Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Rational Convergents to General and Regular Continued Fractions

  1. Jul 5, 2011 #1
    Let a General Continued Fraction (gCF) be given as

    (1) b[itex]_{0}[/itex]+[itex]\frac{a_{1}}{b_{1}+\frac{a_{2}}{b_{2}+\frac{a_{3}}{{b_{3}+...}}}}[/itex]

    or abbreviated as b[itex]_{0}[/itex]+[itex]\frac{a_{1}}{b_{1}+}[/itex] [itex]\frac{a_{2}}{b_{2}+}[/itex] [itex]\frac{a_{3}}{b_{3}+}[/itex] [itex]\frac{a_{4}}{b_{4}+}[/itex] [itex]\frac{a_{5}}{b_{5}+}[/itex] [itex]\frac{a_{6}}{b_{6}+}[/itex] [itex]\frac{a_{7}}{b_{7}+}[/itex] [itex]\frac{a_{8}}{b_{8}+}[/itex] ...

    then n-th rational convergent to (1) is C[itex]_{n}[/itex] := [itex]A_{n}/B_{n}[/itex] with

    A[itex]_{n}[/itex] := b[itex]_{n}[/itex]*A[itex]_{n-1}[/itex] + a[itex]_{n}[/itex]*A[itex]_{n-2}[/itex] and B[itex]_{n}[/itex] := b[itex]_{n}[/itex]*B[itex]_{n-1}[/itex] + a[itex]_{n}[/itex]*B[itex]_{n-2}[/itex]

    (A[itex]_{-1}[/itex] := 1, A[itex]_{0}[/itex] := b[itex]_{0}[/itex], B[itex]_{-1}[/itex] := 0, B[itex]_{0}[/itex] := 1)

    For a Regular Continued Fraction (rCF) all a[itex]_{i}[/itex] are set to +1, giving

    (2) b[itex]_{0}[/itex]+[itex]\frac{1}{b_{1}+\frac{1}{b_{2}+\frac{1}{{b_{3}+...}}}}[/itex]

    or abbreviated [b[itex]_{0}[/itex]; b[itex]_{1}[/itex],b[itex]_{2}[/itex],b[itex]_{3}[/itex],b[itex]_{4}[/itex],b[itex]_{5}[/itex],...]

    and the formula for the n-th rational convergents to (2) C[itex]_{n}[/itex] := [itex]A_{n}/B_{n}[/itex] with

    A[itex]_{n}[/itex] := b[itex]_{n}[/itex]*A[itex]_{n-1}[/itex] + A[itex]_{n-2}[/itex] and B[itex]_{n}[/itex] := b[itex]_{n}[/itex]*B[itex]_{n-1}[/itex] + B[itex]_{n-2}[/itex]

    (A[itex]_{-1}[/itex] := 1, A[itex]_{0}[/itex] := b[itex]_{0}[/itex], B[itex]_{-1}[/itex] := 0, B[itex]_{0}[/itex] := 1)
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Rational Convergents to General and Regular Continued Fractions
  1. Continued fractions (Replies: 4)

  2. Continued fractions (Replies: 1)

  3. Continued fractions (Replies: 3)

Loading...