Showing papers in "Computing in 1978"
TL;DR: To perform this, a new vector product, compatible with the Samelson inverse of a vector, is defined, and conditions for a given order are derived.
Abstract: Some nonlinear methods for solving single ordinary differiential equations are generalized to solve systems of equations. To perform this, a new vector product, compatible with the Samelson inverse of a vector, is defined. Conditions for a given order are derived.
151 citations
TL;DR: A fast iterative method for solving the solution of parabolic control problems by a system of two equations parabolic with respect to opposite orientations is proposed.
Abstract: The solution of parabolic control problems is characterized by a system of two equations parabolic with respect to opposite orientations. In this paper a fast iterative method for solving such problems is proposed.
119 citations
TL;DR: A multi-grid method is applied to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.
Abstract: Multi-grid methods are characterized by the simultaneous use of additional auxiliary grids corresponding to coarser step widths. Contrary to usual iterative methods the speed of convergence is very fast and does not tend to one if the step size approaches zero. The computational amount of one iteration is proportional toN, the number of grid points. Thus, a solution with accuracy ɛ requires 0 (|log ɛ|N) operations. In this paper we apply a multi-grid method to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.
109 citations
TL;DR: It is shown that certain intervals in the methods can be replaced by real numbers, which improves the convergence properties of the methods.
Abstract: Improved forms of some interval Newton Methods are given. It is shown that certain intervals in the methods can be replaced by real numbers. This improves the convergence properties of the methods.
61 citations
TL;DR: An efficient method for solving Linear Bottleneck Assignment problems is described, which starts with a heuristically determined partial assignment and shortest augmenting paths are constructed with the aid of a modification of the algorithm of Dijkstra.
Abstract: An efficient method for solving Linear Bottleneck Assignment problems is described. The method starts with a heuristically determined partial assignment. Then shortest augmenting paths are constructed with the aid of a modification of the algorithm of Dijkstra. Comprehensive numerical investigations are reported and discussed. A FORTRAN IV subroutine can be found in the appendix.
56 citations
TL;DR: In this paper a metal algorithm for the class of IDeC-methods for differential equations is presented and analyzed and conditions are given which guarantee a certain order of accuracy.
Abstract: Iterated Defect Correction (IDeC) is a technique for improving successively an approximate solution of a given problemFy=0. One of the most important fields of application of this principle are differential equations. Here, IDeC can be used as a technique for increasing the order of a discretization method and thus for improving the accuracy. In this paper a metalgorithm for the class of IDeC-methods for differential equations is presented and analyzed. For every component of this metalgorithm conditions are given which guarantee a certain order of accuracy. These conditions are of particular importance for practical applications, as far as the implementation of IDeC-methods is concerned.
44 citations
TL;DR: Several efficient methods for evaluating functions defined by power series expansions by investigating theoretically and physically the convergence rates of the proposed computational schemes.
Abstract: In this paper we present several efficient methods for evaluating functions defined by power series expansions. Simple computer codes for two rapid algorithms are given in a companion paper. The convergence rates of the proposed computational schemes are investigated theoretically and the results are illustrated by numerical examples.
33 citations
TL;DR: The FORTRAN implementation of an efficient algorithm which solves the 0–1 single knapsack problem is given and results are presented, showing the proposed method to be generally superior to the best known algorithms.
Abstract: The FORTRAN implementation of an efficient algorithm which solves the 0–1 single knapsack problem is given. Computational results are presented, showing the proposed method to be generally superior to the best known algorithms.
31 citations
TL;DR: Four steplength algorithms are presented for minimizing a class of nondifferentiable functions which includes functions arising froml1 andl∞ approximation problems and penalty functions arisingfrom constrained optimization problems.
Abstract: Four steplength algorithms are presented for minimizing a class of nondifferentiable functions which includes functions arising from $\ell_1$ and $\ell_\infty$ approximation problems and penalty functions arising from constrained optimization problems Two algorithms are given for the case when derivatives are available wherever they exist and two for the case when they are not available We take the view that although a simple steplength algorithm may be all that is required to meet convergence criteria for the overall algorithm, from the point of view of efficiency it is important that the step achieve as large a reduction in the function value as possible, given a certain limit on the effort to be expended The algorithms include the facility for varying this limit, producing anything from an algorithm requiring a single function evaluation to one doing an exact linear search They are based on univariate minimization algorithms which we present first These are normally at least quadratically convergent when derivatives are used and superlinearly convergent otherwise, regardless of whether or not the function is differentiable at the minimum
25 citations
TL;DR: Then-dimensional generalisation of this theorem, due to C. M. Miranda, is used for the construction of an error estimation procedure for approximate solutions of nonlinear systems of equations.
Abstract: A well known theorem by Bolzano states that, whenever a continuous real function changes sign in some interval, it must have a zero in this interval. Then-dimensional generalisation of this theorem, due to C. Miranda, is used for the construction of an error estimation procedure for approximate solutions of nonlinear systems of equations.
23 citations
TL;DR: A processor scheduling algorithm for increasing response times is described, and it is shown that it generates a processing which meets all deadlines on a one-processor system whenever this is possible.
Abstract: Der Prozessorzuteilungsalgorithmus beeinflust ganz entscheidend die Einhaltung von Zeitbedingungen einer harten Realzeit-Umgebung. Es wird eine Prozessorzuteilung nach aufsteigenden Antwortzeiten zugrundegelegt und gezeigt, das sie bei einem Einprozessorsystem immer eine zeitgerechte Verarbeitung liefert, falls dies uberhaupt moglich ist.
TL;DR: A method is presented for developing Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error with several new 2-nd, 3-rd, and 4-th order algorithms given.
Abstract: A method is presented for developing Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error. Several new 2-nd, 3-rd, and 4-th order algorithms are given. The computation per step of the new algorithms is identical to that of algorithms which provide only an estimate of the local truncation error. Numerical experimentation with the new algorithms shows that the estimated error compares very well with the true accumulated error. Further, the error is of the same order as that incurred using traditional Runge-Kutta algorithms.
TL;DR: A characterization of smoothing splines is derived which leads to procedures using either locally defined bases or small support bases, and a smoothing algorithm which avoids these difficulties by using local defined bases is presented.
Abstract: A characterization of smoothing splines is derived which leads to procedures using either locally defined bases or small support bases. Difficulties in trying to compute the latter for splines in tension are discussed. A smoothing algorithm which avoids these difficulties by using locally defined bases is presented.
TL;DR: The present paper contains a stability concept for discretization methods of a certain, very general classM, which is optimal (in the sense of yielding the best general, two-sided error bounds) without being more restrictive than any of the classical stability definitions.
Abstract: The present paper contains a stability concept for discretization methods of a certain, very general classM, which is optimal (in the sense of yielding the best general, two-sided error bounds) without being more restrictive than any of the classical stability definitions. The optimal stability functional Ψh related to it depends on the linear part of the discretization operator, and has the important property that Ψh [δ] may be of orderq+1, i.e. Ψh [δ] = O(hq+1), even if the local error δ only has orderq, δ = O(hq). This result may be used for the construction of methods with maximum order. Its application to linear cyclic methods, for example, furnishes a new approach to the theory of linearM-cyclick-step methods of maximum order.
TL;DR: A new type of numerical method for the solution of the nonlinear, weakly singular first kind Volterra equation is developed and a basic lemma concerning the boundedness of a special difference inequality is given.
Abstract: Starting from an existence and uniqueness theorem for a generalized nonsingular second kind Volterra equation existence and uniqueness for the solution of the nonlinear, weakly singular first kind Volterra equation is examined A new type of numerical method is developed A basic lemma concerning the boundedness of a special difference inequality is given and order two or three convergence of the method is shown Two numerical examples illustrate the theoretical results
TL;DR: A generalized nonlinear Volterra integral equation, whose kernel contains the unknown function at two different arguments, is solved by collocation with piecewise Hermite-polynomials by the method has order 2m,m ∈ ℕ, if polynomials of degree 2m−1 and appropriate integration formulas are used.
Abstract: Die Arbeit behandelt eine verallgemeinerte nichtlineare Volterrasche Integralgleichung zweiter Art, in deren Kern die unbekannte Funktion mit zwei verschiedenen Argumenten vorkommt. Die Gleichung wird durch einen Kollokationsansatz mit stuckweisen Hermite-Polynomen gelost. Bei Verwendung von Polynomen des Grades 2m−1,m ∈ ℕ, und von geeigneten Quadraturformeln hat das Verfahren die Ordnung 2m. Die Kollokationsstellen mussen im Einklang mit einer Stabilitatsbedingung gewahlt werden.
TL;DR: Two computer codes for calculating the sum of power series belonging to a certain general class are described and the use of the codes for an illustrative test example is explained.
Abstract: This paper describes two computer codes for calculating the sum of power series, belonging to a certain general class. The use of the codes for an illustrative test example is explained.
TL;DR: The problem of finding several eigenfunctions and eigenvalues of the interior Dirichlet problem for Laplace's equation on arbitrary bounded plane regions is considered with two fast algorithms combined: an iterative Block Lanczos method and a capacitance matrix method.
Abstract: The problem of finding several eigenfunctions and eigenvalues of the interior Dirichlet problem for Laplace's equation on arbitrary bounded plane regions is considered. Two fast algorithms are combined: an iterative Block Lanczos method and a capacitance matrix method. The capacitance matrix is generated and factored only once for a given problem. In each iteration of the Block Lanczos method, a discrete Helmholtz equation is solved twice on a rectangle at a cost of the order ofn
2 log2
n operations wheren is the number of mesh points across the rectangle in which the region is imbedded. Es wird uber das Auffinden mehrerer Eigenfunktionen und Eigenwerte des inneren Dirichlet-Problems fur die Laplace-Gleichung mit willkurlich begrenzten ebenen Gebieten berichtet. Zwei schnelle Algorithmen werden miteinander kombiniert. Eine iterative Block-Lanczos-Methode und eine Kapazitats-Matrizen-Methode. Die Kapazitats-Matrix wird berechnet und nur einmal fur ein gegebenes Problem faktorisiert. Bei jedem Iterationsschritt der Block-Lanczos-Methode wird eine diskrete Helmholtz-Gleichung zweimal auf einem Rechteck mit einer zu n2log2
n proportionalen Anzahl von Operationen gelost, wobein die Zahl der Netzpunkte zu dem Rechteck ist, in das das Gebiet eingebettet ist.
TL;DR: A new adaptive algorithm for the integration of analytic functions is presented, obtained by choosing the 6-point Gauß-Legendre integration rule, which is provided with an accurate and very efficient integration process.
Abstract: A new adaptive algorithm for the integration of analytic functions is presented. The algorithm processes the integration interval by generating local subintervals whose length is controlled through a feedback loop. Control is performed by means of a relation derived on an analytical basis and valid for an arbitrary integration rule: two different estimates of an integral are used to compute the interval length necessary to obtain an integral estimate with accuracy within the assigned error bounds. The implied method for local generation of subintervals and an effective assumption of error partition among subintervals give rise to an adaptive algorithm provided with an accurate and very efficient integration process.
TL;DR: A new iteration scheme giving bounds for the positive eigenvector of a nonnegative and irreducible matrix and the corresponding eigenvalue is developed and tested on some examples.
Abstract: Wir geben ein neues Iterationsverfahren an, das Schranken fur den positiven Eigenvektor einer nichtnegativen, irreduziblen Matrix und des zugehorigen Eigenwerts liefert.
TL;DR: For each α ε (0, π/2), the existence of A (α)-stable linear multistep methods with arbitrary order of consistency is shown by an explicit construction.
Abstract: Es wird in einer expliziten Konstruktion gezeigt, das fur jedes α e (0, π/2)A (α)-stabile lineare Mehrschrittverfahren beliebiger Konsistenzordnung existieren. Einige charakteristische Daten der Verfahren und numerische Rechnungen werden angegeben.
TL;DR: It is shown that for continuous functions the related quadratic splines converge without any assumption on the spline grid with a division ratio from 1 - 2 to 1 and the order of convergence is increased.
Abstract: In der Hauptsache wird gezeigt, das quadratische Interpolationssplines ohne zusatzliche Voraussetzungen an das Splinegitter gegen die erzeugende Funktion konvergieren, sofern diese stetig ist. Dabei kann das Interpolationsgitter mit einem zwischen\(1 - \frac{{\sqrt 2 }}{2}\) und\(\frac{{\sqrt 2 }}{2}\) gelegenen Teilungsverhaltnis aus dem Splinegitter erzeugt werden. Im Falle von stetig differenzierbaren Funktionen, bei denen das Teilungsverhaltnis sogar zwischen 0 und 1 gewahlt werden kann, erhoht sich die Konvergenzordnung der zugehorigen Splines und wird bei zweimaliger Differenzierbarkeit voll ausgeschopft. Analoge Resultate werden uber die Konvergenz von Flachenabgleichssplines gewonnen.
TL;DR: The result established in the paper is an extension of the theory given by G. Dahlquist and the recent result of K. Taubert and satisfies the Perron type condition appearing in the uniqueness theory for the Cauchy problem with a nondecreasing comparison function.
Abstract: The general form of a quasilinear nonstationaryk-step method for solving of the Cauchy problem for ordinary differential equations is discussed The convergence theorem is stated under rather weak conditions It is not assumed that the increment function is Lipschitz-continuous but only that it satisfies the Perron type condition appearing in the uniqueness theory for the Cauchy problem with a nondecreasing comparison function The result established in the paper is an extension of the theory given by G Dahlquist and the recent result of K Taubert
TL;DR: For two initial conditions the practical computations show, that the integral equations method yields useful results with relative small effort.
Abstract: Wir betrachten das Anfangs-Randwertproblem fur die Warmeleitungsgleichung im ℝ3 fur ein kompaktes Gebiet mit stetig gekrummtem Rand und geben dafur einen konstruktiven Existenzsatz mit dem Rotheverfahren, einer transversalen Linienmethode, an. Dabei wird die Zeitvariable diskretisiert und die zu jedem Zeitschritt auftretenden elliptischen Randwertprobleme mit einer Integralgleichungsmethode gelost. Die so gewonnenen Naherungslosungen konvergieren gegen die exakte Losung des ursprunglichen Problems. Am Beispiel der Kugel wird dieses Verfahren durchgefuhrt, wobei sich eine einfache Fehlerabschatzung fur die Naherungslosungen ergibt. Fur zwei Anfangstemperaturverteilungen zeigt die praktische Berechnung, das die Integralgleichungsmethode bei verhaltnismasig geringem Aufwand gute Ergebnisse liefert.
TL;DR: It can be shown that the class of context-sensitive graph languages is equivalent to the classOf context- free graph languages and the context-free graph languages properly include the regular graph languages.
Abstract: According to the classification of labelled graph grammars by Nagl [4], it can be shown that the class of context-sensitive graph languages is equivalent to the class of context-free graph languages and the context-free graph languages properly include the regular graph languages.
TL;DR: The theoretical framework of preplanning merges on disks is described, where a number of slots in a cylinder has to be reserved twice, while the first ofm strings is allocated on a disk in order to adjust blocks of strings created later.
Abstract: The theoretical framework of preplanning merges on disks is described. In order to read merge blocks of different strings from a cylinder of a disk in proper sequence their allocation has to be preplanned according to their sequence of exhaustion. This sequence can be predicted in a certain range using the lower and upper limits of the high keys of the merge blocks, which are derived for the general case of ther-way merge. Since this planning information is not definite for all of the merge blocks, a number of slots in a cylinder has to be reserved twice, while the first ofm strings is allocated on a disk in order to adjust blocks of strings created later. This gap size is determined in general for the worst case of preplanning.
TL;DR: The L1 approximation of strictly convex functions by means of first degree splines with a fixed number of knots with main theoretical results a system of equations for the knots, which solves the problem, and an estimate of the approximation error.
Abstract: The L1 approximation of strictly convex functions by means of first degree splines with a fixed number of knots is studied The main theoretical results are a system of equations for the knots, which solves the problem, and an estimate of the approximation error The error estimation allows the determination of bounds for the number of knots needed so that the L1 approximation error does not exceed a given number Finally, an algorithm is used, by means of which a solution to the system can be obtained
TL;DR: It is possible to reduce the Lipschitz constant of the contraction mapping by using the free parameter in the fixed-point equation of the parallel-chord-method to generate the contraction property of the mapping.
Abstract: Mit Hilfe des freien Parameters in der Fixpunktgleichung des Parallelenverfahrens ist es moglich, die Lipschitz-Konstante der kontrahierenden Abbildung zu verkleinern. Durch diese Verkleinerung kann in geeigneten Fallen eine kontrahierende Abbildung erzeugt werden. Ist die Abbildung bereits kontrahierend, ist sie auch kontrahierend fur den berechneten Wert des Parameters. Ferner ergibt sich neben einer verbesserten Fehlerabschatzung fur die Iterationsfolgen auch ein kleinerer Konvergenzfaktor. Das Verfahren wird an Hand eines Beispiels erlautert.
TL;DR: It is shown how a Fourier transform of the ordern=p·m can be reduced top Fourier transforms of orderm by performing essentiallym Fourier transformations of orderp on the data.
Abstract: An elementary and transparent representation of the fast Fourier transform is given. Instead of using the usual and highly algebraic approach it is shown how a Fourier transform of the ordern=p·m can be reduced top Fourier transforms of orderm by performing essentiallym Fourier transforms of orderp on the data. The resulting process is discussed in more detail forn=3q andn=5q. The problem of retrieval of the wanted coefficients from the final data is solved by a simple argument. The generalization for an ordern equal to a product of powers of prime numbers is immediate.
TL;DR: Weak coerciveness of the under lyigg bilinear form for the derivation of hybrid elements is proved by showing the existence of a stationary point.
Abstract: Dieser Beitrag liefert eine Theorie der Finite-Element-Approximation fur eine Klasse von Schalenproblemen. Obgleich diese nur fur den Fall der eingespannten flachen Schale mit regularer Triangulierung ausfuhrlich dargestellt wird, sind die Ergebnisse daruberhinaus von Interesse, da die abgeleiteten Konvergenzordnungen das Verhalten numerischer Naherungslosungen aus anderen Anwendungen der hybriden Finite-Element-Methode erklaren helfen. Die Konvergenzgeschwindigkeit kann nur verbessert werden durch gleichzeitige Erhohung der Ansatzgrade fur die Verschiebungen im Elementinnern und fur die Spannungen auf den Randern. Die Einhaltung einer sogenannten Rangbedingung hat zentrale Bedeutung. Die schwache Koerzitivitat der fur die Ableitung hybrider Elemente verwendeten Bilinearform ist durch den Nachweis der Existenz eines stationaren Werts gesichert.