Exact bounds for complex roots of a polynomial :
complexroot

complexroot takes 2 or 4 arguments : a polynomial and a real number $\epsilon$ and optionnally two complex numbers $\alpha$,$\beta$.
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 [a₁ + ib₁, a₂ + ib₂] then | a₁ - a₂| < $\epsilon$, | b₁ - b₂| < $\epsilon$ and the root a + ib verifies a₁ $\leq$ a $\leq$ a₂ and b₁ $\leq$ b $\leq$ b₂.
  • 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 $\alpha$,$\beta$ as opposite vertices.
  

To find the roots of x³ + 1, input:

complexroot(x^3+1,0.1)

Output :

[[-1,1],[[(4-7*i)/8,(8-13*i)/16],1],[[(8+13*i)/16,(4+7*i)/8],1]]

Hence, for x³ + 1 :

• -1 is a root of multiplicity 1,
• 1/2+i*b is a root of multiplicity 1 with -7/8 $\leq$ b $\leq$ - 13/16,
• 1/2+i*c is a root of multiplicity 1 with 13/1 $\leq$ c $\leq$ 7/8.

To find the roots of x³ + 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]]