[Numpy-discussion] polynomial fromroots
Charles R Harris
charlesr.harris@gmail....
Sat Oct 9 21:21:51 CDT 2010
On Sat, Oct 9, 2010 at 8:05 PM, Charles R Harris
<charlesr.harris@gmail.com>wrote:
>
>
> On Sat, Oct 9, 2010 at 8:01 PM, Charles R Harris <
> charlesr.harris@gmail.com> wrote:
>
>>
>>
>> On Sat, Oct 9, 2010 at 7:47 PM, <josef.pktd@gmail.com> wrote:
>>
>>> I'm trying to see whether I can do this without reading the full manual.
>>>
>>> Is it intended that fromroots normalizes the highest order term
>>> instead of the lowest?
>>>
>>>
>>> >>> import numpy.polynomial as poly
>>>
>>> >>> p = poly.Polynomial([1, -1.88494037, 0.0178126 ])
>>> >>> p
>>> Polynomial([ 1. , -1.88494037, 0.0178126 ], [-1., 1.])
>>> >>> pr = p.roots()
>>> >>> pr
>>> array([ 0.53320748, 105.28741219])
>>> >>> poly.Polynomial.fromroots(pr)
>>> Polynomial([ 56.14003571, -105.82061967, 1. ], [-1., 1.])
>>> >>>
>>>
>>> renormalizing
>>>
>>> >>> p2 = poly.Polynomial.fromroots(pr)
>>> >>> p2/p2.coef[0]
>>> Polynomial([ 1. , -1.88494037, 0.0178126 ], [-1., 1.])
>>>
>>>
>>> this is, I think what I want to do, invert roots that are
>>> inside/outside the unit circle (whatever that means
>>>
>>> >>> pr[np.abs(pr)<1] = 1./pr[np.abs(pr)<1]
>>> >>> p3 = poly.Polynomial.fromroots(pr)
>>> >>> p3/p3.coef[0]
>>> Polynomial([ 1. , -0.54270529, 0.0050643 ], [-1., 1.])
>>>
>>>
>> Wrong function ;) You defined the polynomial by its coefficients. What you
>> want to do is
>>
>> In [1]: import numpy.polynomial as poly
>>
>> In [2]: p = poly.Polynomial.fromroots([1, -1.88494037, 0.0178126 ])
>>
>> In [3]: p
>> Out[3]: Polynomial([ 0.03357569, -1.90070346, 0.86712777, 1. ],
>> [-1., 1.])
>>
>> In [4]: p.roots()
>> Out[4]: array([-1.88494037, 0.0178126 , 1. ])
>>
>>
> Oh, and least squares follows the same convention:
>
> In [5]: x = linspace(-1,1,10)
>
> In [6]: y = (x - 1)*( x + 1.88494037)*(x - 0.0178126)
>
> In [7]: p = poly.Polynomial.fit(x, y, 3)
>
> In [8]: p
> Out[8]: Polynomial([ 0.03357569, -1.90070346, 0.86712777, 1. ],
> [-1., 1.])
>
>
You can also use a scaled and shifted domain, which will change the
coefficient representation but can be used to obtain a polynomial less
subject to roundoff errors.
In [18]: p = poly.Polynomial.fromroots([1, -1.88494037, 0.0178126 ],
domain=[-2,1])
In [19]: p
Out[19]: Polynomial([ 0.3187287 , -0.89681388, -0.42191482, 1. ],
[-2., 1.])
In [20]: p.roots()
Out[20]: array([-1.88494037, 0.0178126 , 1. ])
In [21]: plot(*p.linspace())
Out[21]: [<matplotlib.lines.Line2D object at 0x3ea10d0>]
I attached the plot.
Chuck
>
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