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#include <getopt.h>
#include <iomanip>
#include <random>
#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
#include "eigen/Eigen/Dense"
#include "eigen/unsupported/Eigen/CXX11/Tensor"
#include "eigen/unsupported/Eigen/CXX11/TensorSymmetry"
using Rng = randutils::random_generator<pcg32>;
using Real = double;
using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
class Tensor : public Eigen::Tensor<Real, 3> {
using Eigen::Tensor<Real, 3>::Tensor;
public:
Matrix operator*(const Vector& x) const {
std::array<Eigen::IndexPair<int>, 1> ip20 = {Eigen::IndexPair<int>(2, 0)};
const Eigen::Tensor<Real, 1> xT = Eigen::TensorMap<const Eigen::Tensor<Real, 1>>(x.data(), x.size());
const Eigen::Tensor<Real, 2> JxT = contract(xT, ip20);
return Eigen::Map<const Matrix>(JxT.data(), dimension(0), dimension(1));
}
};
Vector normalize(const Vector& x) {
return x * sqrt(x.size() / x.squaredNorm());
}
class Spherical3Spin {
private:
Tensor J;
public:
unsigned N;
Spherical3Spin(unsigned N, Rng& r) : J(N, N, N), N(N) {
Eigen::StaticSGroup<Eigen::Symmetry<0,1>, Eigen::Symmetry<1,2>> sym123;
for (unsigned i = 0; i < N; i++) {
for (unsigned j = i; j < N; j++) {
for (unsigned k = j; k < N; k++) {
sym123(J, i, j, k) = r.variate<Real, std::normal_distribution>(0, sqrt(12) / N);
}
}
}
}
std::tuple<Real, Vector, Matrix> H_∂H_∂∂H(const Vector& x) const {
Matrix ∂∂H = J * x;
Vector ∂H = ∂∂H * x / 2;
Real H = ∂H.dot(x) / 6;
return {H, ∂H, ∂∂H};
}
};
class ConstrainedHeight {
private:
Vector x0;
Real E;
public:
Spherical3Spin S;
unsigned N;
ConstrainedHeight(unsigned N, Real E, Rng& r) : x0(N), E(E), S(N, r), N(N) {
for (Real& x0ᵢ : x0) {
x0ᵢ = r.variate<Real, std::normal_distribution>();
}
x0 = normalize(x0);
}
Real overlap(const Vector& v) const {
return v.head(N).dot(x0) / N;
}
std::tuple<Vector, Matrix> ∂L_∂∂L(const Vector& v) const {
Vector x = v.head(N);
Real ω₀ = v(N);
Real ω₁ = v(N + 1);
auto [H, ∂H∂x, ∂²H∂²x] = S.H_∂H_∂∂H(x);
Vector ∂L∂x = x0 + ω₀ * x + ω₁ * ∂H∂x;
Real ∂L∂ω₀ = 0.5 * (x.squaredNorm() - N);
Real ∂L∂ω₁ = H - N * E;
Vector ∂L(N + 2);
∂L << ∂L∂x, ∂L∂ω₀, ∂L∂ω₁;
Matrix ∂L²∂x² = ω₀ * Matrix::Identity(N, N) + ω₁ * ∂²H∂²x;
Vector ∂²L∂x∂ω₀ = x;
Vector ∂²L∂x∂ω₁ = ∂H∂x;
Matrix ∂∂L(N + 2, N + 2);
∂∂L <<
∂L²∂x², ∂²L∂x∂ω₀, ∂²L∂x∂ω₁,
∂²L∂x∂ω₀.transpose(), 0, 0,
∂²L∂x∂ω₁.transpose(), 0, 0;
return {∂L, ∂∂L};
}
Vector newtonMethod(const Vector& v0, Real γ = 1) {
Vector v = v0;
Vector ∂L;
Matrix ∂∂L;
while (std::tie(∂L, ∂∂L) = ∂L_∂∂L(v), ∂L.squaredNorm() > 1e-10) {
v -= γ * ∂∂L.partialPivLu().solve(∂L);
v.head(N) = normalize(v.head(N)); // might as well stay on the sphere
}
return v;
}
};
int main(int argc, char* argv[]) {
unsigned N = 10;
double E = 0;
double γ = 1;
unsigned samples = 10;
int opt;
while ((opt = getopt(argc, argv, "N:E:g:n:")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 'E':
E = atof(optarg);
break;
case 'g':
γ = atof(optarg);
break;
case 'n':
samples = atoi(optarg);
break;
default:
exit(1);
}
}
Rng r;
Vector x = Vector::Zero(N);
x(0) = sqrt(N);
std::cout << std::setprecision(15);
for (unsigned sample = 0; sample < samples; sample++) {
Vector v0(N + 2);
v0 << x,
r.variate<Real, std::normal_distribution>(),
r.variate<Real, std::normal_distribution>();
ConstrainedHeight M(N, E, r);
Vector v = M.newtonMethod(v0, γ);
std::cout << M.overlap(v) << std::endl;
}
return 0;
}
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