8.6. Newton’s Method¶
Loops are often used in programs that compute numerical results by starting with an approximate answer and iteratively improving it.
For example, one way of computing square roots is Newton’s method. Suppose
that you want to know the square root of
n. If you start with almost any
approximation, you can compute a better approximation with the following
better = 1/2 * (approx + n/approx)
Execute this algorithm a few times using your calculator. Can you see why each iteration brings your estimate a little closer? One of the amazing properties of this particular algorithm is how quickly it converges to an accurate answer.
The following implementation of Newton’s method requires two parameters. The first is the value whose square root will be approximated. The second is the number of times to iterate the calculation yielding a better result.
Modify the program …
All three of the calls to
newtonSqrt in the previous example produce the correct square root for the first parameter. However, were 10 iterations required to get the correct answer? Experiment with different values for the number of repetitions (the 10 on lines 8, 9, and 10). For each of these calls, find the smallest value for the number of repetitions that will produce the correct result.
Repeating more than the required number of times is a waste of computing resources. So definite iteration is not a good solution to this problem.
In general, Newton’s algorithm will eventually reach a point where the new approximation is no better than the previous. At that point, we could simply stop. In other words, by repeatedly applying this formula until the better approximation gets close enough to the previous one, we can write a function for computing the square root that uses the number of iterations necessary and no more.
This implementation, shown in codelens,
while condition to execute until the approximation is no longer changing. Each time through the loop we compute a “better” approximation using the formula described earlier. As long as the “better” is different, we try again. Step through the program and watch the approximations get closer and closer.
while statement shown above uses comparison of two floating point numbers in the condition. Since floating point numbers are themselves approximation of real numbers in mathematics, it is often
better to compare for a result that is within some small threshold of the value you are looking for.