Problem 19      Due 3/21/01

In this problem you will examine the behavior
of root solving algorithms for functions which
have double roots.  Such functions satisfy the
condition f(x*)=f'(x*)=0, while f"(x*) is not
zero.

        For the first part, determine the rate
of convergence of Newton's method for functions
of this type.  Do this analytically.
(Hint: keep an extra term in the Taylor series,
and expand f'(xi) about x* as well)

        For the second part, use the method of
bisection, Newton's method, and the secant method
to find the repeated root of x*(x-2)^2 and
x*(x-2)^3.  The latter has a triple root.  Determine
the rate of convergence r and the constant C for
all three methods for both functions.  Regression
works well for this purpose.
(Hint: you can't use bisection on the first 
function.  Why?)

        You may adapt the matlab example program.