numpy.polynomial.polynomial.polyfromroots

numpy.polynomial.polynomial.polyfromroots(roots) [source]

Генерировать мономический полином с заданными корнями.

Возвращает коэффициенты полинома

<img alt="p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n)," src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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где r_n — это корни, указанные в roots. Если ноль имеет кратность n, то он должен появляться в roots n раз. Например, если 2 — корень кратности три, а 3 — корень кратности 2, то roots выглядит примерно так [2, 2, 2, 3, 3]. Корни могут появляться в любом порядке.

Если возвращаемые коэффициенты — c, то

<img alt="p(x) = c_0 + c_1 * x + ... + x^n" src="data:image/svg+xml;base64,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Коэффициент последнего члена равен 1 для мономических многочленов в этом виде.

Параметры:	roots : array_like Последовательность, содержащая корни.
Возвращает:	out : ndarray Одномерный массив коэффициентов многочлена. Если все корни вещественные, то out также вещественный, в противном случае он комплексный. (см. примеры ниже).

См. также

chebfromroots, legfromroots, lagfromroots, hermfromroots, hermefromroots

Примечания

Коэффициенты определяются перемножением линейных множителей вида (x - r_i), т.е.

<img alt="p(x) = (x - r_0) (x - r_1) ... (x - r_n)" src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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где n == len(roots) - 1; обратите внимание, что это подразумевает, что 1 всегда возвращается для .

Примеры

>>> from numpy.polynomial import polynomial as P
>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
array([ 0., -1.,  0.,  1.])
>>> j = complex(0,1)
>>> P.polyfromroots((-j,j)) # complex returned, though values are real
array([ 1.+0.j,  0.+0.j,  1.+0.j])