Mathematics 171 – Numerical Analysis 1 Discussion 2 – Newton-Raphson Iteration September 17, 1998

Background

Last time at the end of class we introduced a second root-finding method called the Newton-Raphson iteration method. The idea was this: To generate approximate solutions of an equation f(x), starting from some initial approximation x_{0} (usually chosen “close” to the desired root), we can generate a sequence x_{n}, n ≥ 1 of better and better approximations (we hope!) by computing:

x_{n} = x_{n−1} − \frac{f(x_{n−1})}{f′(x_{n−1})}

for as many n ≥ 1 as we like. f′(x) is the ordinary derivative of f(x); the formula above will fail if f′(x_{n−1}) = 0 for some n.

Discussion Questions

A) Apply Newton-Raphson iteration to try to determine approximate values of the roots of the following equations in the indicated interval. In parts 1,2,3, for x_{0} you can take the midpoint of the interval. Starting from x_{0}, compute x_{1}, x_{2}, x_{3}, x_{4}, x_{5}:

1) x^{3} − 2x^{2} − 5 = 0, root in [1, 4].

2) x − 0.8 − 0.2 sin(x) = 0, root in [0, π/2].

3) \frac{1}{2} + \frac{x^{2}}{4} − x sin(x) − \frac{cos(2x)}{2} = 0, root in [0, 3.6].

4) x^{3} − \frac{x}{2} = 0, starting from x_{0} = \sqrt{.1} Discuss your results. Is there a difference between what happens in 3 and 4 and what happens in 1 and 2?

B) In most root-finding situations, of course, we do not actually know the value of the root we are looking for. Suppose we wanted to determine an approximation to a root that was (at least “probably”) good to a given tolerance, that is:

|exact root - approximate root| < tolerance

Suggest “stopping condition(s)” we could use to decide when to terminate the Newton-Raphson iteration. Think about part 4 of A again. How could you keep from repeating the computation of new x_{n}’s for ever?

C) Look at the Maple code for Newton-Raphson from the handout given out in class on Tuesday, September 15. What “stopping conditions” are used there? How does this relate to what you said in B?

Assignment

Group write-ups for questions A, B, and C due in class Tuesday, September 22.