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Is Newton's iterative method for finding a square root more efficient then the Babylonian method? Considering most roots are irrational, which method would get me within, say 16 decimal places, the fastest? I need a high level of precision so I was curious which would be more efficient.

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The Newton-Raphson process for the case of the square root is in effect the same algorithm as the Babylonian Method, aka Heron's Method.

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There are many clever tricks for computing square roots quickly. This is especially true if you really want the reciprocal of a square root, or if you want a square root of the form $\sqrt{ a^2 + b^2}$. See this page, for starters.

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If speed and robustness is key, check Brent's method

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