Computations with effective real numbers
A real number x is said to be effective if there exists an algorithm which, given a required tolerance epsilon is an element of Z2 returns a binary approximation (x) over tilde is an element of Z2(Z) for x with vertical bar(x) over tilde - x vertical bar < epsilon. Effective real numbers are interesting in areas of numerical analysis where numerical instability is a major problem. One key problem with effective real numbers is to perform intermediate computations at the smallest precision which is sufficient to guarantee an exact end-result. In this paper we first review two classical techniques to achieve this: a priori error estimates and interval analysis. We next present two new techniques: "relaxed evaluations" reduce the amount of re-evaluations at larger precisions and "balanced error estimates" automatically provide good tolerances for intermediate computations, (c) 2005 Elsevier B.V. All rights reserved.
effective real number algorithm error estimates interval analysis