Function: polsturm
Section: polynomials
C-Name: RgX_sturmpart
Prototype: lGDG
Help: polsturm(T,{ab}): number of distinct real roots of the polynomial
 T (in the interval ab = [a,b] if present).
Doc: number of distinct real roots of the real polynomial \var{T}. If
 the argument \var{ab} is present, it must be a vector $[a,b]$ with
 two real components (of type \typ{INT}, \typ{REAL}, \typ{FRAC}
 or  \typ{INFINITY}) and we count roots belonging to that closed interval.

 If possible, you should stick to exact inputs, that is avoid \typ{REAL}s in
 $T$ and the bounds $a,b$: the result is then guaranteed and we use a fast
 algorithm (Uspensky's method, relying on Descartes's rule of sign, see
 \tet{polrootsreal}). Otherwise, the polynomial is rescaled and rounded first
 and the result may be wrong due to that initial error. If only $a$ or $b$ is
 inexact, on the other hand, the interval is first thickened using rational
 endpoints and the result remains guaranteed unless there exist a root
 \emph{very} close to a nonrational endpoint (which may be missed or unduly
 included).
 \bprog
 ? T = (x-1)*(x-2)*(x-3);
 ? polsturm(T)
 %2 = 3
 ? polsturm(T, [-oo,2])
 %3 = 2
 ? polsturm(T, [1/2,+oo])
 %4 = 3
 ? polsturm(T, [1, Pi])  \\ Pi inexact: not recommended !
 %5 = 3
 ? polsturm(T*1., [0, 4])  \\ T*1. inexact: not recommended !
 %6 = 3
 ? polsturm(T^2, [0, 4])  \\ not squarefree: roots are not repeated!
 %7 = 3
 @eprog
Variant: Also available is
 \fun{long}{sturm}{GEN T} (for the case \kbd{ab = NULL}). The function
 \fun{long}{sturmpart}{GEN T, GEN a, GEN b} is obsolete and deprecated.
