UCB Math 128A, Spring 2009: Programming Assignment 1
Due February 18
In this assignment, we will address two issues with the Bisection method and Newton’s method:
•Finding an interval [a,b] for the Bisection method, with f(a) and f(b) having different signs.
•Combining the excellent convergence properties of Newton’s method with the guaranteed
root-finding of the Bisection method.
1. Implement a MATLAB function findbracket of the form
function [a,b]=findbracket(f,x0)
which finds an interval [a, b] around x0such that f(a) and f(b) have different signs. Use the
following strategy:
1. Start with a=b=x0, and dx = 0.001
2. Subtract dx from a, and terminate if f(a)f(b)<0
3. Add dx to b, and terminate if f(a)f(b)<0
4. Multiply dx by 2 and repeat from step 2.
2. Implement a MATLAB function newtonbisection of the form
function p=newtonbisection(f,df,a,b,tol)
combining Newton’s method and the Bisection method according to the following strategy:
1. Start with p=a
2. Attempt a Newton step p=p−f(p)/f 0(p)
3. If pis outside of [a, b], set p= (a+b)/2
4. If f(p)f(b)<0, set a=p, otherwise set b=p
5. Terminate if |f(p)|<tol
6. Repeat from step 2.
Use the functions newton and bisection on the course web page as a starting point, this
function will be like a combination of the two.
3. Run your function newtonbisection using f(x) = sin x−e−xon the interval [1.9,30]:
f=@(x) sin(x)-exp(-x);
df=@(x) cos(x)+exp(-x);
x=newtonbisection(f,df,1.9,30,1e-8);
Present the result in a table showing for each iteration the method used (Newton or Bisect),
a,b,p, and f(p).
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