At 11:21 PM 1/4/2007, Terry Carroll wrote:
>On Wed, 3 Jan 2007, Dick Moores wrote:
>
> > Be that as it may, farey() is an amazing program.
>
>Not to beat this subject to death, but the comment at the bottom of
>http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/52317 about
>continued fractions piqued my interest. I'm no mathematician, but I
>encountered continued fractions a long time ago and was fascinated by
>them. So I read the URL pointed to,
>http://mathworld.wolfram.com/ContinuedFraction.html , and came up with the
>following:
>
>#####################################################
>
>def cf(x, tol=0.0001, Trace=False):
> """
> Calculate rational approximation of x to within tolerance of tol;
> returns a tuple consisting of numerator and denominator p/q
> Trace=True causes iterated results to be shown
> """
> a, r, p, q = [], [], [], []
> Done = False
> n = 0
> if Trace: print "x:%f tol:%f" % (x, tol)
> while not Done:
> a.append(None)
> r.append(None)
> p.append(None)
> q.append(None)
> if n == 0: r[n] = x
> else: r[n] = 1/(r[n-1]-a[n-1])
> a[n] = int(r[n])
> if n == 0:
> p[n] = a[0]
> q[n] = 1
> elif n ==1:
> p[n] = a[n]*p[n-1] + 1
> q[n] = a[n]
> else:
> p[n] = a[n]*p[n-1] + p[n-2]
> q[n] = a[n]*q[n-1] + q[n-2]
> if Trace:
> print "n:%d a:%d p:%d q:%d approx:%f" % \
> (n, a[n], p[n], q[n], float(p[n])/q[n])
> if abs(float(p[n])/q[n] - x) < tol:
> Done = True
> num = p[n]; denom = q[n]
> n += 1
> return (num, denom)
>
>#####################################################
>
>Here's a result for pi:
>
> >>> print cf(3.14159265357989,0.0000001, Trace=True)
>x:3.141593 tol:0.000000
>n:0 a:3 p:3 q:1 approx:3.000000
>n:1 a:7 p:22 q:7 approx:3.142857
>n:2 a:15 p:333 q:106 approx:3.141509
>n:3 a:1 p:355 q:113 approx:3.141593
>n:4 a:292 p:103993 q:33102 approx:3.141593
>(103993, 33102)
>
>i.e., the first 5 approximations it came up with were 3/1, 22/7, 333/106,
>355/113 and a whopping 103993/33102.
>
>For the 0.36 example you used earlier:
>
> >>> print cf(0.36, .01, Trace= True)
>x:0.360000 tol:0.010000
>n:0 a:0 p:0 q:1 approx:0.000000
>n:1 a:2 p:1 q:2 approx:0.500000
>n:2 a:1 p:1 q:3 approx:0.333333
>n:3 a:3 p:4 q:11 approx:0.363636
>(4, 11)
> >>>
>
>it went right from 1/3 to 4/11 (0.363636), skipping the 3/8 (0.375) option
>from the farey series.
>
>But this continued fraction algorithm is ill-suited to answer the question
>"what's the closest fraction with a denominator < N", because it doesn't
>try to find that, it jumps further ahead with each iteration.
>
>Anyway, I thought you might find it interesting based on our discussion.
Terry,
Well, I have to admit I don't understand your code at all. But I see it works.
I modified one of my functions of frac.py, and came up with
===============================================
from __future__ import division
import time, psyco
psyco.full()
def d(number):
import decimal
decimal.getcontext().prec = 16
return decimal.Decimal(str(number))
def bestFracForMinimumError(decimal, minimumError):
denom = 0
smallestError = 10
count = 0
while True:
denom += 1
num = int(round(d(decimal) * d(denom)))
error = abs((((d(num)) / d(denom)) - d(decimal)) /
d(decimal)) * d(100)
if d(error) <= d(smallestError):
count += 1
smallestError = d(error)
q = d(num)/d(denom)
print "%d/%d = %s has error of %s per cent" % (num,
denom, q, smallestError)
if d(smallestError) <= d(minimumError):
print "count is", count
break
=====================================================================
You can see the results of both
bestFracForMinimumError(3.14159265357989, 0.00000002)
(BTW your pi is a bit off but I used yours, instead of math.pi, which
is 3.1415926535897931 . Also, I needed 0.00000002 in order to produce
your 103993/33102)
and
bestFracForMinimumError(.36, .01)
at <http://www.rcblue.com/Python/PartOfReplyToTerryOnTutorList.txt>
Dick
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