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Saturday, August 27, 2011

Continued fractions (2)

A finite continued fraction (FCF) is a map $$f: \mathbf{N}^m \rightarrow \mathbf{Q}$$ $$\left( a_1, a_2, \cdots a_m \right) \mapsto a_1 + \frac{1}{a_2 + \frac{1}{\ddots + \frac{1}{a_m}}}$$

Continued fractions are calculated by creating a table of convergents, as follows:

 $k$ $a_k$ $p_k$ $q_k$ $C_k$ $-1$ $0$ $0$ $1$ $0$ $1$ $a_1$ $a_1 \cdot p_0 + p_{-1}$ $1$ $\frac{p_1}{q_1}$ $k$ $a_k$ $a_k \cdot p_{k-1} + p_{k-2}$ $a_k \cdot q_{k-1} + q_{k-2}$ $\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

See also: Continued fractions (1)

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)