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A [[Continued fraction|continued fraction]]
 
A [[Continued fraction|continued fraction]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314701.png" /></td> </tr></table>
+
$$
 
+
a _ {0} +
in which the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314703.png" /> must satisfy the following conditions:
+
\frac{b _ {1} \mid  }{\mid  a _ {1} }
 +
+ \dots +
  
1) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314705.png" /> are integers; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314706.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314707.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314708.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314709.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147010.png" />;
+
\frac{b _ {n} \mid  }{\mid  a _ {n} }
 +
+ \dots ,
 +
$$
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147012.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147014.png" /> for an infinite set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147015.png" />;
+
in which the sequences  $  \{ a _ {n} \} _ {n=} 0 ^  \omega  $
 +
and  $  \{ b _ {n} \} _ {n=} 1  ^  \omega  $
 +
must satisfy the following conditions:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147017.png" />;
+
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 $;
  
4) the partial fractions of the continued fraction are all irreducible fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147021.png" /> is value of the continued fraction.
+
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 $;
  
For each real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147022.png" /> there exists one and only one diagonal continued fraction with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147023.png" /> as its value; this fraction is periodic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147024.png" /> is a [[Quadratic irrationality|quadratic irrationality]].
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147025.png" /></td> </tr></table>
+
$$
 +
a _ {0} +
 +
\frac{b _ {1} \mid  }{\mid  a _ {1} }
 +
+ \dots +
 +
 
 +
\frac{b _ {n} \mid  }{\mid  a _ {n} }
 +
  =
 +
\frac{P _ {n} }{Q _ {n} }
 +
,
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147028.png" />. These are the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147029.png" /> alluded to in condition . The continued fraction as described above for a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147030.png" /> can be obtained by the nearest integer algorithm, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147035.png" />, etc., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147036.png" /> denotes the nearest integer to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147037.png" />. It is also possible to use the entier function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147038.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147039.png" />. One then has the continued fraction algorithm which is more commonly used.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147041.png" />.
+
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 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.

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)
How to Cite This Entry:
Diagonal continued fraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_continued_fraction&oldid=46639
This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article