As in many branches of science, there has been a change of inner structure taking place in appl.ed Mathematics in recent times, with a marked turn towards abstraction, exemplified by the wide use of methods of functio...
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As in many branches of science, there has been a change of inner structure taking place in appl.ed Mathematics in recent times, with a marked turn towards abstraction, exemplified by the wide use of methods of functional analysis. In this summarizing lecture, an attempt is made to demonstrate the useful appl.cation of abstract notions in Numerical Mathematics, especially by means of the concept of partial order. This concept is used, for instance, in connection withRiesz' partially ordered Banachspaces, pseudometric spaces, intervals, monotone and positive operators, linear and nonlinear optimization. appl.cations of monotone operators are given for benting of beams, boundary value problems for linear and nonlinear elliptic differential equations, extrapolation for initial value problems and eigenvalue problems. Emphasis is laid on the interrelation between various problem areas, e.g. between boundary value problems and optimization problems.
When solving linear equations with elimination methods, pivotal strategies are used to improve numerical precision. In this paper some new pivotal strategies are suggested and compared with known strategies by means o...
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When solving linear equations with elimination methods, pivotal strategies are used to improve numerical precision. In this paper some new pivotal strategies are suggested and compared with known strategies by means of numerical experiments.
Linear econometric models are used for prediction purposes and simulation of economic policy. The paper investigates the stability behaviour of an econometric model with respect to disturbances of initial conditions o...
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Linear econometric models are used for prediction purposes and simulation of economic policy. The paper investigates the stability behaviour of an econometric model with respect to disturbances of initial conditions or changes in the disturbance *** the first part, some theoretical considerations and sufficient conditions for stability are developed, the second part consists of a destription of the macroeconomic model “AUSTRIA I” a research study of the Institute for Advanced Studies, Vienna, and its stability *** the Appendix one will find a modified Hessenberg-Algorithm to compute all eigenvalues of real matrices and the computer program in FORTRAN is added.
There exists a general principle for finite matrices, according to which to every matrix norm corresponds an inclusion theorem for eigenvalues. If the norm is the row-sum norm, we have theGershgorin theorem. Another i...
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There exists a general principle for finite matrices, according to which to every matrix norm corresponds an inclusion theorem for eigenvalues. If the norm is the row-sum norm, we have theGershgorin theorem. Another inclusion theorem is used for obtaining bounds for the deviations of the eigenvalues from the diagonal elements, involving the order of the matrix, the maximal modulus of the off-diagonal elements and the distances of the diagonal elements. These bounds yield estimates for theJacobi method for determination of eigenvalues of symmetric matrices.
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 dele...
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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.
If the cutting plane method is appl.ed formally to the computation of flows with minimal costs then repeatedly negative cycles have to be determined. Up to now this can be done only with algorithms which are relativel...
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If the cutting plane method is appl.ed formally to the computation of flows with minimal costs then repeatedly negative cycles have to be determined. Up to now this can be done only with algorithms which are relatively clumsy. A modification of the algorithm enables us to avoid the search of such cycles and to generate an effective algorithm. At the same time we obtain stimulations for other numerical procedures, e. g., it becomes apparent how to proceed, when the approximation of the (nonlinear) cost function is successively improved.
Direct methods for the solution of linear approximation problems tend to fail in practice because of numerical instabilities. Difficulties arise from an undesired accumulation of round-off errors. In [3] and in chapte...
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Direct methods for the solution of linear approximation problems tend to fail in practice because of numerical instabilities. Difficulties arise from an undesired accumulation of round-off errors. In [3] and in chapter 2 of this paper effective and inexpensive tests are developed to recognize numerical difficulties and stabilize numerical methods by iterative refinement and restart procedures, if necessary. On the ground of this, a stabilized version of a modified revised dual simplex algorithm can be developed capable of solving the approximation problems under consideration. The technical details of this method and a FORTRAN implementation are given in [4]. Numerical examples are discussed to illustrate that the developed method is superior to commonly used algorithms not only by a broader range of appl.cations and more stability, but also by considerably less storage requirement (for large problems), while the execution times are comparable or less.
In this paper quadrature procedures with partially preassigned nodes are described. They are constructed by integration of a general, linear, hermitian, interpolating operator. Those nodes, which are not preassigned, ...
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In this paper quadrature procedures with partially preassigned nodes are described. They are constructed by integration of a general, linear, hermitian, interpolating operator. Those nodes, which are not preassigned, and for the first free are fixed by the requirement, that all weights of first derivatives in these free nodes vanish in the resulting quadrature formulas. In order to get positive weights for the function-values additional conditions must be settled up. For one (ore two) preassigned node(s) the given sufficient positivity-conditions say, that the preassigned nodes are not situated in the interior of the interval of integration, e. g. for strongly monotone basic-functions which are needed in the construction of the interpolating operator. As special cases the quadratures of Radau and Lobatto are involved. Further quadratures having nodes outside the interval of integration and beeing very siutable for connected formulas for large intervals are contained, partially with unusual interpolating conditions. At last it is shown, how to obtain numerically the free nodes of quadratureswith preassigned nodes from thosewithout preassigned nodes (and vice versa) by making use of inbedding-methods.
For list structures as known from the programming languages IPL-V and LISP 1.5, an explicit finite representation by so-called context-free expressions is establiched. Effective procedures are obtained for the constru...
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For list structures as known from the programming languages IPL-V and LISP 1.5, an explicit finite representation by so-called context-free expressions is establiched. Effective procedures are obtained for the construction of the explicit representations and for their reduction according to the number of names involved.
A termination criterion by Nickel (Theorem 6, [1]) for guaranteeing the numerical convergence for locally stable and consistent algorithms is generalized. The assumption |xν+2,xν+1|≤L|xν+1,xν| (0≤L<1,L real c...
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A termination criterion by Nickel (Theorem 6, [1]) for guaranteeing the numerical convergence for locally stable and consistent algorithms is generalized. The assumption |xν+2,xν+1|≤L|xν+1,xν| (0≤L<1,L real constant) of the approximation sequence {xν} to the solution is replaced by the convergence of the progression\(\sum\limits_{v = 1}^\infty {|x,x_{v + 1} |/Q^v (0< Q< 1)} \). Therefore the theorem of this paper is appl.cable to a large number of numerical procedures, for which untill now no termination criterion has been known (for example: Rombergprocedure). In particular this weakening is important for the computation of approximation solutions for integral equations.
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