MATH 441/541 - Numerical Analysis
Third Meeting: Root Finding Algorithms
Thursday, September 6th, 2007
Root Finding (i.e. Solving Nonlinear Equations) in One Variable
•Introductory Thoughts:
1. How are root finding and solving nonlinear equations the same thing?
2. What are the challenges to doing this analytically?
•Section 2.1: The Bisection Method (a.k.a. the Binary Search Method)
1. The Problem:
–Given: Suppose you have a continuous function fdefined on [a, b] with f(a) and f(b) having
opposite signs.
–Some Conclusions: The Intermediate Value Theorem guarantees that fhas at least one
root on [a, b], call it p.
–The Big Question: How do you go about finding the root pgiven this information and the
formula for f?
2. The concept of “Bracketing the Root”
3. The Bisection Method Algorithm:
OBJECTIVE: Given the continuous function fon the interval [a, b] where f(a) and f(b) have
opposite signs, find a solution to f(x) = 0.
INPUT: endpoints aand b; tolerance TOL; maximum number of iterations N(and f).
OUTPUT: approximate solution por message of failure.
4. Other Possible Stopping Criteria
5. Convergence Issues:
–Will it converge?
–How fast will it converge?
–Example: The equation xcos(x) = 3 + 8x−x3has a solution on the interval [2,5]. How
many iterations will you need, at most, to find the solution with an accuracy of 10−4?
6. Strengths and Weaknesses of the Bisection Method:
•Section 2.3: Newton’s Method (Secant Method, and Method of False Position)
1. The Problem:
–Given: Suppose you have a continuous (and possibly continuously differentiable) function
fand some reasonable guesses as to a root of the function.
–The Big Question: How do you go about finding the root pgiven this information and the
formula for f?
2. The Secant Method:
(a) Given the continuous function f(x) and two reasonable guesses to the root, p0and p1.
(b) A Visual Explanation
(c) Deriving the Formula for the New Guess at the Root:
