For the generalized continued fraction
1 + 2
------
3 + 4
--
5
the desired terms are
1
1 + 2
--
3
1 + 2
------
3 + 4
--
5
that is,
0 2 4 cf 1 3 5x
1 5r3 29r19
we can get this with
1 0 1 0 1 # 1 , 1 + %`+/\ 2 3 4 5x
1 5r3 29r19
Why are these the "desired" terms? It is because of their connection with
associated linear fractional transformations t1 =: 2 % 3 + ] and t2 =: 4 % 5
+ ] . The three terms are 1 and 1 + t1 0 and 1 + t1 t2 0 .
--Kip Murray
Sent from my iPad
> On Feb 23, 2014, at 6:03 PM, Roger Hui <[email protected]> wrote:
>
> Sometimes %`+/ and %`+/\ are the expressions, depending on circumstances.
> For example:
>
> x=: ?. 0
> x
> 0.038363
> %`+/ x,,(1+2*i.20),.*:x
> 0.0383442
> 7 o. x
> 0.0383442
>
> Gauss, 1812, according to C.D. Olds, Continued Fractions, MAA, 1963.
>
>
>
>
> On Sun, Feb 23, 2014 at 3:55 PM, Roger Hui <[email protected]>wrote:
>
>> +`%/ and +`%/\ are the expressions you seek. For example:
>>
>> +`%/3,20$4 6
>> 3.60555
>> %: 13
>> 3.60555
>>
>>
>>
>>
>>> On Sun, Feb 23, 2014 at 2:17 PM, km <[email protected]> wrote:
>>>
>>> The generalized continued fraction
>>>
>>> b0 + a1
>>> ---------
>>> b1 + a2
>>> ---------
>>> b2 + a3
>>> --------
>>> b3 + ...
>>>
>>> is the infinite sequence
>>>
>>> b0 , b0 + a1%b1 , b0 + a1%b1 + a2%b2 , b0 + a1%b1 + a2%b2 + a3%b3 , ...
>>>
>>> where between the commas I am assuming J's right-to-left evaluation. The
>>> nth term would be
>>>
>>> b0 + a1%b1 + a2%b2 + a3%b3 + ... + an%bn (still using right to left
>>> evaluation).
>>>
>>> What is a good way to calculate this nth term in J?
>>>
>>>
>>> For a half-page introduction to generalized continued fractions see
>>>
>>> http://people.math.sfu.ca/~cbm/aands/page_19.htm
>>>
>>> Continued fractions for ln(1+z) and ln((1+z)%(1-z)) are given here
>>>
>>> http://people.math.sfu.ca/~cbm/aands/page_68.htm
>>>
>>>
>>> --Kip Murray
>>>
>>> Sent from my iPad
>>>
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> ----------------------------------------------------------------------
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