1. In using Newton's method to find a root of an equation f(x) = 0, with initial guess x0 = 3, we compute 1(x0) = 1, and find that x1 = 5. What is fl(x0)?
2. Let a be a given number and suppose that 0 < e < 1 is also a given number, Show that the equation a = x - e sin x has a unique solution and it can be found by fixed point iteration of an appropriate function with any starting value.
3. The following formula is used to compute a function f:
f (x) = N/ - ¦/r-2
Suggest a more accurate way to compute the same function.
4. The reciprocal of a number R can be computed by the following iteration (no divisions!):
Xn+1 = ZIP - XnR).
This is Newton's method applied to find the zero of a certain function f. What is f?2k-1-1 = Xk 2Xle + 2
n
"le - uj4: z
2
= -Xk + 42k - 2
a) What is p(x)? What are its roots?
1)) One of the iterations will converge to either of the two roots of p(x) if the initial guess is sufficiently close. Each of the others will converge to one root, but can't possibly converge to the other unless the initial guess is exactly equal to the root. Which is which? Give a justification for your answer.
