Continuous fractions in CAS

[Albert Chan]

(09-24-2020 06:19 PM)Albert Chan Wrote:  From giac source, I learned a nice trick to estimate continued fraction convergent error.
Without knowing the convergent ! (see prog.cc, float2continued_frac())

Let y[i] = [a[i]; a[i+1], a[i+2], ..., a[n] + x]. This is the effect of dropping x of y[0]:

\( \displaystyle ε = |y_0 - [a_0;a_1,a_2,\;...,\;a_n] |
≈ \left| {y_1 - [a_1;a_2,a_3,\;...,\;a_n] \over y_1^2 } \right|
≈\;...\;≈ {x \over y_1^2\;y_2^2 \;...\;y_n^2}\)

Flip RHS denominator to LHS, we have rough estimate of when to stop generating CF coefficients.