Re: [Maxima-discuss] how to perform symbolic math with continued fractions

[Hugo C. <hug...@gm...>]

Thanks a lot!

On Sun, 1 Jul 2018, 22:41 Stavros Macrakis (Σταῦρος Μακράκης), <
mac...@al...> wrote:

> The core of the continued fraction routines dates to the 1970s.
> No conceptual work has been done to them since then, though some
> ill-thought-out enhancements have been made.
>
> For example, a minor addition from 2007 supports non-square roots, e.g.,
> cf(2^(1/3)), by using the floor function, which uses approximate evaluation
> (bfloat) and ratsimp. But that isn't even documented, and it is poorly
> integrated: that code lets cf((1/2)^(1/3)) give a correct result, but
> cf((1/2)^(1/2)) gives an error! It also interprets cflength differently
> from the rest of cf (also undocumented).
>
> In fact, the approximation approach works fine for other functions as
> well; it just isn't exposed to the user. For example, cf(sin(1)) gives an
> error, but ?cfnroot(sin(1)),cflength:10 happily returns [0, 1, 5, 3, 4, 19,
> 2, 2, 2, 2]. But you might as well write cf(bfloat(sin(1))) and control the
> precision more accurately. For that matter, ?cfnroot(sin(1)),cflength:30
> gives the error "expt undefined: 0 to a negative exponent".
>
> The cfnroot approximation technique is absurdly inefficient, as well. For
> example, cf(2^(1/10)),cflength:4 takes 56 sec (you can see why by tracing
> ratsimp with cflength:2), whereas cf(bfloat(2^(1/10))) gives 17 terms in <
> 1 ms.
>
> All that to say that there is no one working on CF seriously.
>
> As for checking theorems about CFs, you will probably find cfdisrep useful
> for that.
>
>               -s
>
>
>
>
>
>
>
>
>
>
> On Sun, Jul 1, 2018 at 4:36 AM Hugo Coolens <hug...@gm...> wrote:
>
>>
>>
>> 2018-06-29 19:10 GMT+02:00 Stavros Macrakis (Σταῦρος Μακράκης) <
>> mac...@al...>:
>>
>>> The cf group of functions is a *horribly designed hack*.
>>>
>>> The hack of supporting [...]*[...] within the cf function is an
>>> abomination which wrecks referential transparency:
>>>
>>> qq: [2,3]*[4,5] => [8,15]    (with listarith: true, the default)
>>> cf(qq) => [8,15]
>>> cf([2,3]*[4,5]) => [9,1,4]
>>>
>>>
>>> It is also an abomination because it only supports * and not / and
>>> integer ^.
>>>
>>> It is also an abomination because it gives incorrect results when
>>> there's more than one repeating cf:
>>>
>>>      cf(sqrt(2)*sqrt(5)) => [3,2,1,2]
>>>      cf(sqrt(10)) => [3,6]
>>>
>>> For that matter, it is an abomination that cf(sqrt(...)) doesn't
>>> indicate in any way that the result repeats, or what the repeating group is.
>>>
>>> As I understand it, the "continued fractions part" of Maxima is not well
>> developed at this very moment and barely useable?
>> Is there a chance it will be replaced by something better in the (near)
>> future?
>>
>>
>>
>>> As for performing cf manipulations symbolically, what do you have in
>>> mind?
>>>
>>> I was going to check some theorems concerning cf's symbolically with
>> Maxima
>>
>>
>>
>>> In particular, what is cf([a]/[0,b]) == cf(a/b)? After all, cf(5/5) =
>>> [1], cf(5/7) = [0,1,2,2], cf(5/11) = [0,2,5], cf(5/13) = [0,2,1,1,2],
>>> cf(5/15) = [0,3], etc. Not even the length is constant.
>>>
>>>
>>>
>>>
>>>
>>> On Fri, Jun 29, 2018 at 8:09 AM Hugo Coolens <hug...@gm...>
>>> wrote:
>>>
>>>> Is it possible to perform symbolic math with continued fractions?
>>>> If I try something like this:
>>>> cf([a,b,c]);
>>>> maxima answers with
>>>> Maxima encountered a Lisp error: Condition in MACSYMA-TOP-LEVEL [or a
>>>> callee]: INTERNAL-SIMPLE-TYPE-ERROR: $C is not of type NUMBER:
>>>>
>>>> can I tell maxima a,b and are of type NUMBER?
>>>>
>>>> kind regards,
>>>> Hugo
>>>>
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>>>
>>