Consider the **[Euclidean algorithm][1] (EA)** as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining 

$$q_i = \left\lfloor \frac{r_i}{r_{i+1}} \right\rfloor\qquad (*)$$

$$r_{i+2} = r_i\bmod r_{i+1} $$

with $r_0 = a$, $r_1 = b$. The relative length $b/a$ is then given by the (finite or infinite) continued fraction 

$$\cfrac{1}{q_0 + \cfrac{1}{q_1 + \cfrac{1}{q_2 + \cfrac{1}{\ddots }}}} =:\ [ q_0, q_1, q_2, \ldots  ]^{-1}$$

A rather similar and somehow simpler algorithm is the following which I call **proto-Euclidean algorithm (PEA)**:

$$q_i = \left\lfloor \frac{r_0}{r_{i+1}} \right\rfloor $$

$$r_{i+2} = r_0\bmod r_{i+1} $$

The relative length $b/a$ is then given by the (finite or infinite) continued product

$$\frac{1}{q_0}(1- \frac{1}{q_1}(1- \frac{1}{q_2}(1-\cdots)))  =:\ \langle q_0, q_1, q_2, \ldots \rangle$$

[**Update:** The one and crucial difference between the two algorithms is the numerator in $(*)$ which represents the reference length against which the current "remainder" is measured: in EA it is adjusted in every step to the last "remainder", in PEA it is held fixed to $r_0$.]

For comparison’s sake, with $a=1071$, $b=462$ , the Euclidean algorithm yields 

$$[2, 3, 7]^{-1} = \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{7}}} = \frac{22}{51} $$

while the proto-Euclidean algorithm yields 

$$\langle2,7,25,51\rangle = \frac{1}{2}(1- \frac{1}{7}(1- \frac{1}{25}(1-\frac{1}{51}))) = \frac{22}{51} $$.

> Under which name is the proto-Euclidean algorithm
> known? Where is it investigated and compared to the Euclidean algorithm? Or is it just folklore?

I am especially interested in the following questions:

> - How fast does PEA converge compared to EA?

(Just a side note: the first approximations in the sample above are equal: $[2, 3] = \frac{7}{3} = \langle2,7\rangle^{-1} $).

One advantage of EA over PEA seems to be that it takes fewer steps, and smaller numbers are involved in the course of calculation, since the numerator in $(*)$ decreases.

> - Is PEA significantly less efficient than EA?


  [1]: http://en.wikipedia.org/wiki/Euclidean_algorithm