Embedding universal computer arithmetic in higher programming languages

In numerical computations mainly real and complex numbers, intervals as well as matrices and vectors with such components occur. It is well known that the arithmetic operations with real numbers, complex numbers etc. can be carried over to real floating-point numbers, complex floating-point numbers etc. using roundings. This proceeding results in agreeable arithmetic-, order- and compatibility-properties for an abundance of numerical data types and the accompanying arithmetic operations. Most programming languages however only provide real floating-point numbers; all the other data types and operations have to be simulated, e. g. in the form of arrays and procedure calls, which often causes loss of accuracy and arithmetic properties. Furthermore the complicate notation makes programs difficult to read. Therefore in this article an extension of PASCAL is presented which serves as an example for the way these numerical data types can be embedded into the syntax of a programming language.     

Numerische Lösung gewöhnlicher Differentialgleichungen mit Splinefunktionen

In this paper a general procedure to obtain spline approximations for the solutions of initial value problems for ordinary differential equations is presented. Several well-known spline approximation methods are included as special cases. It is common practice to partition the interval for which the initial value problem is defined into equidistant subintervals and to construct successively the spline approximation; thereby the spline function has to satisfy certain conditions at the knots. In the general procedure presented here additional knots are admitted in every subinterval. At these points which need not be equally spaced the spline approximation has to fulfill analogous conditions as at the original knots. Convergence and divergence theorems are proved; especially the influence of the additional knots on convergence and divergence of the method is investigated.     

Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region

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.     

Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter

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.     

The computer generation of poisson random variables

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.     

Solving linear systems by methods based on a probabilistic interpretation

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.     

On real factorizations of symmetric circulant sparse matrices

For the matricesA mentioned in the headline we determine the limit points up to which there is possible a real factorization of the formA=QQ ^T . HereQ=(q _ij ) is a circulant matrix, where from the elementsq _ij andq _ji withi≠j always one element is vanishing.     

Metrological reliability in measurement

The presentation of the main results attained today in the field of metrological reliability must be preceded by isolating some principal problems. The first concerns the reliability of measurements and the reliability of measuring instruments, while the second concerns the definition of the metrological failure. The third problem is to do with the classification of characteristics of the metrological reliability.The paper will discuss that part of the metrological reliability which deals with the hardware, that is, with instrumental reliability. Basic models used in metrological reliability will be evaluated. A relevant aim of the paper is to describe the present state and to express some new ideas about the future development of metrological reliability.