6.29.2  Exact bounds for complex roots of a polynomial: complexroot

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

• complexroot takes two mandatory arguments and two optional arguments:
  • P, a polynomial.
  • є, a postive real number.
  • Optionally, α, β, two complex numbers.
• complexroot(P,є) 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|<є, |b₁−b₂|<є 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,є,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.
  Input:
  complexroot(x^3+1,0.1)
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

  −1 1 ⎡ ⎣ 0.499999046325680..0.500000953674320 ⎤ ⎦ − ⎡ ⎣ 0.866024494171135..0.866026401519779 ⎤ ⎦ i 1 ⎡ ⎣ 0.499999046325680..0.500000953674320 ⎤ ⎦ + ⎡ ⎣ 0.866024494171135..0.866026401519779 ⎤ ⎦ i 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.
  • 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+2*i.
  Input:
  complexroot(x^3+1,0.1,-1,1+2*i)
  Output:

  ⎡ ⎢ ⎢ ⎣

  −1 1 ⎡ ⎣ 0.499999046325680..0.500000953674320 ⎤ ⎦ + ⎡ ⎣ 0.866024494171135..0.866026401519779 ⎤ ⎦ i 1

  ⎤ ⎥ ⎥ ⎦