Choosing an internal floating-point representation for a binary computer with given word-length is influenced by two factors: the size of the range of admissible numbers and the precision of the respective floating-po...
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Choosing an internal floating-point representation for a binary computer with given word-length is influenced by two factors: the size of the range of admissible numbers and the precision of the respective floating-point arithmetic. In this paper “precision” is defined by a statistical model of rounding errors. According to this definition base 4 floating-point arithmetic on an average produces smaller rounding errors than all other floating-point arithmetics with a base 2k, provided that the ranges of numbers have equal size.
The basic equations of the theory of characteristics for quasilinear initial value problems of hyperbolic type are derived with the help of the matrix calculus, the numbern of the space-coordinates being arbitrary. On...
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The basic equations of the theory of characteristics for quasilinear initial value problems of hyperbolic type are derived with the help of the matrix calculus, the numbern of the space-coordinates being arbitrary. One obtains, hereafter, practically useful difference-methods in rectangular grids. The convergence and the numerical stability of these methods are tested by means of an example (two-dimensional non-linear pressure waves in gases).
Considered are lineark-step methods for the abstract Cauchy problem. For the consistency order of the approximations several uniform conditions are given.
Considered are lineark-step methods for the abstract Cauchy problem. For the consistency order of the approximations several uniform conditions are given.
In this paper the distribution functionF of round-off errors, arising in fixed point multiplication of two random numbers, is determined. The factors of the products are non-negative and have at mostN digits. Each pos...
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In this paper the distribution functionF of round-off errors, arising in fixed point multiplication of two random numbers, is determined. The factors of the products are non-negative and have at mostN digits. Each possible value of each factor is assumed to have the same probability. It is shown that the difference betweenF and a suitable ramp function approches 0 asN→∞.
Reducing the number of multiplications for the evaluation of a polynomial and its derivatives does not necessarily mean that one should expect a commensurate reduction of the total cost of computation. In this paper w...
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Reducing the number of multiplications for the evaluation of a polynomial and its derivatives does not necessarily mean that one should expect a commensurate reduction of the total cost of computation. In this paper we present a cost analysis for a family of algorithms, which computes all derivatives of a polynomial in 3n−2 multiplications or divisions. This represents an improvement over the classical methods, which require 1/2n(n+1) multiplications. The analysis, however, reveals the presence of a multiplications-divisions cost trade-off due to which the cost complexity remainsO(ξ2n3) for all algorithms irrespective of any reduction in the number of arithmetic operations.
For error estimates and convergence proofs frequently the norm of the inverse of a differential operator is needed. In this paper the norm of the inverse of a second order operator is bounded from above by a finite di...
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For error estimates and convergence proofs frequently the norm of the inverse of a differential operator is needed. In this paper the norm of the inverse of a second order operator is bounded from above by a finite difference method. At the same time a sufficient criterion for the existence of the inverse is given.
The numerical solution of a nonlinear operator equation arises the question of finding suitable initial solutions. For general classes of problems the method of “continuation” (“Einbettung”) gives a systematical m...
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The numerical solution of a nonlinear operator equation arises the question of finding suitable initial solutions. For general classes of problems the method of “continuation” (“Einbettung”) gives a systematical means to construct such starting *** given problemT(x)=θ is inbedded in a family of ProblemsT(s, x)=θ wheres∈[0,1]. These problems should have a simple solution fors=0; fors=1T(s, x)=θ must represent the original problem. The solution of the family has to depend continously on the parameters. Starting withs=0 one constructs succesively the solutions fors1
In this paper the mathematical structures occuring in rounded computations with complex numbers and intervals are studied. By means of the special representation of the complex numbers and the properties of their oper...
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In this paper the mathematical structures occuring in rounded computations with complex numbers and intervals are studied. By means of the special representation of the complex numbers and the properties of their operations one can show, that the same structures as in the real case occur again. Nevertheless we can prove formulae for the interval arithmetic wellknown for complex intervals ([1]) which are easy to calculate.
Systems of linear first-order recurrence relations, as well as higher-order scalar recurrence relations, are analyzed with respect to numerical stability. Examples of severe numerical instability are presented involvi...
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Systems of linear first-order recurrence relations, as well as higher-order scalar recurrence relations, are analyzed with respect to numerical stability. Examples of severe numerical instability are presented involving scalar first- and second-order recurrence relations. Devices for counteracting instability are indicated.
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