its distribution functionF(x). Then the inversion methodX←F −1 (U) produces samples fromF(x). A procedure is developed which prepares “guide tables” in order to facilitate this inversion so that sampling becomes efficient for arbitraryF(x). For discrete distributions these tables are small and easy to set up, and the resulting sampling algorithm compares well with known general methods. Continuous distributions require longer set-up times and more space for tables. These are prepared using given probability densitiesf(x). The method can cope with “reasonable”f(x) including most cases which are commonly encountered in statistics. The reported computational experience, on Poisson, Normal, Gamma and Cauchy distributions, indicates that our general routine is almost as fast as the best known sampling algorithms which were specially designed for these distributions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243420
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  • 19
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    An algorithm for computing functions of triangular matrices (1981)
    Springer
    Computing 26 (1981), S. 67-71 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Berechnung von beliebigen Funktionen dreieckiger Matrizen ist rekursiveweise ermöglicht mittels einfacher Eigenschaften, dagelegt in diesem Bericht.
    Notes: Abstract Any function of a triangular matrix can be recursively calculated on the basis of simple properties stated in this paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243424
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  • 20
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    On the natural interpolation formula for cauchy type singular integral equations of the first kind (1981)
    Springer
    Computing 26 (1981), S. 73-77 
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    ISSN: 1436-5057
    Keywords: Cauchy type singular integral equations ; natural interpolation formulae ; Gauss-Chebyshev quadrature rule ; Nyström (quadrature) method for integral equations ; direct numerical solution of Cauchy type singular integral equations ; Primary: 65R20 ; Secondary: 45E05, 45L10, 65D05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Eine singuläre Integralgleichung erster Art vom Cauchy-Typ kann entweder direkt, mittels einer Gaußschen numerischen Integrationsformel, oder durch Reduktion auf eine äquivalente Fredholmsche Integralgleichung zweiter Art, wo die Nyström-Methode anwendbar ist, gelöst werden. In dieser Arbeit wird bewiesen, daß unter geeigneten und sinnvollen Bedingungen die Ausdrücke der unbekannten Funktion der Integralgleichung, die einerseits bei den natürlichen Integrationsformeln der direkten Methode und anderseits bei der Nyström-Methode entstehen, im ganzen Integrationsintervall gleich sind.
    Notes: Abstract A Cauchy type singular integral equation of the first kind can be numerically solved either directly, through the use of a Gaussian numerical integration rule, or by reduction to an equivalent Fredholm integral equation of the second kind, where the Nyström method is applicable. In this note it is proved that under appropriate but reasonable conditions the expressions of the unknown function of the integral equation, resulting from the natural interpolation formulae of the direct method, as well as of the Nyström method, are identical along the whole integration interval.
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    URL: http://dx.doi.org/10.1007/BF02243425
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  • 21
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    Strong solvability of interval linear programming problems (1981)
    Springer
    Computing 26 (1981), S. 79-82 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es werden notwendige und hinreichende Bedingungen für das Problem der linearen Optimierung mit Intervallkoeffizienten angegeben, bei denen jedes Problem der linearen Optimierung, dessen Koeffizienten in gegebenen Intervallen fixiert werden, eine optimale Lösung besitzt.
    Notes: Abstract Necessary and sufficient conditions for a linear programming problem whose parameters (both in constraints and in the objective function) are prescribed by intervals are given under which any linear programming problem with parameters being fixed in these intervals has a finite optimum.
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    URL: http://dx.doi.org/10.1007/BF02243426
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  • 22
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    A note on regular resolution (1981)
    Springer
    Computing 26 (1981), S. 87-89 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Tseitin [3] und Galil [1] haben gezeigt, daß es in der Aussagenlogik für unendlich vielen KlausenmengenS(n) der Längen gibt, so daß jeder reguläre Resolutionsbeweis fürS(n) mindestens 2 cn (für einec〉0) verschiedene Klausen enthält. Wir geben hier einen einfacheren Beweis für die viel schwächere aber dennoch nichtpolynomiale Schranke 2 clog 2 n an.
    Notes: Abstract Tseitin [3] and Galil [1] have proven that for infinitely manyn there is a set of clausesS(n) of the propositional calculus of lengthn such that any regular resolution proof ofS(n) produces at least 2 cn distinct new clauses for somec〉0. We show that it is possible to obtain a much weaker but still non-polynomial bound of 2 clog 2 n by a simpler argument.
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    URL: http://dx.doi.org/10.1007/BF02243428
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  • 23
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    Derivatives of functions by derivatives of linear positive operators (1981)
    Springer
    Computing 26 (1981), S. 83-86 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir wenden den Satz, daß für eine Segmentfunktionf, deren SegmentableitungD(f) auf dem Segment [0, 1] von oben beschränkt ist, der Hausdorff-Abstand zwischenD(f) und der Ableitung eines linearen Operators durch den Modul derH-Stetigkeit vonD(f) und Konstante nach oben beschränkt ist, an auf Bernstein-Polynome und Mirakyan-Szász-Operatoren.
    Notes: Abstract Given a segment functionf, whose segment-derivative is bounded from above on the segment [0, 1] we applicate the theorem, that the Hausdorff distance betweenD(f) and the derivative of a linear operator is bounded from above by the modulus of H-continuity ofD(f) and some constants, to Bernstein polynomials and Mirakyan-Szász-operators.
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    URL: http://dx.doi.org/10.1007/BF02243427
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  • 24
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    An adaptive, multi-level method for elliptic boundary value problems (1981)
    Springer
    Computing 26 (1981), S. 91-105 
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    ISSN: 1436-5057
    Keywords: 65N20 ; Adaptive mesh refinement ; multigrid methods ; finite element methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Das Unterprogramm PLTMG ist ein FORTRAN-Programm zur Lösung selbstadjungierter elliptischer Randwertprobleme für beliebige Bereiche desR 2. Es basiert auf einer stückweise-linearen Finite-Element-Methode, einer adaptiven Gitterverfeinerungsmethode und einer mehrstufigen iterativen Methode zur Lösung des resultierenden Systems linearer Gleichungen. In dieser Arbeit wird die Methode beschrieben, und einige numerische Ergebnisse und Vergleiche werden dargelegt.
    Notes: Abstract Subroutine PLTMG is a Fortran program for solving self-adjoint elliptic boundary value problems in general regions ofR 2. It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. In this work we describe the method and present some numerical results and comparisons.
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    URL: http://dx.doi.org/10.1007/BF02241777
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  • 25
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    On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems (1981)
    Springer
    Computing 26 (1981), S. 123-139 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Benutzung algebraischer Eigenwerte zur näherungsweisen Berechnung der Eigenwerte von Sturm-Liouville-Operatoren ist bekanntlich nur für die Grundschwingung und einige weitere Harmonische zufriedenstellend. In dieser Arbeit zeigen wir, wie man den asymptotischen Fehler, der bei verwandten aber einfachen Sturm-Liouville-Operatoren auftritt, dazu benutzen kann, um gewisse Klassen algebraischer Eigenwerte so zu korrigieren, daß die gleichmäßig gute Approximationen liefern.
    Notes: Abstract The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.
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    Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter (1981)
    Springer
    Computing 26 (1981), S. 107-121 
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    ISSN: 1436-5057
    Keywords: Primary: 65 H ; Secondary: 47 H ; Turning points ; Newton-like methods ; continuation methods ; local convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die RaumkurveL werde implizit durch das nichtlineare (n, n+1)-SystemH(u)=0 definiert. Es wird ein neues direktes Newton-ähnliches Verfahren zur Bestimmung der Rückkehrpunkte vonL beschrieben, das pro Schritt lediglich die Berechnung einer Jacobimatrix und 5 Funktionswerten vonH erfordert. Außerdem ist pro Schritt ein lineares Gleichungssystem der Dimensionn+1 mit 4 verschiedenen rechten Seiten zu lösen. Unter passenden voraussetzungen wird die lokale undQ-quadratische Konvergenz des Verfahrens bewiesen, sofern eine gewisse Diskretisierungsschrittweise geeignet gewählt wird. Zwei numerische Beispiele bestätigen die theoretischen Resultate.
    Notes: Abstract Let the space curveL be defined implicitly by the (n, n+1) nonlinear systemH(u)=0. A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH. Moreover, a linear system of dimensionn+1 with 4 different right hand sides has to be solved per step. Under suitable conditions the method is shown to converge locally withQ-order two if a certain discretization stepsize is appropriately chosen. Two numerical examples confirm the theoretical results.
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    Two general methods for dynamizing decomposable searching problems (1981)
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    Computing 26 (1981), S. 155-166 
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    ISSN: 1436-5057
    Keywords: decomposable searching problem ; dynamization ; merging ; quad-trees ; k-d trees ; halfspaces ; maximal elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir definieren zwei Klassen zerlegbarer Suchprobleme und untersuchen, wie man diese Probleme effizient dynamisch lösen könnte. Für die erste Klasse (DD) zeigen wir, daß man sowohl das Einfügen als auch das Streichen von einzelnen Elementen im Zeitmittel effizient durchführen kann. Für die zweite Klasse (MD) zeigen wir zudem, daß man mit der Benützung einer schnellen Mischungsprozedur die Effizienz noch um ein wenig verbessern kann. Alss Anwendungen zeigen wir z. B. die effiziente Dynamisierung von Quadbäumen und vom gemeinsamen Durchschnitt endlich vieler Halbebenen in der Ebene.
    Notes: Abstract We define two classes of decomposable searching problems and consider ways of efficiently dynamizing them. For the first class, DD, we show that both insertions and deletions can be processed efficiently. For the second class, MD, we exploit a merge technique to obtain better insertion times. We also give a number of examples of problems to which the methods apply, including the dynamic maintenance of quadtrees and of the common intersection of finitely many halfspaces in the plane.
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    A globally and quadratically convergent algorithm for general nonlinear programming problems (1981)
    Springer
    Computing 26 (1981), S. 141-153 
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    ISSN: 1436-5057
    Keywords: 90C30 ; 65K05 ; Nonlinear programming algorithms ; penalty algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Diese Arbeit beschreibt einen Algorithmus zur Minimierung einer nichtlinearen Funktion mit nichtlinearen Ungleichungen und Gleichungen als Nebenbedingungen. Die vorgeschlagene Methode hat die Eigenschaft, daß sie unter schwachen Voraussetzungen gegen einen Kuhn-Tucker-Punkt des betrachteten Optimierungsproblems konvergiert und unter stärkeren Voraussetzungen eine quadratische Konvergenzgeschwindigkeit aufweist. Ähnlich wie eine vor kurzem von Rosen vorgeschlagene Methode benutzt der Algorithmus eine Straffunktion, um einen Punkt in der Nähe der optimalen Lösung zu berechnen und schaltet dann auf Robinsons Methode um. Die neue Methode hat gegenüber dem Verfahren von Rosen zwei neue Eigenschaften. Erstens wird der richtige Wert des Parameters in der Straffunktion automatisch gefunden. Zweitens enthalten die mit der Methode von Robinson gelösten Teilprobleme nur lineare Gleichungen als Nebenbedingungen. Die Teilprobleme können daher besonders leicht gelöst werden. Vorläufige numerische Ergebnisse werden berichtet.
    Notes: Abstract This paper presents an algorithm for the minimization of a nonlinear objective function subject to nonlinear inequality and equality constraints. The proposed method has the two distinguishing properties that, under weak assumptions, it converges to a Kuhn-Tucker point for the problem and under somewhat stronger assumptions, the rate of convergence is quadratic. The method is similar to a recent method proposed by Rosen in that it begins by using a penalty function approach to generate a point in a neighborhood of the optimum and then switches to Robinson's method. The new method has two new features not shared by Rosen's method. First, a correct choice of penalty function parameters is constructed automatically, thus guaranteeing global convergence to a stationary point. Second, the linearly constrained subproblems solved by the Robinson method normally contain linear inequality constraints while for the method presented here, only linear equality constraints are required. That is, in a certain sense, the new method “knows” which of the linear inequality constraints will be active in the subproblems. The subproblems may thus be solved in an especially efficient manner. Preliminary computational results are presented.
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    Solving linear systems by methods based on a probabilistic interpretation (1981)
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    Computing 26 (1981), S. 209-225 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird gezeigt, wie das probabilistische Konzept der Stoppzeit eines Zufallsprozesses für eine iterative Methode zur Lösung eines Systems linearer Gleichungen herangezogen werden kann. Alle bekannten iterativen Näherungsmethoden zur Lösung linearer Gleichungssysteme, wie z. B. die Blockmethoden nach Jakobi und Gauss-Seidel und Überrelaxationsmethoden, entsprechen verschieden gewählten Stoppzeiten. Das probabilistische Verfahren bietet die Möglichkeit, Lösungstechniken für die spezielle Struktur des Problems zu adaptieren. Darüber hinaus führen die vorgeschlagenen Methoden zu einer schnelleren Konvergenz.
    Notes: Abstract In this paper it is demonstrated how the probabilistic concept of a stopping time in a random process may be used to generate an iterative method for solving a system of linear equations. Actually all known iterative approximation methods for solving linear equations are generated by various choices of a stopping time e. g. the point and block Jacobi methods, the point and block Gauss-Seidel Methods and overrelaxation methods are covered. The probabilistic approach offers—in a natural way—the possibility of adapting the solution technique to the special structure of the problem. Moreover, posterior bounds for the solution are constructed, which lead to faster convergence of the approximations than with usual prior bounds.
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    Bounding the slope of polynomial operators and some applications (1981)
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    Computing 26 (1981), S. 227-237 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung In dieser Arbeit werden mehrere systematische Möglichkeiten angegeben, die es erlauben, bei der Verwendung der intervallmäßigen Auswertung der Ableitung in Iterationsverfahren gewisse Intervalle durch reelle Zahlen zu ersetzen. Dadurch wird im allgemeinen die Konvergenz beschleunigt. Die gleiche Problemstellung wurde bereits von einem anderen Ausgangspunkt ausgehend in [5] betrachtet.
    Notes: Abstract We discuss several systematic possibilities for removing certain intervals when using an interval expression of the derivative in iterative methods. In general, the speed of convergence is accelerated in this way. Starting from a different point of view the same problem has been treated recently in [5].
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    Optimal segmented polynomial L s -approximations (1981)
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    Computing 26 (1981), S. 239-246 
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    Source: Springer Online Journal Archives 1860-2000
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    Description / Table of Contents: Zusammenfassung Es werden charakteristische Eigenschaften solcher Approximationen gegeben. Ein Algorithmus, der gute, aber nicht unbedingt beste Approximationen liefert, wird diskutiert.
    Notes: Abstract Characteristic properties of such approximations are given. An algorithm providing good, but not necessarily best, approximations is also discussed.
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    Families of fifth order Nyström methods for y″=f(x, y) and intervals of periodicity (1981)
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    Computing 26 (1981), S. 247-256 
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    Source: Springer Online Journal Archives 1860-2000
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    Description / Table of Contents: Zusammenfassung In dieser Arbeit werden sämtliche Nyström-Methoden fünfter Ordnung füry″=f(x, y) angegeben, welche vier Funktionsauswertungen pro Schritt benötigen. Es wird gezeigt, daß sich darunter nur eine einzige mit einem nichtleeren Periodizitätsintervall befindet.
    Notes: Abstract In this paper all fifth order Nyström methods fory″=f(x, y) based on four evaluations off are presented. Furthermore, we prove that out of these methods there is only one with a non-vanishing interval of periodicity.
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    Bemerkung zur Differentiation von intervallwertigen Funktionen (1981)
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    Computing 26 (1981), S. 271-274 
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    ISSN: 1436-5057
    Keywords: 65 G 10 ; Differentiation ; interval valued functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract This paper is concerned with relations between several versions of differentiation of interval valued functions.
    Notes: Zusammenfassung Es werden Beziehungen zwischen verschiedenen Ableitungsbegriffen für intervallwertige Funktionen angegeben.
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    On the accurate determination of nonisolated solutions of nonlinear equations (1981)
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    Computing 26 (1981), S. 315-326 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine einfache, aber wirkungsvolle Methode zur exakten Lösung eines nichtlinearen Gleichungssystems, dessen Funktionalmatrix in der Lösung singulär ist, vorgestellt. Dies wird durch eine Vergrößerung des Gleichungssystems zu einem höherdimensionalen Gleichungssystem, dessen entsprechende Lösung isoliert ist, erreicht. Diese Lösung kann daher z. B. mit dem Newton-Verfahren, das lokal mindestens quadratisch konvergent und selbstkorrigierend ist, mit großer Genauigkeit bestimmt werden.
    Notes: Abstract A simple but efficient method to obtain accurate solutions of a system of nonlinear equations with a singular Jacobian at the solution is presented. This is achieved by enlarging the system to a higher dimensional one whose solution in question is isolated. Thus it can be computed e. g. by Newton's method, which is locally at least quadratically convergent and selfcorrecting, so that high accuracy is attainable.
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    FORTRAN for contemporary numerical computation (1981)
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    Computing 26 (1981), S. 277-314 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung In numerischen Rechnungen treten neben den ganzen Zahlen häufig auch reelle und komplexe Zahlen, reelle und komplexe Intervalle sowie Vektoren und Matrizen über diesen Mengen auf. In der vorliegenden Arbeit erweitern wir die ProgrammierspracheFORTRAN so, daß Ausdrücke mit Operanden und Operatoren für all diese Typen (Datenmengen) akzeptiert werden. Wir beginnen mit einer kurzen Zusammenstellung dieser Räume und der arithmetischen Verknüpfungen in den Teilmengen, welche auf einem Rechner darstellbar sind. Es folgt dann eine allgemeine Beschreibung der Spracherweiterung sowie der neuen Standardfunktionen und Standardformelfunktionen für die zusätzlichen Datentypen. Im zweiten Teil der Arbeit geben wir dann die vollständige Syntax für die erweiterte Sprache in Form von leicht lesbaren Syntaxdiagrammen an. Wir erläutern auch die Semantik der Spracherweiterung.
    Notes: Abstract In addition to the integers, the real and complex numbers, the real segments (intervals) and complex segments as well as vectors and matrices over all of these comprise the fundamental data types in computation. We extendFORTRAN so that it accepts operands and operators for all of these types as primitives in expressions. We briefly review the spaces corresponding to these data types and the definitions of the arithmetic operations in their computer representable subsets. Then we give a general description of the language extension including the additional basic external functions and intrinsic functions for the new data types. Following this we give the syntax for the extended language in the form of easily traceable syntax diagrams. Comments on the semantics are also included.
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    Asymptotisch optimale Relaxationsparameter bei nichtlinearen SOR-Verfahren (Newton-SOR-Verfahren) (1981)
    Springer
    Computing 26 (1981), S. 327-332 
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    ISSN: 1436-5057
    Keywords: 65 N 20 (Numerical solution of nonlinear boundary value problems)
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract For nonlinear SOR-methods (Newton-SOR-methods) we discuss the problem how to get asymptotic optimal relaxation parameters ω(v). Here the matrix of the linear system at each step of iteration has not to be consistently ordered, we only assume symmetry and definitness. Our parameters are easy to compute and fixed for all steps. As an example nonlinear elliptic differential equations and their numerical solution by the finite element method or classical difference approximation are considered.
    Notes: Zusammenfassung Für nichtlineare SOR-Verfahren (Newton-SOR-Verfahren) wird ein einfaches Verfahren zur Berechnung asymptotisch optimaler Relaxationsparameter ω(v) erörtert. Hierbei muß die Matrix des in jedem Iterationsschritt auftretenden linearen Gleichungssystems nicht notwendig konsistent geordnet sein, wir setzen nur Symmetrie und Definitheit voraus. Es zeigt sich, daß die Parameter unabhängig von der jeweiligen Iterationsstufe ν gewählt werden können. Ihre Berechnung wird im Zusammenhang mit der numerischen Lösung nichtlinearer elliptischer Differentialgleichungen vermittels der Methode der finiten Elemente oder der klassischen Differentenapproximation diskutiert.
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    On the R-order of coupled sequences (1981)
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    Computing 26 (1981), S. 333-342 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Für Folgen aus einem iterativen Näherungsverfahren, die einem mit Hilfe von Potenzfunktionen beschreibbaren System von Ungleichungen genügen, wird gezeigt, daß ihre R-Ordnung mindestens gleich dem Spektralradius der Matrix aus den Exponenten ist, sofern ein zum Spektralradius gehörender Eigenvektor mit ausschließlich positiven Komponenten existiert.
    Notes: Abstract The R-order of sequences, coupled by a system (1) of difference inequalities, is shown to be at least equal to the spectral radius of the matrix of the exponents if a positive eigenvector belonging to the spectral radius exists.
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    A reliable rosenbrock integrator for stiff differential equations (1981)
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    Computing 26 (1981), S. 355-360 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird gezeigt, daß die Sicherheit eines Programmes für gewöhnliche Differentialgleichungen durch einen einfachen Trick wesentlich erhöht werden kann. Mit Hilfe dieser Idee und einer A-stabilen Rosenbrock-Formel haben wir ein kleines Programm geschrieben, welches gute numerische Resultate liefert.
    Notes: Abstract This note points out that the reliability of step-by-step integrators for ordinary differential equations can be increased considerably by a simple trick. We incorporated this idea into a program based on an A-stable Rosenbrock formula. This program comprises about 100 statements only and gives good numerical results.
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    A note on linear expected time algorithms for finding convex hulls (1981)
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    Computing 26 (1981), S. 361-366 
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    ISSN: 1436-5057
    Keywords: Convex hull ; average complexity ; geometrical complexity ; algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir betrachtenn als unabhängig identisch verteilte Zufallsvektoren imR d mit der gemeinsamen Verteilungsdichtef. Die mittlere Konvexität eines Algorithmus zur Bestimmung der konvexen Hülle dieser Punkte seiE (C). Die meisten bekannten Algorithmen genügen für gewisse Klassen von Dichten der BedingungE (C)=0(n). In dieser Mitteilung zeigen wirE (C)=0(n) für Algorithmen, die im Vorlauf einen „Wegwerf-Schritt” benützen, wennf auf jedem nicht ausgearteten Rechteck desR 2 beschränkt ist und positiven Abstand von 0 besitzt.
    Notes: Abstract Considern independent identically distributed random vectors fromR d with common densityf, and letE (C) be the average complexity of an algorithm that finds the convex hull of these points. Most well-known algorithms satisfyE (C)=0(n) for certain classes of densities. In this note, we show thatE (C)=0(n) for algorithms that use a “throw-away” pre-processing step whenf is bounded away from 0 and ∞ on any nondegenerate rectangle ofR 2.
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    The implementation of insertion and deletion algorithms for 1–2 brother trees (1981)
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    Computing 26 (1981), S. 367-378 
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    ISSN: 1436-5057
    Keywords: Searchtrees ; 1–2 brother trees ; insertion ; deletion ; implementation ; PASCAL
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine PASCAL-Implementierung der Einfüge- und Entferne-Algorithmen für 1–2-Bruder-Bäume entwickelt. Die Implementierung ist nicht nur an sich interessant, sondern führte darüberhinaus zu einer Verbesserung des Entferne-Algorithmus.
    Notes: Abstract We derive an implementation in PASCAL of the insertion and deletion algorithms for 1–2 brother trees. The implementation is of interest not only in its own right but also in that it has given rise to an improved deletion algorithm.
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    Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations (1981)
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    Computing 27 (1981), S. 71-80 
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    Source: Springer Online Journal Archives 1860-2000
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    Description / Table of Contents: Zusammenfassung Direkte Methoden zur Lösung von singulären Integralgleichungen vom Cauchy-Typus (S. I. G.) beruhen auf der Gaussschen Regel für numerische Integration, wobei die S. I. G. durch Anwendung der resultierenden Funktionalgleichung an geeignet gewählten Kollokationspunkten auf ein lineares Gleichungssystem reduziert wird. In diesem Artikel wurde die Äquivalenz dieser Methode mit derjenigen, welche auf der Lagrangeschen Interpolations-Approximation der unbekannten Funktion, beruht, gezeigt. Indirekte Methoden zur Lösung von S. I. G. können durch Anwendung derselben numerischen Regel an der Fredholmschen Integralgleichung, auf welche die S. I. G. reduziert wird, erhalten werden. In diesem Artikel wurde gezeigt, daß beide Methoden, im Sinne, daß sie dieselben numerischen Resultate liefern, äquivalent sind. Schließlich wurde mit Hilfe dieser Resultate der, Fehler und die Konvergenz der Methoden festgestellt.
    Notes: Abstract Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.
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    An economical method for random number generation and a normal generator (1981)
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    Computing 27 (1981), S. 113-121 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine Verallgemeinerung, der Methode von Walker für den Fall kontinuierlicher Verteilungen präsentiert. Die vorgeschlagene Methode arbeitet nach dem Rückweisungsverfahren, verwendet aber alle erzeugten Zufallswerte. Die Anwendung des Verfahrens auf den Generator für normalverteilte Zufallszahlen nach Kinderman und Ramage ergibt einen Algorithmus, der zum Algorithmus FL5 von Ahrens und Dieter äquivalent ist.
    Notes: Abstract A generalization of the Walker's method is presented for the case of continuous distributions. The proposed method is similar to the rejection technique but makes use of all the generated values. Application of the method for a normal random number generator of Kinderman and Ramage resulted in an algorithm that is equivalent to the FL5 procedure of Ahrens and Dieter.
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    Interval boxes of solutions of nonlinear systems (1981)
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    Computing 27 (1981), S. 137-144 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine Modifikation des Krawczyk-OperatorsK(X) zur Einschließung von Lösungen nichtlinearer Gleichungssysteme behandelt. Wenn für mindestens eine Komponentei die EinschließungK i (X)⊂int (X i ) gilt, kann der OperatorK(X) verbessert werden. Außerdem wird das verbesserte Verfahren mit anderen Modifikationen verglichen.
    Notes: Abstract A variation of the Krawczyk operator for including the solutions of a nonlinear system is introduced here. It improves the Krawczyk operatorK (X) wheneverK i (X)⊂int (X i ) for somei. A comparison with other variations is also given.
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    Some remarks on the discrete maximum-principle for finite elements of higher order (1981)
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    Computing 27 (1981), S. 145-154 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Diskutiert wird das diskrete Maximumprinzip für Finite-Element-Approximationen von elliptischen Standardproblemen in der Ebene. Schon im Fall Δu=0 treten bei stückweise quadratischen Elementen Verletzungen einer leicht verschärften Version des Prinzips auf, außer in einigen ganz speziellen Triangulierungsgeometrien.
    Notes: Abstract The discrete maximum principle for finite element approximations of standard elliptic problems in the plane is discussed. Even in the case Δu=0 a slightly stronger version of the principle does not hold with piecewise quadratic elements for all but some very special triangularisation geometries.
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    The complexity of basic complex operations (1981)
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    Computing 27 (1981), S. 205-215 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Bekanntlich erfordert die Berechung des Produkts zweier komplexer ZahlenX+iY undU+iV genau 3 reellem/d. In dieser Arbeit wird die algebraische Komplexität einiger anderer Operationen aus dem Bereich der komplexen Arithmetik untersucht. Zum Beispiel zeigen wir, daß die Inversion komplexer Zahlen genau 4 und die Division mindestens 5 reellem/d erfordert. Für das zweite Problem ist 6 als obere Schranke bekannt, so daß die genaue Komplexität der Division ein offenes Problem bleibt. Die Beweise benützen einige Kriterien, die auch allgemein für Komplexitätsbetrachtungen über fest vorgegebene Mengen rationaler Funktionen von Nutzen sein könnten.
    Notes: abstract It is well-known that the product of two complex numbersX+iY andU+iV requires exactly 3 realm/d. We study the algebraic complexity of several other operations from the repertoire of complex arithmetic. We show, for instance, that complex inversion requires exactly 4m/d and that general complex division requires at least 5m/d. For the latter problem an upperbound of 6m/d is known, leaving some speculation as to its precise complexity. The proofs illustrate several criteria which may be of more general use in assessing the complexity of concrete sets of rational functions.
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    Some models of error in floating point multiplication (1981)
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    Computing 27 (1981), S. 227-236 
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    Description / Table of Contents: Zusammenfassung Es werden einige neue Resultate bezüglich des Fehlers bei der Gleitkomma-Multiplikation mitgeteilt. In Abhängigkeit vom Maß für den Fehler wird festgestellt, für welche Basen der mittlere Multiplikationsfehler minimal wird; dabei sind gegenüber früheren Untersuchungen neue Fehlermaße einbezogen worden. Ein Teil der Ergebnisse hat Nutzanwendungen auf den Computerentwurf.
    Notes: Abstract New results are given on error in floating point multiplication. Certain choices of the base minimize the mean multiplicative error. These choices depend on which measure of error is selected. Some measures are included which were not in earlier studies. Some of the results have application to computer design.
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    A remark on gradual underflow (1981)
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    Computing 27 (1981), S. 217-225 
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    Description / Table of Contents: Zusammenfassung Manchmal lassen sich Berechnungsfehler leicht verstehen, wenn erst einmal für jeden berechneten Wert alle Operationen bekannt sind, die mit ihm durchgeführt werden. Mit Hilfe gerichteter Graphen lassen sich diese Abhängigkeiten aufzeigen. Einfache graphentheoretische Argumente zeigen, daß allmählicher Underflow die Ausführung gewisser Rechenvorgänge verbessert, von anderen dagegen nicht. (Einige Rechenvorgänge erfordern eingehendere Untersuchungen.) Hier verwenden wir graphentheoretische Argumente, um zu zeigen, daß bei Ausstattung der Computer-Arithmetik mit einem „denormal zero” die Fehler durch allmählichen Underflow stets mit der Unsicherheit durch Rundungsfehler vergleichbar sind, obwohl dieser Vergleich einen Faktor einschließt, der exponentiell mit der Anzahl der arithmetischen Operationen wachsen kann. Für Berechungen, bei denen ein „denormal zero” unnötig und der exponentielle Anstieg unmöglich ist, vermindert allmälicher Underflow die Verfälschung durch Underflow auf ein Maß, das vernachlässigt werden kann.
    Notes: Abstract Sometimes computational errors are easy to understand once all the uses, as an operand, of each computed value are known. Directed graphs provide a notational device for displaying these dependencies. Simple graph arguments show that gradual underflow improves the performance of certain computational procedures but not others. (Some procedures require deeper analysis.) Here we use graph arguments to show that if computer arithmetic is augmented with a “denormal zero”, then errors from gradual underflow are always comparable to the uncertainty due to rounding error, though the comparison involves a factor that can grow exponentially with the number of arithmetic operations. Thus for procedures where existence of a denormal zero is unnecessary and the exponential growth is impossible, gradual underflow diminishes the noise from underflow to a level that can safely be ignored.
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    Computational experience with an algorithm for discrete L1 approximation (1981)
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    Computing 27 (1981), S. 245-252 
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    Description / Table of Contents: Zusammenfassung Ein Algorithmus für die diskrete, lineareL p -Näherung für 1≤p〈2 wurde von Wolfe [13] vorgeschlagen und analysiert. Er untersuchte speziell die lokale Konvergenz für den Fallp=1. Diese Arbeit berichtet über die Anwendung seiner Methode, verdeutlicht ihr Verhalten an Hand einiger Beispiele und zieht Vergleiche mit der linearen Programmiermethode von Barrodale und Roberts [3].
    Notes: Abstract In a recent paper Wolfe [13] proposes and analyses an algorithm for discrete linearL p approximation with 1≤p〈2. In particular he gives a local convergence result for the casep=1. For this case, we discuss the implementation of the method, illustrate its performance on some examples, and draw some comparisons with the linear programming based method of Barrodale and Roberts [3].
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    An updated version of the Kantorovich theorem for Newton's method (1981)
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    Computing 27 (1981), S. 237-244 
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    Keywords: 65H10 ; 65J15 ; 47H17 ; Nonlinear equations ; Newton method ; Kantorovich theorem ; Error bounds
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die vorgestellte affin-invariante Version des Kantorovich-Theorems für das Newton-Verfahren umfaßt die Gragg-Tapia-Fehlerschranken, neuere optimale und schärfere obere Schranken, neue optimale und schärfere untere Schranken, und neue Ungleichungen zurq-quadratischen Konvergenz, sämtlich ausgedrückt durch die übliche majorisierende Folge. Für die Schranken werden geschlossene Ausdrücke angegeben.
    Notes: Abstract An affine invariant version of the Kantorovich theorem for Newton's method is presented. The result includes the Gragg-Tapia error bounds, as well as recent optimal and sharper upper bounds, new optimal and sharper lower bounds, and new inequalities showingq-quadratic convergence all in terms of the usual majorizing sequence. Closed form expressions for these bounds are given.
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    Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region (1981)
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    Computing 27 (1981), S. 253-284 
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    Description / Table of Contents: Zusammenfassung Es wird ein Algorithmus zur automatischen zweidimensionalen Quadratur beschrieben, der speziell für Integranden mit singulärem Verhalten am Rand des Integrationsgebietes geeignet ist. Eine Implementierung dieses Algorithmus in Standard FORTRAN wird vorgestellt. Der Algorithmus beruht auf einer nichtlinearen Transformation des Integranden in beiden Veränderlichen, die so gewählt, wird, daß der transformierte Integrand und alle seine Ableitungen am Rand des (transformierten) Integrationsgebiets verschwinden. Auf den transformierten Integranden wird die Trapezregel in der Produktform angewendet. An Hand von numerischen Testresultaten wird die Effizienz des vorgestellten Algorithmus bei der automatischen Ermittlung von sehr genauen Näherungswerten für Integrale mit unspezifischen, pathologischen Singularitäten demonstriert.
    Notes: Abstract An automatic quadrature algorithm especially designed for double integration of functions with some form of singular behaviour on the boundary of the integration region is described, and its FORTRAN code is presented. The algorithm is based on the use of the product trapezoidal rule, after a non-linear transformation of the integrand in both variables renders a new integrand function whose derivatives vanish on the (transformed) boundary. Numerical results demonstrate the ability of the algorithm to obtain high accuracies in dealing automatically with pathological singularities of non-specific types.
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    A method for globally minimizing concave functions over convex sets (1981)
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    Mathematical programming 20 (1981), S. 22-32 
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    Keywords: Concave Programming ; Extreme Point Solutions ; Global Optimization ; Nonconvex Programming ; Nonlinear Programming
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    Notes: Abstract A method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear. The algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region. The algorithm does not involve cuts of the feasible region, requires only simplex pivot operations and univariate search computations to be performed, allows the objective function to be lower semicontinuous and nonseparable, and is guaranteed to converge to the global solution. Computational aspects of the algorithm are discussed.
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    Linear max-min programming (1981)
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    Mathematical programming 20 (1981), S. 166-172 
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    Keywords: Non-Differentiable Programming ; Convex Programming ; Max—Min Programming
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    Notes: Abstract Theoretical aspects of the programming problem of maximizing the minimum value of a set of linear functionals subject to linear constraints are explored. Solution strategies are discussed and an optimality condition is developed. An algorithm is also presented.
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    Homeomorphisms of triangulations with applications to computing fixed points (1981)
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    Mathematical programming 20 (1981), S. 213-224 
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    Keywords: Complementary Pivoting Algorithms ; Fixed Points ; Triangulations ; Homeomorphism ; Linearity
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    Notes: Abstract In the past decade various complementary pivoting algorithms have been developed to search for fixed points of certain functions and point to set maps. All these methods generate a sequence of simplexes which are ‘shrinking’ to a point. This paper proposes a new method for shrinking the simplexes. It is shown that under certain conditions, the function whose fixed point is sought may be used to control this shrinking process. A computational method for implementing these ideas is also suggested and several examples are solved using this approach.
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    Connected and alternating vectors: Polyhedra and algorithms (1981)
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    Mathematical programming 20 (1981), S. 233-244 
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    Keywords: Polyhedral Combinatorics ; Combinatorial Optimization ; Total Dual Integrality ; Connected Subgraphs ; Optimal Subtrees
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    Notes: Abstract Given a graphG = (V, E), leta S, S ∈ L, be the edge set incidence vectors of its nontrivial connected subgraphs. The extreme points ofℬ = {x ∈ R E: asx ≤ |V(S)| - |S|, S ∈ L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb k, k ∈ K, ofG. WhenG is a tree, the extreme points ofB ≥ 0,b kx ≤ 1,k ∈ K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated withℬ andA, good algorithms are given and total dual integrality ofℬ andA proven.
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    Call for papers: Third Symposium on Mathematical Programming with Data Perturbations (1981)
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    Mathematical programming 20 (1981), S. 251-252 
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    Announcing Mathematical Programming Study 15 (1981)
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    Mathematical programming 20 (1981), S. 253-253 
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    An advanced implementation of the Dantzig—Wolfe decomposition algorithm for linear programming (1981)
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    Mathematical programming 20 (1981), S. 303-326 
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    Keywords: Large-Scale Systems ; Decomposition Algorithms ; Structured Linear Programs ; Optimization Software
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    Notes: Abstract Since the original work of Dantzig and Wolfe in 1960, the idea of decomposition has persisted as an attractive approach to large-scale linear programming. However, empirical experience reported in the literature over the years has not been encouraging enough to stimulate practical application. Recent experiments indicate that much improvement is possible through advanced implementations and careful selection of computational strategies. This paper describes such an effort based on state-of-the-art, modular linear programming software (IBM's MPSX/370).
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    Mathematical programming study 14 (1981)
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    Mathematical programming 20 (1981), S. 359-363 
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    The octahedral algorithm, a new simplicial fixed point algorithm (1981)
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    Mathematical programming 21 (1981), S. 47-69 
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    Keywords: Fixed Point Algorithm ; Computation of Fixed Points ; Nonlinear Equations ; Variable Dimension Algorithm
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    Notes: Abstract A new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented. It uses the restrart technique of Merrill to improve the accuracy of the solution. The algorithm is shown to converge quadratically under certain conditions. The algorithm should be efficient and relatively easy to implement.
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    Polyhedra related to a lattice (1981)
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    Mathematical programming 21 (1981), S. 70-89 
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    Keywords: Polyhedra ; Lattices ; Trees ; Submodular Functions ; Integral Polyhedra ; Polymatroids
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    Topics: Computer Science , Mathematics
    Notes: Abstract Without using the l.p. duality theorem, we give a new and direct proof that Hoffman's lattice polyhedra, polyhedra from problems of Edmonds and Giles, and others, are integer. These polyhedra are intersections of more simple polyhedra such that every vertex of the initial polyhedron is a vertex of some simple polyhedron. In many cases encountered in combinatorics the simple polyhedra have a totally unimodular constraint matrix. This implies that all vertices of the initial polyhedron are integral. The proof is based on a theorem on submodular functions, which was not known earlier. The method of this paper can be applied to the consideration of the matching polyhedron.
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    Entropy in linear programs (1981)
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    Mathematical programming 21 (1981), S. 137-151 
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    Keywords: Linear Program ; Entropy ; Newton—Kantorovich Method ; Chemical Equilibrium ; Geometric Programming ; Maximum Entropy ; Minimum Information
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    Notes: Abstract This paper treats entropy constrained linear programs from modelling as well as computational aspects. The optimal solutions to linear programs with one additional entropy constraint are expressed in terms of Lagrange-multipliers. Conditions for uniqueness are given. Sensitivity and duality are studied. The Newton—Kantorovich method is used to obtain a locally convergent iterative procedure. Related problems based on maximum entropy or minimum information are discussed.
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    A class of simplicial restart fixed point algorithms without an extra dimension (1981)
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    Mathematical programming 20 (1981), S. 33-48 
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    ISSN: 1436-4646
    Keywords: Triangulation ; Fixed Point ; Labelling ; Approximation
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    Notes: Abstract In an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices. Ideas of that paper are used to construct a class of triangulations ofR n. More precisely, for somek, 1 ≤k ≤ n, and positive integersm 1 ⋯ , mk with sumn, a triangulation ofR n is obtained by triangulating the cells which are formed by taking the product of given triangulations ofR mj, j = 1, ⋯ ,k. The triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inR n, being the starting point of the algorithm. Fork = n we obtain theK′ triangulation originally due to Todd. Each element of the class can be used to find a simplex which approximates a fixed point of a mapping onR n by generating a unique path of adjacent simplices of variable dimension starting with the pointv. We also give convergence conditions. It is indicated how in casek = n a connected set of fixed points can be generated. Moreover, we give some computational experience.
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    A parametric linear complementarity technique for the computation of equilibrium prices in a single commodity spatial model (1981)
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    Mathematical programming 20 (1981), S. 81-102 
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    ISSN: 1436-4646
    Keywords: Arc—arc Weighted Adjacency Matrix ; Computational Results ; Parametric Linear Complementarity ; Spatial Equilibrium Model ; Special-Purpose Algorithm
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    Notes: Abstract This paper presents a parametric linear complementarity technique for the computation of equilibrium prices in a single commodity spatial model. We first reformulate the model as a linear complementarity problem and then apply the parametric principal pivoting algorithm for its solution. This reformulation leads to the study of an “arc—arc weighted adjacency matrix” associated with a simple digraph having weights on the nodes. Several basic properties of such a matrix are derived. Using these properties, we show how the parametric principal pivoting algorithm can be greatly simplified in this application. Finally, we report some computational experience with the proposed technique for solving some large problems.
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    Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations (1981)
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    Mathematical programming 20 (1981), S. 255-282 
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    ISSN: 1436-4646
    Keywords: Vehicle Routing ; Lagrangean Relaxation ; Shortest Spanning Trees ; Dynamic Programming Relaxation
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    Notes: Abstract We consider the problem of routing vehicles stationed at a central facility (depot) to supply customers with known demands, in such a way as to minimize the total distance travelled. The problem is referred to as the vehicle routing problem (VRP) and is a generalization of the multiple travelling salesman problem that has many practical applications. We present tree search algorithms for the exact solution of the VRP incorporating lower bounds computed from (i) shortest spanningk-degree centre tree (k-DCT), and (ii)q-routes. The final algorithms also include problem reduction and dominance tests. Computational results are presented for a number of problems derived from the literature. The results show that the bounds derived from theq-routes are superior to those fromk-DCT and that VRPs of up to about 25 customers can be solved exactly.
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    A unification of two classes ofQ-matrices (1981)
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    Mathematical programming 20 (1981), S. 348-352 
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    Keywords: Class of Matrices ; Linear Complementarity Problem
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    Notes: Abstract This note presents a class ofQ-matrices which includes Saigal's classN ofQ-matrices with negative principal minors and the classE of strictly semi-monotoneQ-matrices.
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    Referees 1980 (1981)
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    Mathematical programming 20 (1981), S. 364-366 
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    A new algorithm for the assignment problem (1981)
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    Mathematical programming 21 (1981), S. 152-171 
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    Keywords: Assignment Problems ; Network Flows ; Hungarian Method ; Computational Complexity
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    Notes: Abstract We propose a new algorithm for the classical assignment problem. The algorithm resembles in some ways the Hungarian method but differs substantially in other respects. The average computational complexity of an efficient implementation of the algorithm seems to be considerably better than the one of the Hungarian method. In a large number of randomly generated problems the algorithm has consistently outperformed an efficiently coded version of the Hungarian method by a broad margin. The factor of improvement increases with the problem dimensionN and reaches an order of magnitude forN equal to several hundreds.
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    An equivalence between two algorithms for quadratic programming (1981)
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    Mathematical programming 20 (1981), S. 152-165 
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    Keywords: Column Generation ; Equivalence of Algorithms ; Quadratic Programming ; Simplicial Decomposition
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    Notes: Abstract In this paper, we demonstrate that the Van de Panne—Whinston symmetric simplex method when applied to a certain implicit formulation of a quadratic program generates the same sequence of primal feasible vectors as does the Von Hohenbalken simplicial decomposition algorithm specialized to the same program. Such an equivalence of the two algorithms extends earlier results for a least-distance program due to Cottle—Djang.
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    Superlinear convergence of Broyden's boundedθ-class of methods (1981)
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    Mathematical programming 20 (1981), S. 196-212 
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    Keywords: Unconstrained Minimization ; Quasi-Newton Methods ; Broyden'sθ-class ; Direct Prediction Case ; Q-Superlinear Convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper the so-called Broyden's boundedθ-class of methods is considered. It contains as a subclass Broyden's restrictedθ-class of methods, in which the updating matrices retain symmetry and positive definiteness. These iteration methods are used for solving unconstrained minimization problems of the following form: $$f(\hat x) = \min _{x \in R^n } f(x)$$ It is assumed that the step-size coefficientτ k = 1 in each iteration and the functionalf : R n → R1 satisfies the standard assumptions, viz.f is twice continuously differentiable and the Hessian matrix is uniformly positive definite and bounded (there exist constantsm, M 〉 0 such that m∥y∥2 ⩽ 〈y, $$m\parallel y\parallel ^2 \leqslant \left\langle {y,f(\hat x)\left. y \right\rangle } \right. \leqslant M\parallel y\parallel ^2 $$ for ally ∈ R n) and satisfies a Lipschitz-like condition at the optimal point $$\hat x$$ , the gradient vanishes at $$\hat x$$ Under these assumptions the local convergence of Broyden's methods is proved. Furthermore, the Q-superlinear convergence is shown.
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    A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles (1981)
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    Mathematical programming 20 (1981), S. 225-232 
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    ISSN: 1436-4646
    Keywords: Combinatorial Optimization ; Vertex Packings ; Graph Theory ; Polynomial Algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n 2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG ≥ 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n 2K+3) algorithm for testing membership inG
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    Summer school on design of numerical algorithms for parallel processing (1981)
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    Mathematical programming 20 (1981), S. 358-358 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    On optimizing functions which are defined in part by an approximation process (1981)
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    Mathematical programming 20 (1981), S. 63-80 
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    ISSN: 1436-4646
    Keywords: Approximating Functions ; Convex Programming ; Inventory Control ; Contractive Maps
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    Topics: Computer Science , Mathematics
    Notes: Abstract This paper studies an optimization problem in which the objective function can not be completely given in closed form. In particular, we assume that some part of the objective function must be computed by an approximation process. This paper develops a technique for solving a class of such problems. Examples demonstrating the technique and problem areas in which it has been successfully applied are also given.
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    Combined lp and quasi-Newton methods for minimax optimization (1981)
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    Mathematical programming 20 (1981), S. 49-62 
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    Keywords: Algorithms ; Optimization ; Minimax ; Quasi-Newton ; Superlinear Convergence
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    Notes: Abstract We present an algorithm for minimax optimization that combines LP methods and quasi-Newton methods. The quasi-Newton algorithm is used only if an irregular solution is detected, in which case second-order derivative information is needed in order to obtain a fast final rate of convergence. We prove that the algorithm can converge only to a stationary point and that normally the final rate of convergence will be either quadratic or superlinear. The performance is illustrated through some numerical examples.
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    Call for papers: International symposium on semi-infinite programming and applications (1981)
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    Mathematical programming 20 (1981), S. 122-122 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    Duality gaps in semi-infinite linear programming—an approximation problem (1981)
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    Mathematical programming 20 (1981), S. 129-143 
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    ISSN: 1436-4646
    Keywords: Semi-infinite Programs ; Duality ; Linear Perturbations ; Limiting Lagrangean
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract This paper builds upon the relationship between the objective function of a semi-infinite linear program and its constraints to identify a class of semi-infinite linear programs which do not have a duality gap. The key idea is to guarantee the approximation of the primal program by a sequence of linear programs where thenth approximating program is to minimize the objective function subject to the firstn constraints. The paper goes on to show that any program not in the identified class can be linearly perturbed into it with the optimal value of the perturbed program converging to the optimal value of the original program. The results are then extended to the case when an uncountable number of constraints are present by reducing this case to the countable case.
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    Integer programming duality: Price functions and sensitivity analysis (1981)
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    Mathematical programming 20 (1981), S. 173-195 
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    ISSN: 1436-4646
    Keywords: Integer Programming ; Duality ; Sensitivity Analysis ; Valid Inequalities ; Branch and Bound Methods ; Group Problems ; Lagrangean Relaxation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently a duality theory for integer programming has been developed. Here we examine some of the economic implications of this theory, in particular the necessity of using price functions in place of prices, and the possibility of carrying out sensitivity analysis of optimal solutions. In addition we consider the form of price functions that are generated by known algorithms for integer programming.
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    A set of staircase linear programming test problems (1981)
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    Mathematical programming 20 (1981), S. 245-250 
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    ISSN: 1436-4646
    Keywords: Staircase Linear Programs ; Test Problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A set of staircase linear programming test problems are made available for computational experiments.
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    Conference: “Optimization: Theory & Algorithms” (1981)
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    Mathematical programming 20 (1981), S. 254-254 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    Formula manipulation in ALGOL 68 and application to Routh's algorithm (1981)
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    Computing 26 (1981), S. 167-187 
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    Keywords: 12-04 ; 12D10 ; 30C15 ; 65L07 ; 68C05 ; Long integer arithmetic ; complex polynomials ; zeros of polynomials ; stability analysis ; multistep methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Dieser Aufsatz gibt eine Implementierung des Routh-Algorithmus und des Schur-Kriteriums zur Bestimmung der Lage der Nullstellen eines komplexen Polynoms in einer Veränderlichen über dem Ring der Polynome in mehreren Veränderlichen mit ganzzahligen Koeffizienten. Beide Algorithmen werden angewandt auf die charakteristischen Polynome von Mehrschrittverfahren. Ferner werden Prozeduren für die arithmetischen Operationen +,−,*,÷ mit beliebig langen ganzzahligen Operanden angegeben.
    Notes: Abstract This paper presents an implementation of Routh's algorithm and the Schur criterion, in order to determine the location of the zeros of a complex polynomial in one variable, over the ring of multivariate polynomials with integral coefficients. Both algorithms are applied to the characteristic polynomials of multistep methods. Moreover, procedures are given for the arithmetic operations +,−,*,÷ with arbitrarily long integral numbers as operands.
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    The computer generation of poisson random variables (1981)
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    Computing 26 (1981), S. 197-207 
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    ISSN: 1436-5057
    Keywords: Random number generation ; Poisson distribution ; the rejection method ; inequalities for Poisson probabilities ; average complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine exakte Computer-methode zur Erzeugung von Poisson-verteilten Zufallsvariablen angegeben. Die durchschnittliche pro Zufallsvariable benötigte Zeit nimmt ab, wenn der Poisson-Parameter gegen unendlich strebt.
    Notes: Abstract An exact method for the generation of Poisson random variables on a computer is presented. The average time required per random variate decreases as the Poisson parameter tends to infinity.
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    Reconstructing pictures from projections: On the convergence of the ART algorithm with relaxation (1981)
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    Computing 26 (1981), S. 189-195 
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    Source: Springer Online Journal Archives 1860-2000
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    Description / Table of Contents: Zusammenfassung ART-Algorithmen sind iterative Methoden zur Rekonstruktion von digitalen Bildern aus ihren Projektionen. Die Konvergenz des additiven und linearen (nicht restringierten) ART-Algorithmus mit Relaxation wird unter weit schwächeren Voraussetzungen über die Relaxationsparameter als bei bisher bekannten Resultaten bewiesen. Ein anderer Beweis zeigt die geometrisch schnelle Konvergenz des linearen relaxierten ART-Algorithmus.
    Notes: Abstract The convergence of the additive and linear ART algorithm with relaxation is proved in a new way and under weaker assumptions on the sequence of the relaation parameters than in earlier works. These algorithms are iterative methods for the reconstruction of digitized pictures from one-dimesional views. A second proof using elementary matrix algebra shows the geometric convergence of the linear ART algorithm with relaxation.
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    Ein implizites Charakteristikenverfahren zur Lösung von Anfangsrandwertaufgaben bei hyperbolischen Systemen und seine Konvergenz (1981)
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    Computing 26 (1981), S. 257-264 
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    Description / Table of Contents: Abstract We present an implicit method of characteristics for the solution of systems of quasilinear hyperbolic differential equations with two independent variables. The computation of the variables is done in a rectangular grid and is not bound by Courants conditions, which means no limitation of time-step. It was possible to give a rather elementary proof of convergence. The method is already applied to the computation of flood waves in rivers with great success.
    Notes: Zusammenfassung In der vorliegenden Arbeit wird ein implizites Charakteristikenverfahren zur Lösung quasilinearer hyperbolischer Differentialgleichungssysteme in zwei unabhängigen Variablen untersucht. Die Berechnung der Unbekannten erfolgt im Rechteckgitter und unterliegt nicht der Bedingung von Courant, d. h. keiner Zeitschritttbegrenzung. Der Nachweis der Konvergenz konnte mit elementaren Mitteln geführt werden. Das Verfahren wird bereits mit großem Erfolg zur Berechnung von Hochwasserwellen in Flüssen verwendet.
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    An algorithm for generating all maximal independent subsets of posets (1981)
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    Computing 26 (1981), S. 343-354 
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    ISSN: 1436-5057
    Keywords: poset ; independent subsets ; comparability graph ; clique
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird ein Algorithmus entwickelt, welcher das SystemU 0(Θ) aller unabhängigen Mengen einer endlichen Ordnung Θ=(A, O) generiert, und zwar aufbauend auf einer (komplizierten) Darstellung vonU 0(Θ) als Wurzelbaum. Hierbei können alle wesentlichen Datenmanipulationen innerhalb zugehöriger bipartiter Graphen durchgeführt werden, woraus eine Komplexität von höchstensO(|A|·| $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} $$ ) pro gefundener maximal unabhängiger Menge resultiert ( $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} $$ bezeichnet die unmittelbaren Ordnungs-relationen). Tatsächlich zeigte eine intensive Untersuchung von Beispielen mit Trägermengen der Kardinalität 10 bis 200 nahezu überhaupt keine Abhängigkeit der durchschnittlichen Rechenzeit von externen Parametern (wie |A|, |O| und | $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} $$ |). Zudem ergaben sich erstaunlich niedrige Rechenzeiten. Insgesamt erweist sich somit der Algorithmus als empfehlenswerte Alternative gegenüber der Anwendung der bekannten graphentheoretischen Verfahren auf den zu Θ gehörenden Vergleichbarkeitsgraphen.
    Notes: Abstract This paper gives an algorithm for determining the setU 0(Θ) of all maximal independent subsets of a finite poset Θ=(A, O), based on a (complex) Θ-induced arrangement ofU 0(Θ) as a rooted tree. By this procedure, all essential data manipulations can be restricted to certain bipartite graphs, thereby leading to a computation complexity not exceedingO(|A|·| $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} $$ ) per maximal independent subset obtained (where $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} $$ denotes the immediate ordering relations). In fact, an intensive study of examples (with |A| ranging from 10–200) even showed almost total independence of external parameters such as |A|, |O| or | $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} $$ | and at the same time led to considerably low computation times. Thus, in particular, the application of standard graph-theoretical methods to the associated comparability graph of Θ would no longer seem a reasonable approach to the considered problem.
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    On the numerical evaluation of derivatives of Cauchy principal value integrals (1981)
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    Computing 27 (1981), S. 81-88 
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    ISSN: 1436-5057
    Keywords: Primary: 65 D 30 ; 65 D 32 ; Secondary: 41 A 55 ; Cauchy type integrals ; principal value integrals ; finite-part integrals ; quadrature rule ; Gaussian quadrature rules ; differentiation under the integral sign ; Taylor's formula ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Quadraturformeln für die approximative Berechnung der Ableitungen Cauchyscher Hauptwertintegrale (bezüglich der freien Variablen der Integrale) können durch formale Differentiationen der rechten Seiten der verwendeten Integrationsformeln (ohne Ableitungen) genommen werden. Die Rechtfertigung dieser Methode unter geeigneten Bedingungen erscheint in dieser kurzen Mitteilung. Die Ergebnisse dieser kurzen Mitteilung sind auch anwendbar für die numerische Berechnung der endlichen Bestandteile gewisser divergenter Integrale und für die numerische Lösung singulärer Integrodifferentialgleichungen.
    Notes: Abstract Quadrature rules for the approximate evaluation of derivatives of Cauchy principal value integrals (with respect to the free variable inside the integral) can be obtained by formal differentiations of the right sides of the corresponding quadrature rules (without derivatives). The justification of this method, under appropriate conditions, is presented in this short communication. The results of this short communication are also applicable to the numerical evaluation of a class of finite-part integrals and to the numerical solution of singular integrodifferential equations.
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    On the solvability of ordinary differential equations with only partially continuous right-hand-side by difference methods (1981)
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    Computing 27 (1981), S. 165-167 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Ein Beispiel zeigt, daß sogar die durch stark stabile Differenzenverfahren bestimmten Näherungslösungen gewöhnlicher Differentialgleichungen nicht gegen eine Lösung der Differentialgleichung konvergieren müssen, falls die rechte Seite der Differentialgleichung nur partiell stetig ist.
    Notes: Abstract An example shows that even strongly stable difference methods need not yield a solution of initial value problems for ordinary differential equations with only partially continuous right-hand-side.
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    On the error control in ODE solvers with local extrapolation (1981)
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    Computing 27 (1981), S. 169-177 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Jackson, Enright und Hull haben auf analytischem Weg die Effektivität von RK-Algorithmen für lineare Differentialgleichungen mit konstanten Koeffizienten bewertet. Ihre Ergebnisse könnten sich auf allgemeinere Probleme übertragen lassen, falls eine gewisse Beziehung zwischen dem geschätzten und dem wahren lokalen Fehler weiter gilt. Die numerische Untersuchung dieses Sachverhalts für einen Code mit “lokaler Extrapolation” gibt einen Einblick in die Wirkungsweise der Fehlerkontrolle bei solchen Codes.
    Notes: Abstract Jackson, Enright, and Hull have analytically assessed the effectiveness of RK-algorithms for linear ODEs with constant coefficients. Their results may carry over to more general problems if a certain relation between the local error estimate and the true local error of the algorithm continues to hold. Numerical investigation of this relation for a code which performs local extrapolation gives some insight in the performance of the error control mechanism of such codes.
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    Convergence of linear multistep methods for volterra first kind equations with k(t,t)≡0 (1981)
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    Computing 27 (1981), S. 189-204 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Diese Arbeit befaßt sich mit Volterraschen Integralgleichungen erster Art, deren Kernk(t,s) fürt=s identisch verschwindet. Die Begriffe Null-Stabilität und schwache Null-Stabilität werden eingeführt und Konvergenzresultate unter der Voraussetzung angegeben, daß der Verfahrensfehler eine asymptotische Entwicklung mit einer gewissen Anzahl von Gliedern besitzt. Weiters werden einfache numerische Beispiele angegeben, die diese Konvergenzraten bestätigen.
    Notes: Abstract This paper is concerned with Volterra integral equations of the first kind whose kernel,k(t,s), is identically zero whent=s. The concepts of zero-stability and weak zero-stability are introduced and convergence results under the assumption that the truncation error has an asymptotic expansion with a certain number of terms are presented. Simple numerical examples verifying these rates of convergence are given.
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    Algorithm for the solution of the Bottleneck Assignment Problem (1981)
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    Computing 27 (1981), S. 179-187 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Für das “Bottleneck Assignment Problem” wird eine FORTRAN-Implementierung eines wirkungsvollen Lösungsverfahrens angegeben. Die angeführten Rechenergebnisse zeigen, daß die vorgeschlagene Methode den derzeit besten bekannten Algorithmen überlegen ist.
    Notes: Abstract The FORTRAN implementation of an efficient algorithm which solves the Bottleneck Assignment Problem is given. Computational results are presented, showing the proposed method to be generally superior to the best known algorithms.
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    A note on quasi-newton formulae for sparse second derivative matrices (1981)
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    Mathematical programming 20 (1981), S. 144-151 
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    ISSN: 1436-4646
    Keywords: Unconstrained Minimization ; Quasi-Newton Methods ; Sparsity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In order to apply quasi-Newton methods to solve unconstrained minimization calculations when the number of variables is very large, it is usually necessary to make use of any sparsity in the second derivative matrix of the objective function. Therefore, it is important to extend to the sparse case the updating formulae that occur in variable metric algorithms to revise the estimate of the second derivative matrix. Suitable extensions suggest themselves when the updating formulae are derived by variational methods [1, 3]. The purpose of the present paper is to give a new proof of a theorem of Dennis and Schnabel [1], that shows the effect of sparsity on updating formulae for second derivative estimates.
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    Multicriteria integer programming: A (hybrid) dynamic programming recursive approach (1981)
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    Mathematical programming 21 (1981), S. 204-223 
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    ISSN: 1436-4646
    Keywords: Multicriteria Optimization ; Integer Programming ; Dynamic Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Dynamic programming recursive equations are used to develop a procedure to obtain the set of efficient solutions to the multicriteria integer linear programming problem. An alternate method is produced by combining this procedure with branch and bound rules. Computational results are reported.
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    On some characterisations of totally unimodular matrices (1981)
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    Mathematical programming 20 (1981), S. 14-21 
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    ISSN: 1436-4646
    Keywords: Totally Unimodular Matrices ; Balanced Matrices ; Hypergraphs ; Perfect Graphs ; Coloring
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A characterisation of totally unimodular matrices is derived from a result of Hoffman and Kruskal. It is similar in spirit to a result of Baum and Trotter. Its relation with some other known characterisations is discussed and in the particular case where the matrices have (0, 1) entries, we derive some properties of the associated unimodular hypergraphs. Similar results for balanced and perfect matrices are also reviewed.
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    Mathematical programming study 13 (1981)
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    Mathematical programming 20 (1981), S. 123-126 
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    The 1980 lanchester prize (1981)
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    Mathematical programming 20 (1981), S. 127-128 
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    Average time behavior of distributive sorting algorithms (1981)
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    Computing 26 (1981), S. 1-7 
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    Keywords: Distributive sorting ; bucket sorting ; average complexity ; expected running time ; 5.31
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir untersuchen die mittlere KomplexitätE(C) von Fachsortier-Algorithmen, die auf eine StichprobeX 1, ...,X n mit der Verteilungsdichtef aufR 1 angewendet werden. Wir nehmen an, daß die Zeit zur Bestimmung des Sortierfachs konstant ist, und daß für die Sortierung innerhalb jedes Fachs ein Algorithmus mit dem mittleren Zeitbedarfg(n) zur Verfügung steht. Dabei istg konvex,g(n)/n↑∞,g(n)/n 2 nichtsteigend undg unabhängig vonf. Wir zeigen: 1) Wennf kompakten Träger hat, dann gilt ∫g(f(x))dx〈∞ genau dann, wennE(C)=0(n). 2) Wennf keinen kompakten Träger hat, dann gilt $$E(C)/n\xrightarrow{n}\infty $$ . Überf benötigen wir keinerlei weitere Voraussetzungen.
    Notes: Abstract In this paper we investigate the expected complexityE(C) of distributive (“bucket”) sorting algorithms on a sampleX 1, ...,X n drawn from a densityf onR 1. Assuming constant time bucket membership determination and assuming the use of an average timeg(n) algorithm for subsequent sorting within each bucket (whereg is convex,g(n)/n↑∞,g(n)/n 2 is nonincreasing andg is independent off), the following is shown: 1) Iff has compact support, then ∫g(f(x))dx〈∞ if and only ifE(C)=0(n). 2) Iff does not have compact support, then $$E(C)/n\xrightarrow{n}\infty $$ . No additional restrictions are put onf.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243418
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  • 95
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    Full table scatter storage parallel searching (1981)
    Springer
    Computing 26 (1981), S. 9-18 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die vorliegende Arbeit geht von der unbestrittenen Tatsache aus, daß in Zukunft immer mehr für häufig vorkommende Aufgaben in einem Computer-System spezielle Prozessoren zur Entlastung der Zentraleinheit eingesetzt werden. Aus Gründen der Leistungsfähigkeit und in Anbetracht der fallenden Kosten für Hardware ist man zur Lösung solcher Teilaufgaben an Algorithmen interessiert, die parallele Verarbeitung erlauben. Im konkreten Beispiel wird untersucht, wie Kollisionsstrategien durch mehrere Prozessoren beim Suchen in einer Hashtabelle ausgeführt werden können. Im besonderen wird eine quadratische Kollisionsstrategie entwickelt, die die ganze Tabelle bestreicht und wesentlich einfacher ist, als die bekannten (sequentiellen) Verfahren. Die Methode beruht auf Ergebnissen aus der Theorie der Quadratreste und ist auch als ein sequentielles Verfahren für Tabellenlängenp=8m±3 effizienter als vergleichbare andere Kollisionsstrategien.
    Notes: Abstract The principle ideas of this paper are based on the uncontested statement that in the near future more and more dedicated hardware devices will be in use in addition to the general central processing unit. For reasons of performance it is desired to use algorithms which allow parallelism. In our particular case we discuss a scatter storage technique and describe appropriate parallel methods for most of the common collision strategies. The methods are based on a possible partition of the range of the tableaddresses 0≦a≦N−1. However, in addition, we present a simple and easily implemented quadratic residue search which in itself can be used as a conventional sequential method. This algorithm is deduced from classical lemmas on the theory of quadratic residues for primes of the form 8m±3.
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    URL: http://dx.doi.org/10.1007/BF02243419
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  • 96
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    Correction to algorithm 39 clusterwise linear regression (1981)
    Springer
    Computing 26 (1981), S. 275-275 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243486
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  • 97
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    Heuristic algorithms for the multiple knapsack problem (1981)
    Springer
    Computing 27 (1981), S. 93-112 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Vorhanden ist eine Menge von Gegenständen mit gegebenem Gewicht und gegebenem Wert, ferner eine Menge von Rucksäcken mit gegebener Tragfähigkeit. Die Gegenstände sollen so in die Rucksäcke gepackt werden, daß die Tragfähigkeit der einzelnen Rucksäcke nicht überschritten wird und der Gesamtwert der verwendeten Gegenstände möglichst groß ist. Die besten bekannten Algorithmen liefern mit vertretbarem Rechenaufwand, Lösungen für höchstens 200 Gegenstände und 4 Rucksäcke. Aber praktische, Anwendungen wie z. B. Teilungs-, Lagerhaltungs- und Ladeaufgaben benötigen oft eine größere Zahl von Variablen und verlangen, daher die Verwendung von Heuristik. Der vorliegende Artikel erläutert Methoden, mit deren Hilfe sich suboptimale Lösungen des Mehr-Rucksack-Problems finden lassen. Numerische Experimente mit Problemen verschiedener, Größe zeigen, daß sich die verschiedenen Algorithmen zufriedenstellend verhalten, sowohl hinsichtlich der Laufzeiten als auch bezüglich der Güte der gefundenen Lösungen. Eine RORTRAN-IV-Version der Algorithmen ist beigefügt.
    Notes: Abstract Given a set of items, each having a profit and a weight, and a set of knapsacks, each having a capacity, we consider the problem of inserting items into the knapsacks in such a way that the subset of inserted, items has the maximum total profit without the total weight in each knapsack exceeding its capacity. The best algorithms for the exact solution of this problem can be applied, with acceptable running times, to cases with a maximum of 200 items and 4 knapsacks, but real world applications (such as, for example, cutting stock and loading problems), often involving a greater number of variables, call for the use of heuristics. This paper presents methods for finding suboptimal, solutions to the Multiple Knapsack Problem. An extensive computational experience was carried out both on small-size and large-size randomly generated problems; the results indicate that the proposed algorithms have a satisfactory behaviour with regard both to running times and quality of the solutions found. A Fortrans IV implementation of the algorithms is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02243544
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  • 98
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    Interpolation methods in one dimensional optimization (1981)
    Springer
    Computing 27 (1981), S. 155-163 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Numerische Vergleiche zwischen rationalen Funktionen als Interpolationsfunktionen bei eindimensionalen Optimierungsaufgaben. Als Text-Zielfunktionen wurden Funktionen von drei Typen ausgewählt: symmetrische und nichtsymmetrische Funktionen und Funktionen mit Badewannengestalt.
    Notes: Abstract Various rational functions are compared as interpolation functions in one dimensional optimization. Three types of objective functions are employed for numerical tests—quadratic type, asymmetric type and bath-tub shaped functions.
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    URL: http://dx.doi.org/10.1007/BF02243549
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  • 99
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    Piecewise linear optimization: Duality and numerical results (1981)
    Springer
    Computing 27 (1981), S. 285-298 
    Details
    ISSN: 1436-5057
    Keywords: 90 C 05 ; 90 C 25 ; Piecewise linear programming ; convex programming ; duality ; Stückweise lineare Optimierung ; konvexe Optimierung ; Dualität
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die zu einem Optimierungs-problem mit stückweise linearer, konvexer und stetiger Zielfunktion und linearen Nebenbedingungen duale Aufgabe wird untersucht. Es zeigt sich, daß das duale Problem vom selben Typ wie das primale ist und mit denselben Verfahren gelöst werden kann, wie sie in einer früheren Arbeit des Autors vorgeschlagen wurden. Erfahrungen die bei der Anwendung verschiedener Varianten dieser Verfahren zur Lösung einer Anzahl von Beispielen gemacht wurden, wobei sowohl von der primalen als auch von der dualen Version ausgegangen wurde, werden berichtet.
    Notes: Abstract The dual of an optimization problem with piecewise linear, convex, and continous objective function and linear restrictions is investigated. It is shown that the dual problem is of the same type as the primal and can be solved by the same methods as were proposed in a previous work of the author. Experiences arising from the application of different variants of these methods to solve a number of examples, where the primal as well as the dual version was employed, are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02277180
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  • 100
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    The centered form for interval polynomials (1981)
    Springer
    Computing 27 (1981), S. 339-348 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es werden zentrierte Formen entwickelt, die den Wertebereich eines Intervallpolynoms über einen Intervall abschätzen. Mehrere Methoden werden vorgeschlagen und an Intervallpolynomen, deren Koeffizienten mit einem Zufallsgenerator erzeugt worden sind, getestet. Theoretische und numerische Ergebnisse zeigen, daß keine der vorgeschlagenen Methoden optimal ist. Die Methoden werden auch mit dem Hornerschema verglichen.
    Notes: Abstract Centered forms for computing the range of values of a real interval polynomial over an interval are considered. A number of methods are proposed and tested on randomly generated interval polynomials. Theoretical and numerical results show that none of the suggested methods produce the optimal results in all cases. The methods are also compared to evaluation using the Horner scheme
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    URL: http://dx.doi.org/10.1007/BF02277183
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