Square Root


 

Complete Source Code

/* This program will approximate the square root of a given nonnegative real number This approximation can then be compared to the value which is computed by the built in function Math.sqrt(). The algorithm that is implemented here can be described as follows. First x represents the given number. If the square of x is greater than x, then we know that x is not the square root. So, we shall try the square of x/2 instead. If this value is also greater than x, then we shall try the square of x/4 next. If instead, this value is less than x, then we shall square the number which lies halfway between x/2 and x (i.e. 3x/4) and see if that's greater than x or not. By stepping back and forth in this fashion and squaring all the midpoints, we will obtain increasingly better approximations.

In order to implement this procedure, we need a variable s which stores the current approximation, a variable t which s can get arbitrarily close to with each successive midpoint trial, and a variable temp which keeps track of the previous approximation. This variable is needed in order to shift the direction of approach back and forth. The approximations are then carried out by two nested while loops which are meant to handle each direction in turn. */


public class SquareRoot {


   public static void main(String[] args) {

      double x, s, t, temp;

      System.out.println();

      while (true) {

         System.out.print("Please enter a nonnegative real number: ");
         x = TextIO.getlnDouble();

         if (x >= 0) break;

      }

      s = x;
      temp = 0;

      if (x > 0 && x < 1) s = 1/s;

      while (Math.abs(s*s-x) > 0.00000001) {

         t = temp;

         while (s*s > x) {

            temp = s;
            s = (s+t)/2;

         }

         t = temp;

         while (s*s < x) {

            temp = s;
            s = (s+t)/2;

         }

      }

      System.out.println("\nThe approximate square root of " + x + " is " + s + ".");
      System.out.println("The actual square root of " + x + " is " + Math.sqrt(x) + ".");

   }


}