Newton's iteration is an algorithm for computing the square root of a number via the recurrence equation
(1)
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where . This recurrence converges quadratically as .
Newton's iteration is simply an application of Newton's method for solving the equation
(2)
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For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....
The first few approximants , , ... to are given by
(3)
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These can be given by the analytic formula
(4)
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(5)
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These can be derived by noting that the recurrence can be written as
(6)
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which has the clever closed-form solution
(7)
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Solving for then gives the solution derived above.
The following table summarizes the first few convergents for small positive integer