How to calculate the root of the nth degree?

Question

Delphi

Tell me how to calculate the n-th root with EPS accuracy using Newton's iterative formula and the number of iterations for which this accuracy is achieved.

Answer 1

So, we need to solve this problem.
x ^b = a, a > 0, or x ^b - a = 0
Newton's method says:
x _n+1 = x _n − f(x) / f′(x)
x _n+1 = x _n − (x _n ^b — a) / (b x _n ^b−1 ) = (x n ₋  a/(x _n ^b−1 ₎ )/ b Ideal hacks with fractional numbers (for example, get the order and take the initial guess 2 ^[ord/b] for negative order and 2 ^{[ord/b] + 1}
for non-negative.
[x] - truncation of a fractional number, with an integer b [ord/b] = ord div b. You can use Delphi's Frexp and Ldexp functions to get the order, play around with it, and assemble it back into a machine fractional, they are very fast.
End when |x _n+1 − x _n | less e. Since the convergence of the method is quadratic, we are almost guaranteed to get the accuracy we need.
Since we need two approximations, n and n + 1, to check the accuracy, we store them.
And the last. Believe me, beautiful lady, no one will solve your educational problem to the end.

Answer 2

Here is a true calculator

Answer 3