Sensitivities for guiding refinement in arbitrary-precision arithmetic

Abstract

Programmers often develop and analyze numerical algorithms assuming that they operate on real numbers, but implementations generally use floating-point approximations. Arbitrary precision arithmetic enables developers to write programs that operate over reals: given an output error bound, the program will produce a result within that bound. A key drawback of arbitrary-precision arithmetic is its speed. Fast implementations of arbitrary-precision arithmetic use interval arithmetic (which provides a lower and upper bound for all variables and expressions in a computation) computed at successively higher precisions until the result is within the error bound. Current approaches refine computations at precisions that increase uniformly across the computation rather than changing precisions per-variable or per-operator. This thesis proposes a novel definition and implementation of derivatives through interval code that I use to create a sensitivity analysis. I present and analyze the critical path algorithm, which uses sensitivities to guide precision refinements in the computation. Finally, I evaluate this approach empirically on sample programs and demonstrate its effectiveness..