This program uses the famous method by Isaac Newton to iteratively find the square root of a number. It relies upon an inital guess which is then improved until the answer is accurate within 0.000001.

```
/**
* Computes square root of a number (Newton's Method)
* @author Bradley Carey
* @version 16-Oct-2009
*/
import java.util.Scanner;
public class NewtonsMethod
{
public static void main(String[] args)
{
Scanner myScanner = new Scanner(System.in);
boolean done = false;
double guess=0.0;
while (true)
{
done = false;
System.out.print("\nEnter the number to take the square root of: ");
String answer = myScanner.nextLine();
double x = Double.parseDouble(answer);
if (x<=0)
{
System.out.print("Error: only positive inputs are valid.\n");
return;
}
else
{
System.out.print("Enter a guess for the square root: ");
answer = myScanner.nextLine();
guess = Double.parseDouble(answer);
}
while (!done)
{
double newGuess = guess-(guess*guess-x)/(2*guess);
System.out.print("newGuess = " + newGuess + "\n");
if (Math.abs(guess-newGuess)<0.00001)
{
System.out.print("To within .000001, The square root of " + x + " is " + newGuess);
System.out.print("\nDo another? y/n ");
answer = myScanner.nextLine();
done=true;
if(answer.equals("n"))
{
System.out.print("Good Bye!!\n");
System.exit(0);
}
}
else
{
guess = newGuess;
}
}
}
}
}
```