[Numpy-discussion] polynomial fromroots

[josef.pktd at gmail.com]

On Sat, Oct 9, 2010 at 10:01 PM, Charles R Harris
<charlesr.harris at gmail.com> wrote:
>
>
> On Sat, Oct 9, 2010 at 7:47 PM, <josef.pktd at 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

My coefficients are from a lag-polynomial in time series analysis
(ARMA), and they really are the (estimated) coefficients. It is
essentially the same as the model for scipy.signal.lfilter.
I just need to check the roots to see whether the process is
stationary and invertible. If one of the two lag-polynomials (moving
average) has roots on the wrong side of the unit circle, then I can
invert them.

I'm coding from memory of how this is supposed to work, so maybe I'm
back to RTFM and RTFTB (TB=text book).

(I think what I really would need is a z-transform, but I don't have
much of an idea how to do this on a computer)

Thanks, the main thing I need to do is check the convention or
definition for the normalization. And as btw, I like that the coef are
in increasing order
e.g. seasonal differencing multiplied with 1 lag autoregressive
poly.Polynomial([1.,0,0,-1])*poly.Polynomial([1,0.8])

(I saw your next message: Last time I played with function
approximation,  I didn't figure out what the basis does, but it worked
well without touching it)

Josef

>
> 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.        ])
>
> Chuck
>
>
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