Theorem: Let and be the Sturm sequence of and (x). Let be the number of sign variation of for . Then, for any , is the number of roots in ]a,b[. 
This leads to the following schema of algorithms, (which applies event to polynomials with multiple roots):
Algorithm: Isolation of the roots of on the interval . INPUT: A polynomial .

SlvSturm
. The default value is . If we want a specific precision eps
, we can use the following construction: solve(p, SlvSturm<O,C>(eps))
.O
, corresponding to the first parameter O
of this class.C
is the type of coefficients used during the computation. The default type is the type QQ
of rational numbers. \p
of type UPOL
, is supposed to be given with exact coefficients. It is converted to a polynomial of type UPolDse<C>
, if needed.synaps/usolve/sturm/SlvSturm.h
.Seq< Interval<O> > solve(const UPOL& p, IslSturm<O,C> mth);
O
, containing one and only one root of p.
C
is the type of coefficients used during the computation. The default type is the type QQ
of rational numbers. \p
of type UPOL
, is supposed to be given with exact coefficients. It is converted to a polynomial of type UPolDse<C>
, if needed.synaps/usolve/sturm/IslSturm.h
.#include <synaps/upol.h> #include <synaps/arithm/QQ.h> #include <synaps/usolve/sturm/IslSturm.h> int main() { using std::cout; using std::endl; UPolDse<QQ> P("2*x^7+3*x^6+2*x^5+x^4x^32*x^23*x+1"); cout <<"Testing: "<<P<<endl; // Testing: 2*x^7+3*x^6+2*x^5+1*x^41*x^32*x^23*x+1 cout <<solve(P,IslSturm<QQ>())<<endl; // [ 5/2,0 ], [ 0,5/8 ], [ 5/8,5/4 ] cout <<solve(P,IslSturm<QQ>(),0,1)<<endl; // [ 0,1/2 ], [ 1/2,1 ] }