The command realroot(poly, widthgoal) returns a list of isolating intervals for all real roots of the univariate polynomial poly. The width of the interval is less than or equal to the optional parameter widthgoal, a positive number.  If widthgoal is omitted, the most convenient width is used for each interval returned.

Each interval is returned as a list of two rational numbers. The list $\left[a\&comma;a\right]$ represents the single point $a$.  The list $\left[a\&comma;b\right]$ with $a<b$ represents the open real interval defined by $a$ and $b$. Multiplicity information is not included. The denominators of $a$ and $b$ are integer powers of $2$.

The intervals are sorted in ascending order.

The algorithm behind realroot uses Descartes' rule of signs.

$\mathrm{realroot}\left(x^8+5x^7-4x^6-20x^5+4x^4+20x^3\&comma;\frac{1}{1000}\right)$

$\left[\left[\mathrm{-5}\&comma;\mathrm{-5}\right]\&comma;\left[-\frac{6074001001}{4294967296}\&comma;-\frac{24296003995}{17179869184}\right]\&comma;\left[0\&comma;0\right]\&comma;\left[\frac{185363}{131072}\&comma;\frac{92683}{65536}\right]\right]$

Loos and Collins, Computer Algebra, B. Buchberger, ed.