Difference between revisions of "Diagonal continued fraction"
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A [[Continued fraction|continued fraction]] | A [[Continued fraction|continued fraction]] | ||
− | + | $$ | |
− | + | a _ {0} + | |
− | + | \frac{b _ {1} \mid }{\mid a _ {1} } | |
+ | + \dots + | ||
− | + | \frac{b _ {n} \mid }{\mid a _ {n} } | |
+ | + \dots , | ||
+ | $$ | ||
− | + | in which the sequences $ \{ a _ {n} \} _ {n=} 0 ^ \omega $ | |
+ | and $ \{ b _ {n} \} _ {n=} 1 ^ \omega $ | ||
+ | must satisfy the following conditions: | ||
− | + | 1) the numbers $ a _ {n} $ | |
+ | and $ b _ {n} $ | ||
+ | are integers; $ | b _ {n} | = 1 $; | ||
+ | $ a _ {n} \geq 1 $ | ||
+ | if $ n \geq 1 $; | ||
+ | $ a _ \omega \geq 2 $ | ||
+ | if $ 0 < \omega < \infty $; | ||
− | + | 2) $ b _ {n} + a _ {n} \geq 1 $ | |
+ | for all $ n $; | ||
+ | if $ \omega = \infty $, | ||
+ | then $ b _ {n} + a _ {n} \geq 2 $ | ||
+ | for an infinite set of indices $ n $; | ||
− | + | 3) $ Q _ {n} < Q _ {n+} 1 $ | |
+ | for all $ n $; | ||
+ | 4) the partial fractions of the continued fraction are all irreducible fractions $ A / B $ | ||
+ | such that $ | r - A/B | < 1 / 2B ^ {2} $ | ||
+ | and $ B > 0 $, | ||
+ | where $ r $ | ||
+ | is value of the continued fraction. | ||
+ | For each real number $ r $ | ||
+ | there exists one and only one diagonal continued fraction with $ r $ | ||
+ | as its value; this fraction is periodic if $ r $ | ||
+ | is a [[Quadratic irrationality|quadratic irrationality]]. | ||
====Comments==== | ====Comments==== | ||
After truncation and evaluation one obtains | After truncation and evaluation one obtains | ||
− | + | $$ | |
+ | a _ {0} + | ||
+ | \frac{b _ {1} \mid }{\mid a _ {1} } | ||
+ | + \dots + | ||
+ | |||
+ | \frac{b _ {n} \mid }{\mid a _ {n} } | ||
+ | = | ||
+ | \frac{P _ {n} }{Q _ {n} } | ||
+ | , | ||
+ | $$ | ||
− | with | + | with $ P _ {n} , Q _ {n} \in \mathbf Z $, |
+ | $ ( P _ {n} , Q _ {n} ) = 1 $, | ||
+ | $ Q _ {n} > 0 $. | ||
+ | These are the numbers $ Q _ {n} $ | ||
+ | alluded to in condition . The continued fraction as described above for a real number $ x _ {0} $ | ||
+ | can be obtained by the nearest integer algorithm, that is, $ a _ {0} = \langle x _ {0} \rangle $, | ||
+ | $ x _ {1} = 1 / ( x _ {0} - a _ {0} ) $, | ||
+ | $ a _ {1} = \langle x _ {1} \rangle $, | ||
+ | $ x _ {2} = 1 / ( x _ {1} - a _ {1} ) $, | ||
+ | $ a _ {2} = \langle x _ {2} \rangle $, | ||
+ | etc., where $ \langle x\rangle $ | ||
+ | denotes the nearest integer to $ x $. | ||
+ | It is also possible to use the entier function $ [ x] $ | ||
+ | instead of $ \langle x\rangle $. | ||
+ | One then has the continued fraction algorithm which is more commonly used. | ||
− | The adjective "diagonal" stems from the fact that | + | The adjective "diagonal" stems from the fact that $ b _ {n} = \pm 1 $ |
+ | for all $ n $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> O. Perron, "Die Lehre von den Kettenbrüchen" , '''I''' , Teubner (1977)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> O. Perron, "Die Lehre von den Kettenbrüchen" , '''I''' , Teubner (1977)</TD></TR></table> |
Latest revision as of 17:33, 5 June 2020
$$ a _ {0} + \frac{b _ {1} \mid }{\mid a _ {1} } + \dots + \frac{b _ {n} \mid }{\mid a _ {n} } + \dots , $$
in which the sequences $ \{ a _ {n} \} _ {n=} 0 ^ \omega $ and $ \{ b _ {n} \} _ {n=} 1 ^ \omega $ must satisfy the following conditions:
1) the numbers $ a _ {n} $ and $ b _ {n} $ are integers; $ | b _ {n} | = 1 $; $ a _ {n} \geq 1 $ if $ n \geq 1 $; $ a _ \omega \geq 2 $ if $ 0 < \omega < \infty $;
2) $ b _ {n} + a _ {n} \geq 1 $ for all $ n $; if $ \omega = \infty $, then $ b _ {n} + a _ {n} \geq 2 $ for an infinite set of indices $ n $;
3) $ Q _ {n} < Q _ {n+} 1 $ for all $ n $;
4) the partial fractions of the continued fraction are all irreducible fractions $ A / B $ such that $ | r - A/B | < 1 / 2B ^ {2} $ and $ B > 0 $, where $ r $ is value of the continued fraction.
For each real number $ r $ there exists one and only one diagonal continued fraction with $ r $ as its value; this fraction is periodic if $ r $ is a quadratic irrationality.
Comments
After truncation and evaluation one obtains
$$ a _ {0} + \frac{b _ {1} \mid }{\mid a _ {1} } + \dots + \frac{b _ {n} \mid }{\mid a _ {n} } = \frac{P _ {n} }{Q _ {n} } , $$
with $ P _ {n} , Q _ {n} \in \mathbf Z $, $ ( P _ {n} , Q _ {n} ) = 1 $, $ Q _ {n} > 0 $. These are the numbers $ Q _ {n} $ alluded to in condition . The continued fraction as described above for a real number $ x _ {0} $ can be obtained by the nearest integer algorithm, that is, $ a _ {0} = \langle x _ {0} \rangle $, $ x _ {1} = 1 / ( x _ {0} - a _ {0} ) $, $ a _ {1} = \langle x _ {1} \rangle $, $ x _ {2} = 1 / ( x _ {1} - a _ {1} ) $, $ a _ {2} = \langle x _ {2} \rangle $, etc., where $ \langle x\rangle $ denotes the nearest integer to $ x $. It is also possible to use the entier function $ [ x] $ instead of $ \langle x\rangle $. One then has the continued fraction algorithm which is more commonly used.
The adjective "diagonal" stems from the fact that $ b _ {n} = \pm 1 $ for all $ n $.
References
[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 |
[a2] | O. Perron, "Die Lehre von den Kettenbrüchen" , I , Teubner (1977) |
Diagonal continued fraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_continued_fraction&oldid=46639