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To: python-list@python.org
From: Terry Reedy <tjreedy@udel.edu>
Subject: Re: Numeric root-finding in Python
Date: Sun, 12 Feb 2012 16:19:31 -0500
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Xref: x330-a1.tempe.blueboxinc.net comp.lang.python:20295

On 2/12/2012 5:10 AM, Eelco wrote:
> On Feb 12, 7:41 am, Steven D'Aprano<steve
> +comp.lang.pyt...@pearwood.info>  wrote:
>> This is only peripherally a Python problem, but in case anyone has any
>> good ideas I'm going to ask it.
>>
>> I have a routine to calculate an approximation of Lambert's W function,
>> and then apply a root-finding technique to improve the approximation.
>> This mostly works well, but sometimes the root-finder gets stuck in a
>> cycle.
>>
>> Here's my function:
>>
>> import math
>> def improve(x, w, exp=math.exp):
>>      """Use Halley's method to improve an estimate of W(x) given
>>      an initial estimate w.
>>      """
>>      try:
>>          for i in range(36):  # Max number of iterations.
>>              ew = exp(w)
>>              a = w*ew - x
>>              b = ew*(w + 1)
>>              err = -a/b  # Estimate of the error in the current w.
>>              if abs(err)<= 1e-16:
>>                  break
>>              print '%d: w= %r err= %r' % (i, w, err)
>>              # Make a better estimate.
>>              c = (w + 2)*a/(2*w + 2)
>>              delta = a/(b - c)
>>              w -= delta
>>          else:
>>              raise RuntimeError('calculation failed to converge', err)
>>      except ZeroDivisionError:
>>          assert w == -1
>>      return w
>>
>> Here's an example where improve() converges very quickly:
>>
>> py>  improve(-0.36, -1.222769842388856)
>> 0: w= -1.222769842388856 err= -2.9158979924038895e-07
>> 1: w= -1.2227701339785069 err= 8.4638038491998997e-16
>> -1.222770133978506
>>
>> That's what I expect: convergence in only a few iterations.
>>
>> Here's an example where it gets stuck in a cycle, bouncing back and forth
>> between two values:
>>
>> py>  improve(-0.36787344117144249, -1.0057222396915309)
>> 0: w= -1.0057222396915309 err= 2.6521238905750239e-14
>> 1: w= -1.0057222396915044 err= -2.6521238905872001e-14
>> 2: w= -1.0057222396915309 err= 2.6521238905750239e-14
>> 3: w= -1.0057222396915044 err= -2.6521238905872001e-14
>> 4: w= -1.0057222396915309 err= 2.6521238905750239e-14

>> 35: w= -1.0057222396915044 err= -2.6521238905872001e-14
>> Traceback (most recent call last):
>>    File "<stdin>", line 1, in<module>
>>    File "<stdin>", line 19, in improve
>> RuntimeError: ('calculation failed to converge', -2.6521238905872001e-14)
>>
>> (The correct value for w is approximately -1.00572223991.)
>>
>> I know that Newton's method is subject to cycles, but I haven't found any
>> discussion about Halley's method and cycles, nor do I know what the best
>> approach for breaking them would be. None of the papers on calculating
>> the Lambert W function that I have found mentions this.
>>
>> Does anyone have any advice for solving this?

> Looks like floating point issues to me, rather than something
> intrinsic to the iterative algorithm. Surely there is not complex
> chaotic behavior to be found in this fairly smooth function in a +/-
> 1e-14 window. Otoh, there is a lot of floating point significant bit
> loss issues to be suspected in the kind of operations you are
> performing (exp(x) + something, always a tricky one).

To investigate this, I would limit the iterations to 2 or 3 and print 
ew, a,b,c, and delta, maybe in binary(hex) form

> I would start by asking: How accurate is good enough? If its not good
> enough, play around the the ordering of your operations, try solving a
> transformed problem less sensitive to loss of significance; and begin
> by trying different numeric types to see if the problem is sensitive
> thereto to begin with.

-- 
Terry Jan Reedy