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Section 22.1 Repeating Calculations

While calculating an approximate square root with Newton’s method, we often want to use the formula \(1/2((N/G) + G)\) multiple times to keep improving our approximation. Earlier, we defined a procedure to make it easier to reuse the formula. But if we want to apply the formula a whole bunch of times in a row, a loop makes sense. Try running this program with the Codelens:
Each time we pass through the loop, we update the guess and it becomes a little closer to the truest value for the square root we can represent (we can never write a perfect representation of the answer as a decimal, so we have to settle for being correct to a significant number of digits). The for loop is initially set to repeat 4 times, which gets us pretty close to the true value of the square root - 10.48808848170152.

Checkpoint 22.1.1.

Modify the loop to repeat more times. What is the minimum number of repetitions you need to get an answer that is as accurate as possible? (One that displays as 10.48808848170152)
The number of repetitions needed to get the most accurate possible square root for a given number is not always the same. If you change number to 2, it would only take 5 iterations of the loop to find the best possible guess for the square root. Without experimenting, we don’t really know how many repetitions are needed to get the best possible answer. Currently, our program says something like “repeat this calculation 5 times”. What we would like to say is something more like “repeat this calculation while it keeps producing a new value”.
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