[Numpy-discussion] Unrealistic expectations of class Polynomial or a bug?
eat
e.antero.tammi@gmail....
Sun Jan 29 11:03:55 CST 2012
On Sat, Jan 28, 2012 at 11:14 PM, Charles R Harris <
charlesr.harris@gmail.com> wrote:
>
>
> On Sat, Jan 28, 2012 at 11:15 AM, eat <e.antero.tammi@gmail.com> wrote:
>
>> Hi,
>>
>> Short demonstration of the issue:
>> In []: sys.version
>> Out[]: '2.7.2 (default, Jun 12 2011, 15:08:59) [MSC v.1500 32 bit
>> (Intel)]'
>> In []: np.version.version
>> Out[]: '1.6.0'
>>
>> In []: from numpy.polynomial import Polynomial as Poly
>> In []: def p_tst(c):
>> ..: p= Poly(c)
>> ..: r= p.roots()
>> ..: return sort(abs(p(r)))
>> ..:
>>
>> Now I would expect a result more like:
>> In []: p_tst(randn(123))[-3:]
>> Out[]: array([ 3.41987203e-07, 2.82123675e-03, 2.82123675e-03])
>>
>> be the case, but actually most result seems to be more like:
>> In []: p_tst(randn(123))[-3:]
>> Out[]: array([ 9.09325898e+13, 9.09325898e+13, 1.29387029e+72])
>> In []: p_tst(randn(123))[-3:]
>> Out[]: array([ 8.60862087e-11, 8.60862087e-11, 6.58784520e+32])
>> In []: p_tst(randn(123))[-3:]
>> Out[]: array([ 2.00545673e-09, 3.25537709e+32, 3.25537709e+32])
>> In []: p_tst(randn(123))[-3:]
>> Out[]: array([ 3.22753481e-04, 1.87056454e+00, 1.87056454e+00])
>> In []: p_tst(randn(123))[-3:]
>> Out[]: array([ 2.98556327e+08, 2.98556327e+08, 8.23588003e+12])
>>
>> So, does this phenomena imply that
>> - I'm testing with too high order polynomials (if so, does there exists a
>> definite upper limit of polynomial order I'll not face this issue)
>> or
>> - it's just the 'nature' of computations with float values (if so,
>> probably I should be able to tackle this regardless of the polynomial order)
>> or
>> - it's a nasty bug in class Polynomial
>>
>>
> It's a defect. You will get all the roots and the number will equal the
> degree. I haven't decided what the best way to deal with this is, but my
> thoughts have trended towards specifying an interval with the default being
> the domain. If you have other thoughts I'd be glad for the feedback.
>
> For the problem at hand, note first that you are specifying the
> coefficients, not the roots as was the case with poly1d. Second, as a rule
> of thumb, plain old polynomials will generally only be good for degree < 22
> due to being numerically ill conditioned. If you are really looking to use
> high degrees, Chebyshev or Legendre will work better, although you will
> probably need to explicitly specify the domain. If you want to specify the
> polynomial using roots, do Poly.fromroots(...). Third, for the high degrees
> you are probably screwed anyway for degree 123, since the accuracy of the
> root finding will be limited, especially for roots that can cluster, and
> any root that falls even a little bit outside the interval [-1,1] (the
> default domain) is going to evaluate to a big number simply because the
> polynomial is going to h*ll at a rate you wouldn't believe ;)
>
> For evenly spaced roots in [-1, 1] and using Chebyshev polynomials, things
> look good for degree 50, get a bit loose at degree 75 but can be fixed up
> with one iteration of Newton, and blow up at degree 100. I think that's
> pretty good, actually, doing better would require a lot more work. There
> are some zero finding algorithms out there that might do better if someone
> wants to give it a shot.
>
> In [20]: p = Cheb.fromroots(linspace(-1, 1, 50))
>
> In [21]: sort(abs(p(p.roots())))
> Out[21]:
> array([ 6.20385459e-25, 1.65436123e-24, 2.06795153e-24,
> 5.79026429e-24, 5.89366186e-24, 6.44916482e-24,
> 6.44916482e-24, 6.77254127e-24, 6.97933642e-24,
> 7.25459208e-24, 1.00295649e-23, 1.37391414e-23,
> 1.37391414e-23, 1.63368171e-23, 2.39882378e-23,
> 3.30872245e-23, 4.38405725e-23, 4.49502653e-23,
> 4.49502653e-23, 5.58346913e-23, 8.35452419e-23,
> 9.38407760e-23, 9.38407760e-23, 1.03703218e-22,
> 1.03703218e-22, 1.23249911e-22, 1.75197880e-22,
> 1.75197880e-22, 3.07711188e-22, 3.09821786e-22,
> 3.09821786e-22, 4.56625520e-22, 4.56625520e-22,
> 4.69638303e-22, 4.69638303e-22, 5.96448724e-22,
> 5.96448724e-22, 1.24076485e-21, 1.24076485e-21,
> 1.59972624e-21, 1.59972624e-21, 1.62930347e-21,
> 1.62930347e-21, 1.73773328e-21, 1.73773328e-21,
> 1.87935435e-21, 2.30287083e-21, 2.48815928e-21,
> 2.85411753e-21, 2.85411753e-21])
>
Thanks,
for a very informative feedback. I'll study those orthogonal polynomials
more detail.
Regards,
- eat
>
>
> Chuck
>
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