\(\newcommand{\bx}{\textbf{x}} \newcommand{\bo}{\textbf{0}} \newcommand{\bv}{\textbf{v}} \newcommand{\bu}{\textbf{u}} \newcommand{\bq}{\textbf{q}} \newcommand{\by}{\textbf{y}} \newcommand{\bb}{\textbf{b}} \newcommand{\ba}{\textbf{a}} \newcommand{\grad}{\boldsymbol{\nabla}} \newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pdd}[2]{\frac{\partial^2 #1}{\partial #2^2}} \newcommand{\pddm}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}} \newcommand{\deriv}[2]{\frac{d #1}{d #2}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Section3.1The Bisection Method

The following (partial) algorithm (known as the Bisection Method) uses the Intermediate Value Theorem to systematically approximate solutions to the algebraic equation \(f(x) = 0\).

Assume that \(f(x)\) is continuous on the interval \([a,b]\). To make approximations of the solutions to the equation \(f(x) = 0\), do the following:

  1. Check to see if \(f(a)\) and \(f(b)\) have opposite signs (why is this important?).

  2. Compute the midpoint \((a+b)/2\) and evaluate \(f(\frac{a+b}{2})\).

  3. Compare the signs of \(f(a)\), \(f(b)\), and \(f(\frac{a+b}{2})\). Replace one of the endpoints with \((a+b)/2\) … which one and why?

  4. Repeat steps 2 and 3

  5. Stop when \(a\) and \(b\) are close enough to each other