5.32.1 Exact bounds for complex roots of a polynomial :
complexroot
complexroot takes 2 or 4 arguments : a polynomial and a real
number є and optionally two complex numbers α,β.
complexroot returns a list of vectors.

If complexroot has 2 arguments,
the elements of each vector are

either an interval (the
boundaries of this interval are the opposite vertices of a rectangle with sides
parallel to the axis and containing a complex root of the polynomial) and the
multiplicity of this root.
Let the interval be [a_{1}+ib_{1},a_{2}+ib_{2}] then a_{1}−a_{2}<є,
b_{1}−b_{2}<є and the root a+ib verifies
a_{1}≤ a ≤ a_{2} and b_{1}≤ b ≤ b_{2}.
 or the value of an exact complex root of
the polynomial and the multiplicity of this root
 If complexroot has 4 arguments, complexroot returns a list of
vectors as above, but only for the roots lying in
the rectangle with sides parallel to the axis having α,β as
opposite vertices.
To find the roots of x^{3}+1, input:
complexroot(x^
3+1,0.1)
Output :
[[1,1],[[(47*i)/8,(813*i)/16],1],[[(8+13*i)/16,(4+7*i)/8],1]]
Hence, for x^{3}+1 :

1 is a root of multiplicity 1,
 1/2+i*b is a root of multiplicity 1 with −7/8≤ b ≤
−13/16,
 1/2+i*c is a root of multiplicity 1 with 13/16≤ c ≤
7/8.
To find the roots of x^{3}+1 lying inside the rectangle
of opposite vertices −1,1+2*i, input:
complexroot(x^
3+1,0.1,1,1+2*i)
Output :
[[1,1],[[(8+13*i)/16,(4+7*i)/8],1]]