11.3.6  Exact bounds for complex roots of a polynomial

The complexroot command finds bounds for the complex roots of a polynomial.

• complexroot takes two mandatory arguments and two optional arguments:
  • P, a polynomial.
  • ε, a postive real number.
  • Optionally, α, β, two complex numbers.
• complexroot(P,ε) returns a list of vectors, where the elements of each vector are one of:
  • 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.

    Suppose the interval is [a₁+ib₁,a₂+ib₂] then |a₁−a₂|<ε, |b₁−b₂|<ε and the root a+ib satisfies a₁≤ a ≤ a₂ and b₁≤ b ≤ b₂.

  • the value of an exact complex root of the polynomial and the multiplicity of this root.
• complexroot(P,ε,α,β) 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.

Examples

Find the roots of x³+1.

Hence, for x³+1, −1 is a root of multiplicity 1, a+ib is a root of multiplicity 1 with 0.499999046325680 ≤ a ≤ 0.500000953674320 and −0.866026401519779 ≤ b ≤ −0.866024494171135, and c+id is a root of multiplicity 1 with 0.499999046325680 ≤ c ≤ 0.500000953674320 and 0.866024494171135 ≤ d ≤ 0.866026401519779.

Find the roots of x³+1 lying inside the rectangle with opposite vertices −1,1+2i.