Numerical Analysis

Math 370 Fall 1998  MWF 11:30am - 12:25pm
\begin{matrix}c \\ ◯ \end{matrix}1998 Ron Buckmire  Fowler 127

Class 11: Monday September 28

SUMMARY Introduction to Root Finding READING Burden & Faires, 47–54

Example Consider a ball constructed of wood which has a density of ρ = 0.638 grams per cubic cm and the radius is r = 10 cm. How much of the ball will be submerged when it is in water (with unit density)?

M_{w} = Mass of water displaced = ∫_{0}^{d} π(r^{2} − (x − r)^{2})dx

M_{b} = Mass of ball = 4πr^{3}ρ/3 What’s the equation which must be solved to find d, the distance below the surface the ball will float? (Produce an equation for d of the form f(d) = 0 with d being the only letter present.)

\underline{Question} How would you solve this equation for d?

Root-Finding

We will be looking at algorithms for the solution of equations of one variable, i.e. equations of the form f(x) = 0. This is often referred to as finding the roots of the equation f(x) = 0 or finding the zeroes of the function f(x).

Bracketing The Root

How do we know where the roots of a function f(x) are? How can we “bracket” a zero of f(x)?

\underline{Groupwork} The Matlab function brackplo will do this for us. Go to the computers and run brackplo on the function you need to find zeroes of to find d. I have made a function called sphere.m which you can use to help you. What do you see? How many roots are there? What range did you ask brackplo to search on?

The Bisection Method of Bolzano

The bisection algorithm produces a sequence of approximations {p_{n}} to the zero of the function f(x)

where p_{n} = a_{n} + \frac{b_{n} − a_{n}}{2} = \frac{a_{n} + b_{n}}{2} and the n-th bracket is described by [a_{n}, b_{n}] Write down the Bisection Algorithm in pseudocode here:

bisect.m

In the NMM Toolbox, we have an implementation of the bisection algorithm in bisect.m. Use Matlab to find the value of d which we have been looking for which tells us how much of the pine sphere is submerged.

d =