Mark
Suppose I’m really good at computing decimal approximations to the logarithm but not necessarily so good at computing the exponential. Well, I can use the fact that
z=\log(w) \implies w = e^z
together with Newton’s method to help me to help me compute the exponential in terms of the logarithm. For example, I can compute the decimal approximation to e^2 by solving \log(w)-2=0 using Newton’s method.
- Write down the Newton’s method iteration function you’d use to estimate e^2.
- Perform two Newton steps from w_0=1.0.
- Explain why this technique can’t work for e^{5i}.
- Choose an appropriate branch of the logarithm so that we can compute e^{5i}.