// 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;
}
MatrixWrapper::SymmetricMatrix
NonLinearAnalyticConditionalGaussian_Ginac::CovarianceGet() const
{
if (cond_size!=0)
{
MatrixWrapper::ColumnVector u_num (u_size);
MatrixWrapper::ColumnVector x_num (x_size);
GiNaC::ex substitute (func_size);
MatrixWrapper::Matrix D (func_size,cond_size);
u_num = ConditionalArgumentGet(1);
x_num = ConditionalArgumentGet(0);
for (unsigned int i=0; i<cond_size; i++)
{
// temp variable to substitute in
substitute = dfunc_dcond[i];
// substitute all u_sym with u_num
for (unsigned int j=0; j<u_size; j++)
substitute = substitute.subs( u_sym[j]==u_num(j+1) );
// substitute all x_sym with x_num
for (unsigned int j=0; j<x_size; j++)
substitute = substitute.subs( x_sym[j]==x_num(j+1) );
// convert substitute back to matrix
GiNaC::matrix substitute_matrix = GiNaC::ex_to<GiNaC::matrix>(substitute);
// build matrix D
for (unsigned int j=0; j<func_size; j++)
D(j+1,i+1) = GiNaC::ex_to<GiNaC::numeric>( substitute_matrix(j,0).evalf() ).to_double();
}
//cout << "D: " << D << endl;
//cout << "CondCov:\n" << (Matrix)cond_covariance << endl;
MatrixWrapper::Matrix temp = D * (MatrixWrapper::Matrix)AdditiveNoiseSigmaGet() * D.transpose();
// convert func_covariance_matrix to symmetric matrix
MatrixWrapper::SymmetricMatrix additiveNoise(temp.rows());
temp.convertToSymmetricMatrix(additiveNoise);
return additiveNoise;
}
else
{
return AdditiveNoiseSigmaGet();
}
}
Expr Expr::subs(std::vector<std::pair<Expr, Expr> > Subs) {
GiNaC::ex Ex = Expr_;
for (auto& E : Subs) {
//dbgs() << "Replacing " << E.first.getExpr() << " with " << E.second.getExpr() << " in " << Ex;
Ex = Ex.subs(E.first.getExpr() == E.second.getExpr());
//dbgs() << " giving " << Ex << "\n";
}
return Expr(Ex);
}
const flattened_tensor& CovariantRiemannB3Cache::get()
{
if(this->B3) return *this->B3;
auto args = res.generate_cache_arguments(printer);
this->B3 = std::make_unique<flattened_tensor>(res.fl.get_flattened_size<field_index>(RESOURCE_INDICES::RIEMANN_B3_INDICES));
const auto max = res.share.get_max_field_index(variance::covariant);
const auto max_l = res.share.get_max_field_index(variance::contravariant);
SubstitutionMapCache subs_cache(res, printer);
DerivativeSymbolsCache deriv_cache(res, res.share, printer);
for(field_index i = field_index(0, variance::covariant); i < max; ++i)
{
for(field_index j = field_index(0, variance::covariant); j < max; ++j)
{
for(field_index k = field_index(0, variance::covariant); k < max; ++k)
{
unsigned int index = res.fl.flatten(i,j,k);
GiNaC::ex subs_expr = 0;
if(!res.cache.query(expression_item_types::Riemann_B3_item, index, args, subs_expr))
{
timing_instrument timer(res.compute_timer);
auto& deriv_syms = deriv_cache.get();
GiNaC::ex expr = 0;
for(field_index l = field_index(0, variance::contravariant); l < max_l; ++l)
{
auto Rie_ijk = (*res.Rie_T)(static_cast<unsigned int>(k), static_cast<unsigned int>(i),
static_cast<unsigned int>(j), static_cast<unsigned int>(l));
auto Rie_ikj = (*res.Rie_T)(static_cast<unsigned int>(k), static_cast<unsigned int>(j),
static_cast<unsigned int>(i), static_cast<unsigned int>(l));
auto Rie_sym = (Rie_ijk + Rie_ikj) / 2;
expr += Rie_sym * deriv_syms[res.fl.flatten(l)];
}
// get substitution map
GiNaC::exmap& subs_map = subs_cache.get();
subs_expr = expr.subs(subs_map, GiNaC::subs_options::no_pattern);
res.cache.store(expression_item_types::Riemann_B3_item, index, args, subs_expr);
}
(*this->B3)[index] = subs_expr;
}
}
}
return *this->B3;
}
MatrixWrapper::ColumnVector
NonLinearAnalyticConditionalGaussian_Ginac::ExpectedValueGet() const
{
MatrixWrapper::ColumnVector u_num (u_size);
MatrixWrapper::ColumnVector x_num (x_size);
MatrixWrapper::ColumnVector func_num(func_size);
GiNaC::ex substitute (func_size);
MatrixWrapper::ColumnVector expected(func_size);
u_num = ConditionalArgumentGet(1);
x_num = ConditionalArgumentGet(0);
// use Mu of additive noise
if (cond_size!=0)
for (unsigned int i=0; i<u_size; i++)
for (unsigned int j=0; j<cond_size; j++)
if (u_sym[i] == cond_sym[j])
u_num(i+1) += (this->AdditiveNoiseMuGet())(j+1);
// evaluate func
for (unsigned int i=0; i<func_size; i++)
{
// temp variable to substitute in
substitute = func_sym(i,0);
// substitute all u_sym with u_num
for (unsigned int j=0; j<u_size; j++)
substitute = substitute.subs( u_sym[j]==u_num(j+1) );
// substitute all x_sym with x_num
for (unsigned int j=0; j<x_size; j++)
substitute = substitute.subs( x_sym[j]==x_num(j+1) );
// build matrix func_num
func_num(i+1) = GiNaC::ex_to<GiNaC::numeric>( substitute.evalf() ).to_double();
}
expected = func_num;
if (cond_size==0)
expected += AdditiveNoiseMuGet();
return expected;
}
// NegativeOrdinate
// Returns the ranges for which the ordinate is negative.
vector<pair<GiNaC::ex, GiNaC::ex> >
NegativeOrdinate(vector<double> Vec, GiNaC::symbol Sym, GiNaC::ex Ex) {
vector<pair<GiNaC::ex, GiNaC::ex> > Negs;
// No real roots - it's either all positive or all negative.
if (Vec.empty()) {
GiNaC::ex Subs = Ex.subs(Sym == 0);
if (GiNaC::is_a<GiNaC::numeric>(Subs) &&
GiNaC::ex_to<GiNaC::numeric>(Subs).is_negative())
Negs.push_back(make_pair(GiNaC::inf(-1), GiNaC::inf(1)));
return Negs;
}
sort(Vec.begin(), Vec.end());
GiNaC::ex FirstSubs = Ex.subs(Sym == GiNaC::numeric(Round(Vec[0] - 1.0f)));
// Negative at -inf to the first root minus one.
if (GiNaC::is_a<GiNaC::numeric>(FirstSubs) &&
GiNaC::ex_to<GiNaC::numeric>(FirstSubs).is_negative())
Negs.push_back(make_pair(GiNaC::inf(-1), GiNaC::numeric(Vec[0] - 1.0f)));
for (unsigned Idx = 1; Idx < Vec.size(); ++Idx) {
long int E = Round(Vec[Idx]);
// Check if it's negative to the right.
GiNaC::ex Subs = Ex.subs(Sym == GiNaC::numeric(E + 1));
if (GiNaC::is_a<GiNaC::numeric>(Subs))
if (GiNaC::ex_to<GiNaC::numeric>(Subs).is_negative()) {
// Last root, it's negative all the way to +inf.
if (Idx == Vec.size() - 1)
Negs.push_back(make_pair(GiNaC::numeric(E + 1), GiNaC::inf(1)));
// Not the last root, it's negative from this root to the next.
else
Negs.push_back(make_pair(GiNaC::numeric(E + 1),
GiNaC::numeric(Round(Vec[Idx + 1] - 1))));
}
}
return Negs;
}
MatrixWrapper::Matrix
NonLinearAnalyticConditionalGaussian_Ginac::dfGet(unsigned int i) const
{
// Check if i = 0, since this is the old df_dxGet method!
assert(i == 0);
// evaluate function
MatrixWrapper::ColumnVector u_num (u_size);
MatrixWrapper::ColumnVector x_num (x_size);
GiNaC::ex substitute (func_size);
MatrixWrapper::Matrix F (func_size, x_size);
u_num = ConditionalArgumentGet(1);
x_num = ConditionalArgumentGet(0);
// numeric evaluation of derivative: dfunc_dx = F
for (unsigned int i=0; i<x_size; i++)
{
// temp variable to substitute in
substitute = dfunc_dx[i];
// substitute all u_sym with u_num
for (unsigned int j=0; j<u_size; j++)
substitute = substitute.subs( u_sym[j]==u_num(j+1) );
// substitute all x_sym with x_num
for (unsigned int j=0; j<x_size; j++)
substitute = substitute.subs( x_sym[j]==x_num(j+1) );
// convert substitute to matrix. Now all elements in matrix are accessible
GiNaC::matrix substitute_matrix = GiNaC::ex_to<GiNaC::matrix>(substitute);
// build matrix F
for (unsigned int j=0; j<func_size; j++)
F(j+1,i+1) = GiNaC::ex_to<GiNaC::numeric>( substitute_matrix(j,0).evalf() ).to_double();
}
return F;
}
double eval_at(const math::ex &expr, double x_val, double y_val)
{
math::ex temp = expr.subs(x == x_val).subs(y == y_val).evalf();
return math::ex_to<math::numeric>( temp ).to_double();
}