Exact bounds for real roots of a polynomial : realroot

realroot has 2 or 4 arguments : a polynomial and a real number $\epsilon$ and optionnally two reals numbers $\alpha$,$\beta$.
realroot returns a list of vectors.

• If realroot has 2 arguments, the elements of each vector are
  • either a real interval containing a real root of the polynomial and the multiplicity of this root. Let the interval be [a₁, a₂] then | a₁ - a₂| < $\epsilon$ and the root a verifies a₁ $\leq$ a $\leq$ a₂.
  • or the value of an exact real root of the polynomial and the multiplicity of this roots
• If realroot has 4 arguments, realroot returns a list of vectors as above, but only for the roots inside the interval [$\alpha$,$\beta$].

To find the real roots of x³ + 1, input:

realroot(x^3+1, 0.1)

Output :

[[-1,1]]

To find the real roots of x³ - x² - 2x + 2, input:

realroot(x^3-x^2-2*x+2, 0.1)

Output :

[[1,1],[[(-3)/2,(-45)/32],1],[[45/32,3/2],1]]

To find the real roots of x³ - x² - 2x + 2 in the interval [0;2], input:

realroot(x^3-x^2-2*x+2, 0.1,0,2)

Output :

[[1,1],[[11/8,23/16],1]]