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Consider the Euclidean algorithm (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: $[2, 3] 3]^{-1} = \frac{7}{3} frac{3}{7} = \langle2,7\rangle^{-1} langle2,7\rangle$).

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?
4 added 283 characters in body

Consider the Euclidean algorithm (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 (inverse of the) relative length $b/a$ is then given by the (finite or infinite) continued fraction

$$q_0 \cfrac{1}{q_0 + \cfrac{1}{q_1 + \cfrac{1}{q_2 + \cfrac{1}{\ddots }}} }}} =:\ [ q_0, q_1, q_2, \ldots ]$$^{-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} $$. Under which name is the proto-Euclidean algorithm known? Where is it investigated and compared to the Euclidean algorithm? Or is it just folklore? I am especially interested in the following questions: • How fast does PEA converge compared to EA? (Just a side note: the first approximations in the sample above are equal: [2, 3] = \frac{7}{3} = \langle2,7\rangle^{-1} ). One advantage of EA over PEA seems to be that it takes fewer steps, and smaller numbers are involved in the course of calculation, since the numerator in (*) decreases. • Is PEA significantly less efficient than EA? 3 added 16 characters in body Consider the Euclidean algorithm (EA) as a way to measure the relative length b/a of a shorter stick b compared to a longer one a by recursively determining$$q_i = \left\lfloor \frac{r_i}{r_{i+1}} \right\rfloor\qquad (*)r_{i+2} = r_i\bmod r_{i+1} $$with r_0 = a, r_1 = b. The (inverse of the) relative length b/a is then given by the (finite or infinite) continued fraction$$q_0 + \cfrac{1}{q_1 + \cfrac{1}{q_2 + \cfrac{1}{\ddots }}} =:\ [ q_0, q_1, q_2, \ldots ]$$A rather similar and somehow simpler algorithm is the following which I call proto-Euclidean algorithm (PEA):$$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$$For comparison’s sake, with a=1071, b=462 , the Euclidean algorithm yields$$[2, 3, 7] 7]^{-1} = 2 \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{7}} cfrac{1}{7}}} = \frac{51}{22} 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?
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