We present an efficient method for computing roots of mappings on ℝn in the case where the Jacobian has the rankn−1 at the root. For the accurate determination of such a rootx*∈ℝn an auxiliary system ofn equations in...
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We present an efficient method for computing roots of mappings on ℝn in the case where the Jacobian has the rankn−1 at the root. For the accurate determination of such a rootx*∈ℝn an auxiliary system ofn equations inn+1 variables is constructed which possesses (x*, 1) as a turning point. This turning point can be computed by direct methods. We use an adapted method which requires only the solution of (n+1)-dimensional systems of linear equations and the evaluation of one Jacobian and 5 function values per step. This techniques is successfully appl.ed to compute simple bifurcation points by means of a suitable system of nonlinear equations which has the properties mentioned above.
This paper is concerned with some basic notions of intervall arithmetic, particularly with the definitionsindependent intervals, dependent intervals, interdependent intervals, and with ideas of the extended interval a...
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This paper is concerned with some basic notions of intervall arithmetic, particularly with the definitionsindependent intervals, dependent intervals, interdependent intervals, and with ideas of the extended interval arithmetic, *** andKulisch, [1] and [2]. These notions will be investigated from a formal point of view and put into a logically satisfactory frame. We shall also demonstrate that the set of all intervals which aredependent on A and whose generating function is apoint function does not form a field, contrary to a theorem in [1]. Furthermore we shall consider two formal ambiguities resulting from a certainidentification as well as from aspecial form of representing rational interval functions. In this connection we shall also formulate several requirements that thederivative of an interval function should satisfy. In the appendix to the paper we shall propose a more precise and logically correct form of the simple and the extended interval arithmetic.
The construction of timetables is a long and arduous task with may possibilities for errors, and use of digital computers to do this work has been studied for some years. Some practical success already has been achiev...
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The construction of timetables is a long and arduous task with may possibilities for errors, and use of digital computers to do this work has been studied for some years. Some practical success already has been achieved. However, it is still difficult to obtain computer producted timetables which can beused in schools. Nevertheless a practical use of the computer for generating school timetables is possible today. This is shown by the present paper, giving details of a programm for scheduling problems and its appl.cation to the construction of a timetable for a german school. This timetable is in use during the academic year 1968–1969.
Generalized norms are used to define hypernormballs in linear spaces. It turns out that the intervals, intervalvectors, intervalmatrices and intervalfunctions of Interval Analysis are special hypernormballs. Then the ...
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Generalized norms are used to define hypernormballs in linear spaces. It turns out that the intervals, intervalvectors, intervalmatrices and intervalfunctions of Interval Analysis are special hypernormballs. Then the induced algebraic structure in the space of hypernormballs is investigated.
Least-squares-solutions are defined and appropriate iterative methods are derived for the numerical solution of under-determined systems of non-linear equations. Especially an heuristic derivation of a damped Newton-l...
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Least-squares-solutions are defined and appropriate iterative methods are derived for the numerical solution of under-determined systems of non-linear equations. Especially an heuristic derivation of a damped Newton-like method is given in which the multiplicative damping factors are considered as additional unknowns to be computed during the iterations in the least-squares sense. A semilocal convergency theorem is proved for a related modification.
For some types of linear homogeneous differential equations approximations in the uniform sense (instead of asymptotic expansions) for large values of the independent variable are derived. In this first part only solu...
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For some types of linear homogeneous differential equations approximations in the uniform sense (instead of asymptotic expansions) for large values of the independent variable are derived. In this first part only solutions with a decreasing exponential factor are considered. Exact statements concerning the deviation from the best approximation by polynomials inx−1 and good numerical approximations as to be used in setting up subroutines for digital computers are obtained. As an example serves the normal distribution of Gauss.
Assuming the existence of an isolated solution of the given boundary value problem, we show the convergence of the shooting method combined with iteration methods of regula-falsi-type. Nonlinear, Fréchet-differen...
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Assuming the existence of an isolated solution of the given boundary value problem, we show the convergence of the shooting method combined with iteration methods of regula-falsi-type. Nonlinear, Fréchet-differentiable boundary conditions are admissable. The efficiency of the method is demonstrated by several numerical examples.
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 g...
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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 appl.ed to the computation of flood waves in rivers with great success.
In this second part there is given a description of the macro-economic model “Austria I”, a research study of the Institute for Advanced Studies, Vienna, and its stability behaviour. In the appendix one will find a ...
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In this second part there is given a description of the macro-economic model “Austria I”, a research study of the Institute for Advanced Studies, Vienna, and its stability behaviour. In the appendix one will find a modified Hessenberg-algorithm to compute all eigenvalues of real matrices and the computer program in FORTRAN is added.
The problem considered is to partition an edge-valued, node-weighted, finite, connected, simple graph (called a network) into disjoint subgraphs, each of which has a total node weight that does not exceed a weight con...
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The problem considered is to partition an edge-valued, node-weighted, finite, connected, simple graph (called a network) into disjoint subgraphs, each of which has a total node weight that does not exceed a weight constraint, and so that the total value of edges interconnecting the subgraphs is minimized. This paper presents a new approach which is effective in solving the problem, and fast enough to be practical in finding optimal partitions of large networks. It uses the concept of “divide and conquer” to partition the problem into several subproblems, which can be efficiently solved one after the other by using depth-first search and branch-and-bound principle. The operations required under the algorithms are additions, comparisons, and logical operations on binary vectors.
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