Brigham Young University - Idaho



Brigham Young University - Idaho

College of Physical Sciences and Engineering

Department of Mechanical Engineering

Class Prep Notes #5

Root Finding and Iterative Solutions

Finding the roots of an equation is a common engineering application. Many strategies exist for doing this. We will focus on three techniques in this class. The three techniques are:

Plotting

Bisection Method

Excel’s Goal Seek

Plotting

Creating a graph is a relatively quick and easy process to approximate the roots of an equation. It is also commonly used as a starting process to an iterative method. Once the graph is created, the scale may easily be refined to zoom the plot in on the area of interest.

Bisection Method

You may note when using a graph to locate roots that f(x) changed signs on opposite sides of the root. In general, if f(x) is real and continuous in the interval from f(xlow) to f(xhi) and

f(xlow)*f(xhi) < 0

then there is at least one real root between xlow and xhi.

The bisection method is one type of incremental search method in which the interval is always divided in half. If a function changes sign over an interval, the function value at the midpoint of the interval is evaluated. The location of the root is then determined to lie within the subinterval in which the sin change occurs. The process is repeated by refining or halving the estimates.

A simple algorithm for the bisection calculation follows:

1. Choose initial guesses, xlow and xhi, such that the function changes sign over the interval. This can be checked by ensuring that f(xlow)*f(xhi) ................
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