/** * Determines the number of iterations of Newton's method * required to approximate the square root of x within * the given bound. Newton's method starts out by * setting the initial approximate answer to x. * Then in each iteration, answer is replaced by * the quantity (answer + x / answer) / 2.0. * The process stops when the difference between * x and (answer * answer) is strictly less than * the given bound err. The method * returns the number of iterations required. * The given value x must be non-negative. *

* For example, given x = 10 the first three iterations * of Newton's method produce the approximate values * 5.5, 3.66, and 3.20. Those three values squared are * 30.29, 13.39, and 10.21, respectively. * Therefore countIterations(10, 1.0) * returns 3, since it takes 3 iterations to get the result 10.21 * that is within 1.0 of x. * On the other hand, countIterations(10, 200) returns 0, * since 10 * 10 = 100 is already within 200 units of x = 10. * @param x * value whose square root is to be approximated * @param err * given bound for the approximation * @return * number of iterations required to get an approximation * whose square is within the given bound of x */ public static int countIterations(double x, double err) { // TODO return 0; }