7.2.7  Continued fraction representation of a real

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

, 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 2.5.7, item 2.5.7.
• 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 10.1.10) 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. For example, if dfc(a) is equal to [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

or:

With epsilon=1e−9 in the CAS configuration, we obtain:

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

Input to verify:

With epsilon=1e−9 in the CAS configuration, we obtain:

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

If floats are hardware floats, e.g. for Digits=12, we obtain:

or:

or, with epsilon=1e−9 in the CAS configuration:

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