// GetCoeffs
vector<GiNaC::ex> GetCoeffs(GiNaC::ex Ex, GiNaC::symbol Sym) {
assert(Ex.is_polynomial(Sym));
vector<GiNaC::ex> Coeffs;
unsigned Degree = Ex.degree(Sym);
for (unsigned Idx = 0; Idx <= Degree; ++Idx)
Coeffs.push_back(Ex.coeff(Sym, Idx));
return Coeffs;
}
// Solve
vector<double> Solve(GiNaC::ex Ex, GiNaC::symbol Sym) {
vector<double> Roots;
unsigned Degree = Ex.degree(Sym);
auto Coeffs = GetCoeffs(Ex, Sym);
// Bhaskara.
if (Degree == 2) {
GiNaC::ex A = Coeffs[2];
GiNaC::ex B = Coeffs[1];
GiNaC::ex C = Coeffs[0];
GiNaC::ex Delta = B*B - 4 * A * C;
// Guaranteed real roots.
if (GiNaC::is_a<GiNaC::numeric>(Delta) &&
!GiNaC::ex_to<GiNaC::numeric>(Delta).is_negative()) {
GiNaC::ex Delta = GiNaC::sqrt(B*B - 4 * A * C).evalf();
GiNaC::ex One = ((-B) + Delta)/(2*A);
GiNaC::ex Two = ((-B) - Delta)/(2*A);
if (GiNaC::is_a<GiNaC::numeric>(One))
Roots.push_back(GiNaC::ex_to<GiNaC::numeric>(One.evalf()).to_double());
if (GiNaC::is_a<GiNaC::numeric>(Two))
Roots.push_back(GiNaC::ex_to<GiNaC::numeric>(Two.evalf()).to_double());
}
}
// Cardano.
else if (Degree == 3) {
GiNaC::ex A = Coeffs[3];
GiNaC::ex B = Coeffs[2];
GiNaC::ex C = Coeffs[1];
GiNaC::ex D = Coeffs[1];
GiNaC::ex Delta0 = B*B - 3 * A * C;
GiNaC::ex Delta1 = 2 * B*B*B - 9 * A * B * C + 27 * A * A * D;
GiNaC::ex CD = Delta1 + GiNaC::sqrt(Delta1 * Delta1 - 4 * GiNaC::pow(Delta0, 3));
CD = CD/2;
CD = GiNaC::pow(CD, GiNaC::numeric(1)/3);
GiNaC::symbol U("u");
GiNaC::ex Var = GiNaC::numeric(-1)/(3 * A) * (B + U * CD + Delta0/(U * CD));
GiNaC::ex One = Var.subs(U == 1);
GiNaC::ex Two = Var.subs(U == ((-1 + GiNaC::sqrt(GiNaC::numeric(-3)))/2));
GiNaC::ex Three = Var.subs(U == ((-1 - GiNaC::sqrt(GiNaC::numeric(-3)))/2));
if (GiNaC::is_a<GiNaC::numeric>(One))
Roots.push_back(GiNaC::ex_to<GiNaC::numeric>(One.evalf()).to_double());
if (GiNaC::is_a<GiNaC::numeric>(Two))
Roots.push_back(GiNaC::ex_to<GiNaC::numeric>(Two.evalf()).to_double());
if (GiNaC::is_a<GiNaC::numeric>(Three))
Roots.push_back(GiNaC::ex_to<GiNaC::numeric>(Three.evalf()).to_double());
}
return Roots;
}
bool TestExpr::runOnModule(Module&) {
Expr A("a");
Expr B("b");
assert(A + B == B + A);
assert(A * B == B * A);
assert(A - B != B - A);
assert(A / B != B / A);
//map<GiNaC::symbol, int> GetMaxExps(GiNaC::ex Ex);
GiNaC::symbol GA("a");
GiNaC::symbol GB("b");
GiNaC::ex EZ = 5 * GA * GA * GA + 4 * GB + 1;
assert(EZ.is_polynomial(GA));
assert(EZ.degree(GA) == 3);
assert(EZ.coeff(GA, 3) == GiNaC::numeric(5));
assert(EZ.coeff(GA, 2) == GiNaC::numeric(0));
assert(EZ.coeff(GA, 1) == GiNaC::numeric(0));
assert(EZ.coeff(GA, 0) == 4 * GB + 1);
assert(EZ.is_polynomial(GB));
assert(EZ.degree(GB) == 1);
assert(EZ.coeff(GB, 1) == GiNaC::numeric(4));
assert(EZ.coeff(GB, 0) == 5 * GA * GA * GA + 1);
auto SEZ = GetSymbols(EZ);
assert(SEZ.size() == 2);
assert(SEZ.count(GA) && SEZ.count(GB));
auto CEZGA = GetCoeffs(EZ, GA);
assert(CEZGA.size() == 4);
assert(CEZGA[3] == GiNaC::numeric(5));
assert(CEZGA[2] == GiNaC::numeric(0));
assert(CEZGA[1] == GiNaC::numeric(0));
assert(CEZGA[0] == 4 * GB + 1);
auto CEZGB = GetCoeffs(EZ, GB);
assert(CEZGB.size() == 2);
assert(CEZGB[1] == GiNaC::numeric(4));
assert(CEZGB[0] == 5 * GA * GA * GA + 1);
auto EF = GA * GA - 1;
auto EFSolve = Solve(EF, GA);
assert(EFSolve.size() == 2);
assert(Round(EFSolve[0]) == -1 || Round(EFSolve[1]) == -1);
assert(Round(EFSolve[0]) == 1 || Round(EFSolve[1]) == 1);
auto EFNegs = NegativeOrdinate(EFSolve, GA, EF);
assert(EFNegs.empty());
EF = GA * GA - 2 * GA + 1;
EFSolve = Solve(EF, GA);
assert(EFSolve.size() == 2);
assert(Round(EFSolve[0]) == 1 && Round(EFSolve[1]) == 1);
EFNegs = NegativeOrdinate(EFSolve, GA, EF);
assert(EFNegs.empty());
EF = (-1) * GA * GA + GA - 1;
EFSolve = Solve(EF, GA);
assert(EFSolve.size() == 0);
EFNegs = NegativeOrdinate(EFSolve, GA, EF);
assert(EFNegs.size() == 1);
assert(EFNegs[0].first.is_equal(GiNaC::inf(-1)));
assert(EFNegs[0].second.is_equal(GiNaC::inf(1)));
return false;
}