*** 2nd try *** (hopefully getting the formatting right this time) ...
Seems that up til now I have been totally unaware of the existence of
the Stern-Brocot tree of fractions or the Mediant ... Thanks for
opening up that door for me. --
After some reading I now accept that going through the tree I could
approximate any real number with fractions (taking mediants on the
way). But doesn't that mean I have to know the target in the first place..?
In the case of pi approximation (in Ancient Chinese Math) the real
number (target) isn't known ...
If I started out with the interval [3;4] written as fractions 3/1 and
4/1 (or 'coefficients' 3 1 and 4 1 as they called it, stating
circumference-diameter pairs)
piacm 3 1;4 1;0
3 4
1 1
and repeatedly take the mediant of the lower interval boundary and
the upper (replacing the upper with the mediant), after 7 steps I'll
be past the (famous) first reasonable approximation 22/7
piacm 3 1;4 1;7
3 4 7 10 13 16 19 22 25
1 1 2 3 4 5 6 7 8
Taking the last two thus obtained values as interval boundaries and
continue the process for another 15 steps
piacm 22 7;25 8;15
22 25 47 69 91 113 135 157 179 201 223 245 267 289 311 333 355
7 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113
I'll come up with the (famous) better approximation 355/113, which
coincides with the partial sum of the continued fraction [3;7,15,1]
ecf 3 7 15 1
3.14159
x: ecf 3 7 15 1
355r113
[Almost all of these values are shown in the animated picture
contained in the Wikipedia article.]
Now my question is:
In that algorithm, what would be the indicator (or break-off
condition) for having hit a reasonable result, and which new
'summand' to use in improving it..?
-M
ps:
Citing from the Martzloff book (p. 281):
---
"... Zu Chongzhi:
'... [Hence] the "close models" mi lu, 113 for the diameter and 355
for the circumference, and the "rough models" yue lu, 7 for the
diameter and 22 for the circumference [respectively].' ..."
---
So the terms weak | strong fraction hasn't its origin in that text ...
At 2018-01-29 21:01, you wrote:
Guess you are right.
Brocot (from the Stern-Brocot tree of fractions) came up with the
algo to produce from two fractions p/q and r/s the fraction
(p+r)/(q+s), which is in between the two parent fractions. Starting
with 0/1 and 1/0 you can converge to any positive real number x,
where (p+r)/(q+s) becomes the new 'weak' or 'strong' fraction
dependent of its position w.r.t. x.
R.E. Boss
> -----Original Message-----
> From: Chat [mailto:[email protected]] On Behalf Of
> Roger Hui
> Sent: maandag 29 januari 2018 18:50
> To: Chat Forum <[email protected]>
> Subject: Re: [Jchat] weak | strong fraction ...
>
> Guess: p<x and x<q. p and q are rational. p is a "weak
fraction" and q is a
> "strong fraction". If you have a process where at each step the weak
> becomes less weak and the strong becomes less strong, then you are in
> business.
>
>
>
> On Mon, Jan 29, 2018 at 9:05 AM, Martin Kreuzer <[email protected]>
> wrote:
>
> > Hi -
> >
> > This question goes to you Math people:
> >
> > Looking at this article
> > https://en.wikipedia.org/wiki/Mil%C3%BC
> > I found (in the last paragraph) the terms "weak" and "strong" fractions.
> >
> > I've never before seen this mentioned anywhere and I'm in doubt
> > whether it is a valid concept.
> > [nb: I do own the Martzloff (1996) book.]
> >
> > Any enlightenment welcome ...
> >
> > -M
> >
> > ----------------------------------------------------------------------
> > For information about J forums see
> http://www.jsoftware.com/forums.htm
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
