john gabriel's nth root algorithm

In numerical analysis, an N^th root algorithm is an algorithm for determining n-th roots, that is, solving the equation. Originally, this algorithm for the square root was developed by the Greek mathematian Heron of Alexandria. It has nothing to do with John Gabriel, and is certainly not named after him. See also N-th root algorithm

ⁿ√a = x

Let

s_0 = 1

. The iterative method is defined by

s_\left(i+1\right) = \frac{\frac{a}{s_i^\left(n-1\right)} + \left(n-1\right)s_i}{n}

(s_i)_{i\in\R}

converges against ⁿ√a. This method is a special case of Newton's method for the positive solution of

f(x) := x^n-a = 0

. Performance: This algorithm uses 'averaging' and works much faster than the shifting n^th-root algorithm.

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