What is the fastest way to compute the square-root of a real, positive number x to a desired number of correct digits? For example, in number theory, $\sqrt{x}$ for nonnegative integers $x$ can be approximated to desired accuracy as a convergent of the continued fraction of $\sqrt{x}$ if this is known. In calculus, $\sqrt{t}$ can be found using the bisection method. This method is slow. Now, $\sqrt{t}$ can be computed faster by applying Newtons' method to the polynomial $x^2-t$ for any initial guess $k$ not equal to zero. When I say "fastest" I mean "least amount of computations" :) I also learned about Taylor polynomials; maybe using a Taylor polynomial about some $x$ close to to where $\sqrt{x}$ is known can be a fast method?
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If you want square root of $A$ then
$$ x_{n+1} = x_{n}-\frac{x^2-A}{2x},$$
by Newton Raphson successive iterations... best results with least computation.
Narasimham
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Not necessarily. There are higher order methods that converge faster. – marty cohen Sep 29 '17 at 04:10
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OP clarifies that ..When I say "fastest" I mean "least amount of computations" – Narasimham Sep 29 '17 at 05:44
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You have to consider how fast the convergence is and the number of operations per iteration. – marty cohen Sep 29 '17 at 05:58
(Also, Newton's method proof relies on Taylor's formulas, so I guess the part on Taylor polynomials is contained in this)
– Junkyards Sep 28 '17 at 21:55