Continued fraction representation of a real: dfc

Any real number a can be written as a continued fraction:

a = a₀ +

a₁ +

a₂ +

a₃ + ⋯

, which is often abbreviated [a₀;a₁,a₂,a₃,…]. The dfc command writes a real number as a continued fraction.

dfc takes one mandatory argument and one optional argument:
- a, a real number.
- Optionally, n an integer or epsilon, a positive real number.
dfc(a) returns the list of the continued fraction representation of a with precision epsilon, which is given by Section 3.5.7, item 9.
dfc(a,epsilon) returns the list of the continued fraction representation which approximates a or evalf(a) with the specified precision epsilon.
dfc(a,n) returns the list of the continued fraction representation of a of order n.

Remarks.

The convert command with the option confrac (see Section 6.23.26) has a similar functionality: in that case the value of epsilon is the value defined in the cas configuration and the answer may be stored in an optional third argument.
If the last element of the result is a list, the representation is ultimately 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. So If dfc(a)=[a0,a1,a2,[b0,b1]] then:
a=a0+
1

a1+
1

a2+
1

b0+
1

b1+
1

b0+…
if the last element of the result is not an integer, it represents a remainder r (a=a0+1/…+1/an+1/r). So if dfc(a)=[a0,a1,a2,r] then:
a=a0+
1

a1+
1

a2+
1

r

Be aware that this remainder has lost most of its accuracy.

Examples.

Input:
dfc(sqrt(2),5)
Output:
⎡
⎣ 1,2, ⎡
⎣ 2 ⎤
⎦ ⎤
⎦
Input:
dfc(evalf(sqrt(2)),1e-9)
or:
dfc(sqrt(2),1e-9)
Output:
⎡
⎣ 1,2,2,2,2,2,2,2,2,2,2,2,2 ⎤
⎦
Input:
convert(sqrt(2),confrac,’dev’)
Output (if in the cas configuration epsilon=1e-9):
⎡
⎣ 1,2,2,2,2,2,2,2,2,2,2,2,2 ⎤
⎦

and [1,2,2,2,2,2,2,2,2,2,2,2,2] is stored in dev.
Input:
dfc(9976/6961,5)
Output:
⎡
⎢
⎢
⎣ 1,2,3,4,5,
43

7

⎤
⎥
⎥
⎦

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

9976

6961
Input:
convert(9976/6961,confrac,’l’)
Output (if in the cas configuration epsilon=1e-9):
⎡
⎣ 1,2,3,4,5,6,7 ⎤
⎦

and [1,2,3,4,5,6,7] is stored in l.
Input:
dfc(pi,5)
Output:
⎡
⎢
⎢
⎣ 3,7,15,1,292,
−113 π +355

33102 π −103993

⎤
⎥
⎥
⎦
Input:
dfc(evalf(pi),5)
Output (if floats are hardware floats, e.g. for Digits=12):
⎡
⎣ 3,7,15,1,292,1.57581843574 ⎤
⎦
Input:
dfc(evalf(pi),1e-9)
or:
dfc(pi,1e-9)
or (if in the cas configuration epsilon=1e-9):
convert(pi,confrac,’ll’)
Output:
⎡
⎣ 3,7,15,1,292 ⎤
⎦

and [3,7,15,1,292] is stored in ll.