numpy.polynomial.polynomial.polyfromroots

polynomial.polynomial.polyfromroots(roots)

Generate a monic polynomial with given roots.

Return the coefficients of the polynomial

\[p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),\]

where the r_n are the roots specified in roots. If a zero has multiplicity n, then it must appear in roots n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then roots looks something like [2, 2, 2, 3, 3]. The roots can appear in any order.

If the returned coefficients are c, then

\[p(x) = c_0 + c_1 * x + ... + x^n\]

The coefficient of the last term is 1 for monic polynomials in this form.

Parameters:

rootsarray_like

Sequence containing the roots.

Returns:

outndarray

1-D array of the polynomial’s coefficients If all the roots are real, then out is also real, otherwise it is complex. (see Examples below).

See also

numpy.polynomial.chebyshev.chebfromroots

numpy.polynomial.legendre.legfromroots

numpy.polynomial.laguerre.lagfromroots

numpy.polynomial.hermite.hermfromroots

numpy.polynomial.hermite_e.hermefromroots

Notes

The coefficients are determined by multiplying together linear factors of the form (x - r_i), i.e.

\[p(x) = (x - r_0) (x - r_1) ... (x - r_n)\]

where n == len(roots) - 1; note that this implies that 1 is always returned for \(a_n\).

Examples

>>> 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])