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Briefly, a continued logarithm is an arbitrarily long bit string approximating a real number arbitrarily well, and supports arbitrarily precise bit-at-a-time algorithms for rational functions of these numbers. (See http://www.tweedledum.com/rwg/cfup.htm , p 47+)

A six-bits-per-term $\pi$ series:

Product[MatrixForm@{{512 k^3, 0}, {32 k^3 (-37 + 42 k), (-1 + 2 k)^3}}, {k, ?}] == MatrixForm[{{oo ?, 0}, {oo, "?"}}]

Where ? means matrix product, not Mathematica product, and oo is some quantity that blows up with the number of product terms, the same way you compute continued fractions with 2*2 matrices. An incorrect form of this series is derived in https://dspace.mit.edu/handle/1721.1/6088 . We initialize the work matrix m to the first term of the matrix product:

MatrixForm[ m = {{512 k^3, 0}, {32 k^3 (-37 + 42 k), (-1 + 2 k)^3}} /. k -> 1]

This represents the function

Divide @@ (m.{t, 1})

where t is the tail of the series, starting with k=2 rather than 1. It should be easy to show that, in general,

3/8/k < t < 3/8/(k + 1)

giving bounds

m /. {{t -> 3/8/1.}, {t -> 3/8/2.}}

{3.14754, 3.09677}

Since these both exceed 2, we commence output with

Style[cl@? = {1}, 22]

and left multiply m by the divide-by-2 matrix:

MatrixForm[m = {{1, 0}, {0, 2}}.m]

It costs almost nothing to remove the common power of 2:

MatrixForm[m = m/2]

representing the function

Divide @@ (m.{t, 1})

We still are on input term k=1, and can still use

% /. {{t -> 3/8/1.}, {t -> 3/8/2.}}

{1.57377, 1.54839}

which is smack between 1 and 2, which we celebrate with

Style[AppendTo[cl@?, 0], 22]

and left multiply m by the subtract-1-and-reciprocate matrix:

(m = {{0, 1}, {1, -1}}.m) // MatrixForm

representing

Divide @@ (m.{t, 1})

Again using

% /. {{t -> 3/8/1.}, {t -> 3/8/2.}}

{1.74286, 1.82353}

which dictates another

Style[AppendTo[cl@?, 0], 22]

and

(m = {{0, 1}, {1, -1}}.m) // MatrixForm

representing

Divide @@ (m.{t, 1})

% /. {{t -> 3/8/1.}, {t -> 3/8/2.}}

{1.34615, 1.21429}

Style[AppendTo[cl@?, 0], 22]

(m = {{0, 1}, {1, -1}}.m) // MatrixForm

Divide @@ (m.{t, 1})

% /. {{t -> 3/8/1.}, {t -> 3/8/2.}}

{2.88889, 4.66667}

This unambiguously exceeds 2, so

Style[AppendTo[cl@?, 1], 22]

MatrixForm[m = {{1, 0}, {0, 2}}.m]

Divide @@ (m.{t, 1})

% /. {{t -> 3/8/1.}, {t -> 3/8/2.}}

{1.44444, 2.33333}

Our interval of uncertainty contains 2! At last we gobble (right multiply) the k=2 term.

MatrixForm[ m = m.{{512 k^3, 0}, {32 k^3 (-37 + 42 k), (-1 + 2 k)^3}} /. k -> 2]

Divide @@ (m.{t, 1})

(Remembering to bump k !)

% /. {{t -> 3/8/2.}, {t -> 3/8/3.}}

{1.51515, 1.51938}

(Giving six more bits of precision!)

Style[AppendTo[cl@?, 0], 22]

(m = {{0, 1}, {1, -1}}.m) // MatrixForm

Divide @@ (m.{t, 1})

% /. {{t -> 3/8/2.}, {t -> 3/8/3.}}

{1.9412, 1.92538}

Style[AppendTo[cl@?, 0], 22]

(m = {{0, 1}, {1, -1}}.m) // MatrixForm

Divide @@ (m.{t, 1})

% /. {{t -> 3/8/2.}, {t -> 3/8/3.}}

{1.06248, 1.08064}

Style[AppendTo[cl@?, 0], 22]

(m = {{0, 1}, {1, -1}}.m) // MatrixForm

Divide @@ (m.{t, 1})

% /. {{t -> 3/8/2.}, {t -> 3/8/3.}}

{16.0056, 12.4004}

Whoa, a burst of three 1s!

Style[AppendTo[cl@?, 1]; AppendTo[cl@?, 1]; AppendTo[cl@?, 1], 22]

MatrixForm[m = {{1, 0}, {0, 8}}.m]

MatrixForm[m = m/2]

(Since we knew it was three ones and not four, we could have output 1,1,1,0 and left multiplied m by

MatrixForm[{{0, 1}, {1, -1}}.{{1, 0}, {0, 8}}]

This process may be continued indefinitely, or until the integers in m overflow.