Chapter 10.1, Problem 80E

Fixed-point iteration A method for estimating a solution to the equation x = f(x), known as fixed-point iteration, is based on the following recurrence relation. Let x₀ = c and x_n+1 = f(x_n), for n = 1, 2, 3, ... and a real number c. If the sequence ${\left\{x_n\right\}}_{n=0}^{\infty }$ converges to L, then L is a solution to the equation x = f(x) and L is called a fixed point of f. To estimate L with p digits of accuracy to the right of the decimal point, we can compute the terms of the sequence ${\left\{x_n\right\}}_{n=0}^{\infty }$ until two successive values agree to p digits of accuracy. Use fixed-point iteration to find a solution to the following equations with p = 3 digits of accuracy using the given value of x₀.

80. $x=\frac{\sqrt{x^3+1}}{20};x_0=5$