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).