Question
Suppose we wish to develop an Iterative method to compute the square root of a given positive number nu, i.e., to solve the nonlinear equation f(x) = x2 - y = 0 given the value of y. Each of the functions g1 and g2 listed below gives a fixed-point problem that is equivalent to the equation f(x) = 0. For each of three functions, determine whether the corresponding fixed-point iteration scheme xk +1 = g1(xk) is locally convergent to y if y = 3. Explain your reasoning in each case. g1(x) = y + x - x2.
Explanation / Answer
The way to determine if it will be a fixed point is to take the derivative of the function. This gives you the convergence ratio. As with any ratio test, it is necessary that the absolute value of the ratio be less than or equal to 1 and sufficient if the ratio is less than 1. In this case, the derivative is 1 -2x. This equals 1 - 2 * sqrt(3) = -2.46410161513775. The absolute value of this is greater than 1, so there is no convergence. Thus, clearly, if y is less than 1 and x begins less than 1, it will converge (abs(1-2x) < 1.)