Читать книгу Fundamentals of Numerical Mathematics for Physicists and Engineers - Alvaro Meseguer - Страница 14

Suppose we want to locate the zeros or roots of a given function that is continuous within the interval . Mathematically, the main goal of exact root‐finding of in is as follows:

Root‐finding (Exact): Find such that .

However, computers work with finite precision and the condition cannot be exactly satisfied, in general. Therefore, we need to reformulate our problem:

Root‐finding (Approximate): For a given , find such that for some .

This reformulation introduces a new component in the problem: the positive constant , usually termed as tolerance, whose meaning is outlined in the plot on the right. Since the root condition cannot be satisfied exactly, we must provide an interval containing the root . In the figure on the right, the root lies within the interval .