Arbitrary accuracy with variable precision arithmetic

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).KeywordsWord LengthTrue SolutionInterval ArithmeticDecimal PointFloating Point ArithmeticThese 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.