[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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