(41C) and (42S) Arithmetic-Geometric Mean

[Albert Chan]

Originally, Werner's code return converged AM sequence.

(06-09-2020 01:14 PM)Albert Chan Wrote:  If we assume non-zero AGM arguments, AM sequence always converge.

We can shown GM sequence also converge. (even with rounding errors)

Assuming it does not, but alternate between lo, hi.
In other words, GM sequence = GM₁, GM₂, ..., lo, hi, lo, hi, ...

But, √(lo * hi) = √(hi * lo)

Thus, we have only 2 possibilities:
1). GM₁, GM₂, ..., lo, hi, lo, lo, ...
2). GM₁, GM₂, ..., lo, hi, hi, hi, ...

→ assumption were wrong, GM *will* converge.

Bonus: GM convergence does not require non-zero AGM arguments.

→ AGM(x, 0) = AGM(0, x) = 0