2.2

Question 1

What is a fixed point for a function?

Answer 1

A number p such that f(p) = p

Question 2

What is the relationship between roots and finding fixed points?

Answer 2

1. Given the root finding problem f(p) = 0, the fixed point problem is g(x) = x + f(x)
2. Given the fixed point problem g(p) = p, the root finding problem is f(x) = g(x) + x

Question 3

What are some equivalent fixed point problems to the root finding problem x^4 -x -10 = 0

Answer 3

g(x) = x^4 -10
       = (x + 10)^(1/4)
       = 10/(x^3-1)
       = ((10 + x)/x)^(1/3)
       = (x + 10)^(1/2)/x^2

just make p = g(x)

Question 4

What is a fixed point iteration?

Answer 4

The sequence {pn} from n=0 to inf where pn = g(p(n-1))

Question 5

How do you know which fixed point function to choose in order to maximize the rate of convergence?

Answer 5

Maximize the fixed point functions and the one with the lowest max will converge the quickest

Question 6

When g(x) has a _______ slope, it converges quicker

Answer 6

smaller

Question 7

The fixed point iteration has _______ order of convergence

Answer 7

linear. However, when k < 1/2, the fixed point iteration converges faster than the bisection algorithm