[Tutor] Need help with rewriting script to use Decimal module

[Dick Moores]

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