Document

polyfromroots(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 :None:None:`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.

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 $a_n$ .

Parameters

roots : array_like: Sequence containing the roots.

Returns

out : ndarray: 1-D array of the polynomial's coefficients If all the roots are real, then :None:None:`out` is also real, otherwise it is complex. (see Examples below).

Generate a monic polynomial with given roots.

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])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

numpy.polynomial.hermite.hermfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.hermite_e.hermefromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.polynomial.polyvalfromroots numpy.polynomial.laguerre.lagfromroots

Local connectivity graph

Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.

Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)

SVG is more flexible but power hungry; and does not scale well to 50 + nodes.

All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them