Question

Q. I would like you to use a fixed point method to solve the positive real quadratic root of 4 by solving h(x) = x^4 − 4 = 0. The standard method manipulates h(x) = 0 into g(x) = x so that the iterative scheme becomes xn+1 = g(xn). The iterative scheme will converge to the required solution if the root is in the interval defined by |g '(x)| < 1

(i) We begin by adding x to both sides of the equation to form x = x^ 3− 4+ x such that g(x) = x ^3 + x − 4. Will the root be found? (ii) We now try adding −2x to both sides of the equation to form −2x = x ^4− 4 − 2x such that g(x) = −1/ 2 (x ^4− 2x − 4 ) Will the root be found? (iii) What is the smallest value k that will guarantee convergence if we add −kx to both sides of the equatio

Answer 1