Numerical Analysis

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

Class 12: Wednesday September 30

SUMMARY Analyzing Root Finding Algorithms: Bisection READING Burden & Faires, 47–54

General Root-Finding Algorithm

1. Plot the function, in order to get an initial guess for the root and to check for problems 2. Select an initial guess [or bracket ] 3. Iteratively refine your initial guess 4. Decide you are ”converged” 5. Stop

demobisect.m

There is another implementation of Bisection Algorithm in q:/mfiles/math370/demobise.m Modify this m-file to find the root of f(d) = 2552 − 30d^{2} + d^{3} How many steps does it take to converge? Using what initial bracket?

Analyzing Convergence of Bisection

Write down an expression for the size of |b_{n} −a_{n}| which depends on b−a and the n-th iterate (note: |b_{0} − a_{0}| = b − a)

Solve this formula for n.

Try and predict how many iterations it will take Bisection to find the zero of f(x) = log(x) − 5 + x on the interval [1,9] to 5 decimal places

Go to the computer and see how many iterations demobisect.m actually takes to converge. Explain.

Convergence Criteria

There are a number of different ways to consider that a method has “converged” There is convergence criteria on f(x) and convergence criteria on x

Question There is also relative convergence versus absolute convergence. Which do you think is the “best” method of assessessimg convergence?