A real algebraic number is a real root of a polynomial, whose coefficients are integers. We represent such a real algebraic number, by a square-free polynomial and an isolating interval, that is an interval with rational endpoints, which contains only one root of the polynomial:

`NT`

, corrresponding to the output type used to represent the algebraic numbers. It containts the following definitions: template < typename NT > struct Algebraic { typedef NumberTraits<NT> NTR; // Number type traits typedef typename NTR::RT RT; // Ring number type (typically typedef typename NTR::FT FT; // Field number type (typically typedef typename NTR::XT XT; // Approximate number type // (typically double) typedef typename NTR::XIT RIT; // Ring interval type typedef typename NTR::FIT FIT; // Field interval type typedef typename NTR::XIT XIT; // Approximate interval type typedef UPolDse<RT> Poly; // Univariate polynomial type typedef UPolDse<RT> RT_Poly; // Ring polynomial type typedef UPolDse<FT> FT_Poly; // Field polynomial type typedef UPolDse<XT> XT_Poly; // Approximate polynomial type typedef UPolDse<RL> RL_Poly; // Real polynomial type typedef ALGEBRAIC::root_of<RT, Poly> root_t; // The root_of type };

**See also:**`synaps/usolve/Algebraic.h`

.

Seq< Algebraic<ZZ>::root_t > solve(const UPolDse<ZZ> & P, Algebraic<ZZ> mth);

`P`

. If one is interested by a specific root, one can also use the following instructions: Algebraic<ZZ>::root_t solve(const UPolDse<ZZ> & P, Algebraic<ZZ> mth, int i);

`P`

, if it exists. Otherwise, it produces an error. For polynomials of degree up to 4, we use precomputed discrimination systems in order to determine the square-free polynomial and we compute the isolating interval as function in the coefficients of the polynomial.For polynomials of higher degree, we use a Sturm-like solver in order to isolate the roots of the polynomial.

- comparison: operators
`<,>, <=, >=, ==, !=`

- sign of a polynomial at a real algebraic number:
`sign_at(P,alpha)`

.

#include <synaps/upol.h> #include <synaps/arithm/gmp.h> #include <synaps/usolve/Algebraic.h> typedef Algebraic<ZZ>::root_t root_t; int main() { using std::cout; using std::endl; UPolDse<ZZ> p1("2*x^7+3*x^6+2*x^5+1*x^4-1*x^3-2*x^2-3*x+1"), p2("x^4-3*x^3+2*x^3-2*x^2-5*x+10"), p3("x^4-3*x^3+2*x^3-2*x^2-5*x+9"); root_t alpha= solve(p1,Algebraic<ZZ>(),1), beta = solve(p2,Algebraic<ZZ>(),0); cout<<"alpha: "<<alpha<<endl; //| root_of( 2*x^7+3*x^6+2*x^5+1*x^4-1*x^3-2*x^2-3*x+1, 1) //| in [ 0,5/8 ], approx = 0.278212 cout<<"beta : "<<beta <<endl; //| root_of( 1*x^4-1*x^3-2*x^2-5*x+10, 0 ) //| in [ 0,559/342 ], approx = 1.43343 cout<<(alpha<beta)<<endl; //| 1 cout <<sign_at(p1,beta)<<endl; //| 1 cout <<sign_at(p2,beta)<<endl; //| 0 cout <<sign_at(p3,beta)<<endl; //| -1 }