8000 Perturbations in Polynomial.roots() · Issue #14186 · numpy/numpy · GitHub
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f380cedric opened this issue Aug 2, 2019 · 3 comments
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Perturbations in Polynomial.roots() #14186

f380cedric opened this issue Aug 2, 2019 · 3 comments
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component: numpy.polynomial

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@f380cedric
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The roots are well outside of the expected values. But, that is not the case with np.roots(). Is that expected?

Reproducing code example:

import numpy as np

def roots1(coef, x, y):
    for i in range(y.size):
        coefN = coef.copy()
        coefN[0] -= y[i]
        print(np.roots(coefN[::-1]) - x[i])

def roots2(coef, x, y):
    for i in range(y.size):
        print((np.polynomial.polynomial.Polynomial(coef) - y[i]).roots() - x[i])

x = np.arange(4) * 0.1
alpha, beta, gamma = 0.16, 1, 1e-15
coef = np.array([alpha, beta, gamma])
y = alpha + beta*x + gamma*x**2
print("roots")
roots1(coef, x, y)
print("Polynomial")
roots2(coef, x, y)
roots
[-1.e+15  0.e+00]
[-1.e+15  0.e+00]
[-1.00000000e+15  2.77555756e-17]
[-1.00000000e+15  5.55111512e-17]
Polynomial
[-1.e+15  0.e+00]
[-1.0e+15  2.5e-02]
[-1.0e+15 -7.5e-02]
[-1.e+15 -5.e-02]

Numpy/Python version information:

1.16.4 3.7.4 (default, Jul 9 2019, 16:32:37)
[GCC 9.1.1 20190503 (Red Hat 9.1.1-1)]
@ilayn
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ilayn commented Aug 7, 2019

I get identical results via version numbers

NumPy : 1.18.0.dev0+4f7d5eb
Sys :  3.6.5 (v3.6.5:f59c0932b4, Mar 28 2018, 17:00:18) [MSC v.1900 64 bit (AMD64)]

@pmli
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pmli commented Aug 9, 2019

It seems the differences are due to the choice of the companion matrix form. Polynomial.roots in NumPy 1.16 and earlier uses the "standard" form, while in NumPy 1.17 and later it uses the "rotated" form (see #13202). On the other hand, np.roots seems to have always used the "rotated" form (see here).

Just as a remark, there might be more accurate methods for computing roots of polynomials. See, e.g., https://doi.org/10.1137/140983434.

@f380cedric
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Since it was solved on 1.17, I'll close.

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@rgommers @pmli @ilayn @f380cedric
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