a program that repeatedly asks the user whether they wish to calculate another square root. If the response is "y", then the program should proceed; if it is anything else, then the program should quit. Whenever it proceeds, the program should prompt the user for a number (a positive double, and your program may simply assume the input is consistent with this requirement) and then report the square root of that number to within a relative error of no more than
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The intuitive idea of Newton iteration for computing square roots is fairly straightforward. Suppose you have a guess r for x1/2 that is too large; the argument is similar if it is too small. If r is too large to be the square root of x, then x/r must be too small, so the average of r and x/r should be a better guess than either r or x/r. This suggests that if you repeatedly replace your current guess r by (r + x/r)/2, then your sequence of guesses should converge to x1/2. And indeed it can be proved that it does. A good initial guess for x1/2 is simply r = x. If you continue updating r until |r2 – x |/x < ε2, then the relative error of the guess r will be less than ε.
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