The procedure sturmseq computes a Sturm sequence for the polynomial p in x.  It returns the Sturm sequence as a list of polynomials and replaces multiple roots with single roots.

The procedure sturm uses Sturm's theorem to return the number of real roots in the interval (a,b] of polynomial p in x. The first argument to sturm should be a Sturm sequence for p.  This may be computed by sturmseq. Note: The interval excludes the lower endpoint a and includes the upper endpoint b (unless it is $\mathrm{\infty }$).

While sturmseq uses some heuristics to detect zero when the input polynomial has floating point coefficients, the problem of computing the Sturm sequence is numerically ill-conditioned, so the result may be incorrect. Recomputing the input at a higher precision and increasing Digits, or converting all coefficients to rational numbers may help.

$s≔\mathrm{sturmseq}\left(\mathrm{expand}\left(\left(x-1\right)\left(x-2\right)\left(x-3\right)\right)\,x\right)$

$s≔\left[x^3-6x^2+11x-6\,x^2-4x+\frac{11}{3}\,x-2\,1\right]$

$\mathrm{sturm}\left(s\,x\,\frac{3}{2}\,4\right)$

$2$

$\mathrm{sturm}\left(s\,x\,1\,2\right)$

$1$

$\mathrm{sturm}\left(s\,x\,-\mathrm{\infty }\,\mathrm{\infty }\right)$

$3$

$\mathrm{sturmseq}\left(x^3-\mathrm{sqrt}\left(2\right)x+1\,x\right)$

$\left[x^3-\sqrt{2}x+1\,x^2-\frac{\sqrt{2}}{3}\,x-\frac{3\sqrt{2}}{4}\,\mathrm{-1}\right]$

The sturm and sturmseq commands were updated in Maple 2018.

The p and a parameters were updated in Maple 2018.