2.8.7  Continued fraction representation of a real : dfc

dfc takes as argument a real or a rational or a floating point number a and an integer n (or a real epsilon).
dfc returns the list of the continued fraction representation of a of order n (or with precision epsilon i.e. the continued fraction representation which approachs a or evalf(a) with precision epsilon, by default epsilon is the value of the epsilon defined in the cas configuration with the menu Cfg▸Cas Configuration).
convert with the option confrac has a similar functionnality: in that case the value of epsilon is the value of the epsilon defined in the cas configuration with the menu Cfg▸Cas Configuration (see 2.21.23) and the answer may be stored in an optionnal third argument.

Remarks

• If the last element of the result is a list, the representation is ultimaltely periodic, and the last element is the period. It means that the real is a root of an equation of order 2 with integer coefficients.
• if the last element of the result is not an integer, it represents a remainder r (a=a0+1/....+1/an+1/r). Be aware that this remainder has lost most of it’s accuracy.

If dfc(a)=[a0,a1,a2,[b0,b1] that means :

If dfc(a)=[a0,a1,a2,r] that means :

Input :

Output :

Input :

Or :

Output :

Input :

Output (if in the cas configuration epsilon=1e-9) :

and [1,2,2,2,2,2,2,2,2,2,2,2,2] is stored in dev.
Input :

Output :

Input to verify:
1+1/(2+1/(3+1/(4+1/(5+7/43))))
Output :
9976/6961
Input :

Output (if in the cas configuration epsilon=1e-9) :

and [1,2,3,4,5,6,7] is stored in l
Input :

Output :

Input :

Output (if floats are hardware floats, e.g. for Digits=12) :

Input :

Or :

Or (if in the cas configuration epsilon=1e-9) :

Output :