T. J. Dekker

A floating-point technique for extending the available precision

A technique is described for expressing multilength floating-point arithmetic in terms of singlelength floating point arithmetic, i.e. the arithmetic for an available (say: single or double precision) floating-point number system. The basic algorithms are exact addition and multiplication of two singlelength floating-point numbers, delivering the result as a doublelength floating-point number. A straight-forward application of the technique yields a set of algorithms for doublelength arithmetic which are given as ALGOL 60 procedures.

Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function

Two algorithms are presented for finding a zero of a real continuous function defined on a given interval. The methods used are mixtures of linear interpolation, rational interpolation, and bisectmn. The asymptotic behavior of these algorithms is completely satisfactory. The munber of function evaluations needed to find a zero of a function is bounded by four or five times the number needed by bisection and is usually considerably smaller.

Rehabilitation of the Gauss-Jordan algorithm

SummaryIn this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.

Parallel algorithms for solving large linear systems

The solution of linear systems continues to play an important role in scientific computing. The problems to be solved often are of very large size, so that solving them requires large computer resources. To solve these problems, at least supercomputers with large shared memory or massive parallel computer systems with distributed memory are needed. This paper gives a survey of research on parallel implementation of various direct methods to solve dense linear systems. In particular are considered: Gaussian elimination, Gauss-Jordan elimination and a variant due to Huard (19791, and an algorithm due to Enright (19781, designed in relation to solving (stiff) ODES, such that stepsize and other method parameters can easily be varied. Some theoretical results are mentioned, including a new result on error analysis of Huard’s algorithm. Moreover, practical considerations and results of experiments on supercomputers and on a distributed-memory computer system are presented.

The shifted QR algorithm for Hermitian matrices

THE SHIFTED QR ALGORITHM FOR HERMITIAN MATRICES by T. J. Dekker Mathematical Centre, Amsterdam and J. F. Traub* University of Washington, Seattle Technical Report No. 70-11-07, Computer Science Group University of Washington Seattle, Washington 98105 November 1970 *Some of the material in this paper was presented by J. F. Traub in an invited talk at the Gatlinburg Symposium on Numerical Algebra, April 1969-

Stability of the Gauss-Huard algorithm with partial pivoting

This paper considers elimination methods to solve dense linear systems, in particular a variant of Gaussian elimination due to Huard [13]. This variant reduces the system to an equivalent diagonal system just like Gauss-Jordan elimination, but does not require more floating-point operations than Gaussian elimination. To preserve stability, a pivoting strategy using column interchanges, proposed by Hoffmann [10], is incorporated in the original algorithm. An error analysis is given showing that Huard’s elimination method is as stable as Gauss-Jordan elimination with the appropriate pivoting strategy. This result is proven in a similar way as the proof of stability for Gauss-Jordan elimination given in [4]. Numerical experiments are reported which verify the theoretical error analysis of the Gauss-Huard algorithm.ZusammenfassungWir betrachten Eliminationsverfahren zur Lösung linearer Gleichungssysteme mit voll besetzter Koeffizientenmatriz und zwar besonders eine von Huard eingeführte Variante des Gauss’schen Algorithmus [13]. Dabei wird das Gleichungssystem auf Diagonalform reduziert wie in der Gauss-Jordan Variante des Eliminationsverfahrens, aber es werden nicht mehr Operationen benötigt als im klassischen Algorithmus von Gauss. Um die Stabilität zu garantieren, wird eine von Hoffmann entwickelte Pivotstrategie verwendet [10]. Eine Fehleranalyse zeigt, daß der Gauss-Huard Algorithmus kombiniert mit dieser Pivotstrategie genau so stabil ist wie der Gauss-Jordan Algorithmus. Der Beweis ist analog zum Beweis für die Stabilität des Gauss-Jordan Algorithmus in [4]. Numerische Resultate bestätigen die theoretisch gefundene Fehleranalyse.

Program correctness and machine arithmetic

The purpose of this paper is to give some insight in the construction of correct programs, especially numerical software, and in proving their correctness. After a brief survey of general program correctness axioms, the paper deals with various sets of axioms for machine arithmetic and some of its desirable features including a proposed standard for binary floating-point arithmetic. Moreover, a brief discussion is devoted to interval arithmetic. Finally, as an example of proving correctness, some algorithms for finding a zero of a real function in a real interval are considered.