IMath 1985: Interval Mathematics 1985 pp 95-101

# Arbitrary accuracy with variable precision arithmetic

• Fritz Krückeberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 212)

## Abstract

For the calculation of interval-solutions Y including the true solution y of a given problem we need not only that y∈Y holds. Furthermore we are interested in the value of span(Y). So we should get that for an a priori and arbitrarily given bound ɛ>0 the calculation yields that the error remains below ɛ or that span(Y) < ɛ. It is possible to realize span(Y) < ɛ for arbitrary ɛ>0 by using an interval-arithmetic with variable word length within a three-layered methodology, including validation/verification of the solution. The three-layered methodology consists of
• •-Computer algebra procedures.

• •-the numerical algorithm.

• •-an interval arithmetic with variable and controllable word length.

Examples are given in the domain of linear equations and ordinary differential equations (initial value problems).

## Keywords

Word Length True Solution Interval Arithmetic Decimal Point Floating Point Arithmetic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. Demmel J W, Krückeberg F (1985) An Interval Algorithm for Solving of Linear Equations to Prespecified Accuracy. Computing 34: 117–129.Google Scholar
2. Krier N, Spelluci P (1975) Untersuchungen der Grenzgenauigkeit von Algorithmen zur Auflösung linearer Gleichungssysteme mit Fehlererfassung. In: Interval Mathematics, Lecture Notes in Computer Science 29: 288–297 ed. by Nickel K, Springer Verlag, Berlin Heidelberg, New York.Google Scholar
3. Krückeberg F, Leisen R (1985) Solving Initial Value Problems of Ordinary Differential Equations to Arbitrary Accuracy with Variable Precision Arithmetic. In: Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computation: 111–114 ed. by Wahlstrom B, Henriksen R, Sundby N P, Oslo, Norway.Google Scholar
4. Leisen R (1985) Zur Erzielung variabel vorgebbarer Fehlereinschließungen für gewöhnliche Differentialgleichungen mit Anfangswertmengen mittels dynamisch steuerbarer Arithmetik. Diplomarbeit, Universität Bonn.Google Scholar
5. Pascoletti K H (1982) Ein intervallanalytisches Iterationsverfahren zur Lösung von linearen Gleichungssystemen mit mehrparametriger Verfahrenssteuerung. Diplomarbeit, Universität Bonn.Google Scholar