A general, novel methodology for studying the generation of finite precision error in any algorithm

Abstract

In this work, a new general approach is introduced for the study of the generation, propagation and accumulation of the quantization error in any algorithm. This methodology employs a number of fundamental propositions demonstrating the way the four operations addition, multiplication, division and subtraction, influence quantization error generation and transmission. Using these, one can obtain knowledge of the exact number of erroneous digits with which all quantities of any algorithm are computed at each step of it. This methodology offers understanding of the actual cause of the generation and propagation of finite precision error in any computational scheme.