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#pragma once
#include <eigen3/Eigen/Dense>
#include <iterator>
#include "types.hpp"
#include "tensor.hpp"
#include "factorial.hpp"
template <typename Derived>
Vector<typename Derived::Scalar> normalize(const Eigen::MatrixBase<Derived>& z) {
return z * sqrt((Real)z.size() / (typename Derived::Scalar)(z.transpose() * z));
}
template <class Scalar, int p>
std::tuple<Scalar, Vector<Scalar>, Matrix<Scalar>> hamGradHess(const Tensor<Scalar, p>& J, const Vector<Scalar>& z) {
Matrix<Scalar> Jz = contractDown(J, z); // Contracts J into p - 2 copies of z.
Vector<Scalar> Jzz = Jz * z;
Scalar Jzzz = Jzz.transpose() * z;
Real pBang = factorial(p);
Matrix<Scalar> hessian = ((p - 1) * p / pBang) * Jz;
Vector<Scalar> gradient = (p / pBang) * Jzz;
Scalar hamiltonian = Jzzz / pBang;
return {hamiltonian, gradient, hessian};
}
template <class Scalar, int p>
Scalar getHamiltonian(const Tensor<Scalar, p>& J, const Vector<Scalar>& z) {
Scalar H;
std::tie(H, std::ignore, std::ignore) = hamGradHess(J, z);
return H;
}
template <class Scalar, int p>
Vector<Scalar> getGradient(const Tensor<Scalar, p>& J, const Vector<Scalar>& z) {
Vector<Scalar> dH;
std::tie(std::ignore, dH, std::ignore) = hamGradHess(J, z);
return dH;
}
template <class Scalar, int p>
Matrix<Scalar> getHessian(const Tensor<Scalar, p>& J, const Vector<Scalar>& z) {
Matrix<Scalar> ddH;
std::tie(std::ignore, std::ignore, ddH) = hamGradHess(J, z);
return ddH;
}
template <class Scalar>
Real getThresholdEnergyDensity(unsigned p, Scalar κ, Scalar ε, Real a) {
Real apm2 = pow(a, p - 2);
Scalar δ = κ / apm2;
Real θ = arg(κ) + 2 * arg(ε);
return (p - 1) * apm2 / (2 * p) * pow(1 - norm(δ), 2) / (1 + norm(δ) - 2 * abs(δ) * cos(θ));
}
template <class Scalar, int p>
Real getProportionOfThreshold(Scalar κ, const Tensor<Scalar, p>& J, const Vector<Scalar>& z) {
Real N = z.size();
Scalar ε = getHamiltonian(J, z) / N;
Real a = z.squaredNorm() / N;
return norm(ε) / getThresholdEnergyDensity(p, κ, ε, a);
}
template <class Scalar>
Vector<Scalar> zDot(const Vector<Scalar>& z, const Vector<Scalar>& dH) {
return -dH.conjugate() + (dH.dot(z) / z.squaredNorm()) * z.conjugate();
}
template <class Scalar, int p>
std::tuple<Real, Vector<Scalar>> WdW(const Tensor<Scalar, p>& J, const Vector<Scalar>& z) {
Vector<Scalar> dH;
Matrix<Scalar> ddH;
std::tie(std::ignore, dH, ddH) = hamGradHess(J, z);
Vector<Scalar> dzdt = zDot(z, dH);
Real a = z.squaredNorm();
Scalar A = (Scalar)(z.transpose() * dzdt) / a;
Scalar B = dH.dot(z) / a;
Real W = dzdt.squaredNorm();
Vector<Scalar> dW = ddH * (dzdt - A * z.conjugate())
+ 2 * (conj(A) * B * z).real()
- conj(B) * dzdt - conj(A) * dH.conjugate();
return {W, dW};
}
template <class Scalar>
Matrix<Scalar> dzDot(const Vector<Scalar>& z, const Vector<Scalar>& dH) {
Real z² = z.squaredNorm();
return (dH.conjugate() - (dH.dot(z) / z²) * z.conjugate()) * z.adjoint() / z²;
}
template <class Scalar>
Matrix<Scalar> dzDotConjugate(const Vector<Scalar>& z, const Vector<Scalar>& dH, const Matrix<Scalar>& ddH) {
Real z² = z.squaredNorm();
return -ddH + (ddH * z.conjugate()) * z.transpose() / z²
+ (z.dot(dH) / z²) * (
Matrix<Scalar>::Identity(ddH.rows(), ddH.cols()) - z.conjugate() * z.transpose() / z²
);
}
template <class Scalar>
Tensor<Scalar, 3> ddzDot(const Vector<Scalar>& z, const Vector<Scalar>& dH) {
Tensor<Scalar, 1> zT = Eigen::TensorMap<Tensor<const Scalar, 1>>(z.data(), {z.size()});
Tensor<Scalar, 1> dHT = Eigen::TensorMap<Tensor<const Scalar, 1>>(dH.data(), {dH.size()});
Eigen::array<Eigen::IndexPair<int>, 0> ei = {};
Scalar z² = z.squaredNorm();
return - zT.conjugate().contract(dHT.conjugate(), ei).contract(zT.conjugate(), ei) / pow(z², 2)
- dHT.conjugate().contract(zT.conjugate(), ei).contract(zT.conjugate(), ei) / pow(z², 2)
+ zT.conjugate().contract(zT.conjugate(), ei).contract(zT.conjugate(), ei) * (2.0 * dH.dot(z) / pow(z², 3));
}
template <class Scalar>
Tensor<Scalar, 3> dcdzDot(const Vector<Scalar>& z, const Vector<Scalar>& dH, const Matrix<Scalar>& ddH) {
Tensor<Scalar, 1> zT = Eigen::TensorMap<Tensor<const Scalar, 1>>(z.data(), {z.size()});
Tensor<Scalar, 1> dHT = Eigen::TensorMap<Tensor<const Scalar, 1>>(dH.data(), {dH.size()});
Tensor<Scalar, 2> ddHT = Eigen::TensorMap<Tensor<const Scalar, 2>>(ddH.data(), {dH.size(), dH.size()});
Matrix<Scalar> I = Matrix<Real>::Identity(z.size(), z.size());
Tensor<Scalar, 2> IT = Eigen::TensorMap<Tensor<const Scalar, 2>>(I.data(), {z.size(), z.size()});
Eigen::array<Eigen::IndexPair<int>, 0> ei = {};
Scalar z² = z.squaredNorm();
return ddHT.conjugate().contract(zT.conjugate(), ei) / z²
+IT.contract(dHT.conjugate(), ei).shuffle((std::array<int, 3>){0,2,1}) / z²
-zT.contract(dHT.conjugate(), ei).contract(zT.conjugate(), ei) / pow(z², 2)
-ddHT.conjugate().contract(zT, ip00).contract(zT.conjugate(), ei).contract(zT.conjugate(), ei) / pow(z², 2)
-IT.contract(zT.conjugate(), ei) * (dH.dot(z) / pow(z², 2))
-IT.contract(zT.conjugate(), ei).shuffle((std::array<int, 3>){0,2,1}) * (dH.dot(z) / pow(z², 2))
+zT.contract(zT.conjugate(), ei).contract(zT.conjugate(), ei) * (2.0 * dH.dot(z) / pow(z², 3))
;
}
template <class Scalar>
Tensor<Scalar, 3> ddzDotConjugate(const Vector<Scalar>& z, const Vector<Scalar>& dH, const Matrix<Scalar>& ddH, const Tensor<Scalar, 3>& dddH) {
Tensor<Scalar, 1> zT = Eigen::TensorMap<Tensor<const Scalar, 1>>(z.data(), {z.size()});
Tensor<Scalar, 1> dHT = Eigen::TensorMap<Tensor<const Scalar, 1>>(dH.data(), {dH.size()});
Tensor<Scalar, 2> ddHT = Eigen::TensorMap<Tensor<const Scalar, 2>>(ddH.data(), {z.size(), z.size()});
Matrix<Scalar> I = Matrix<Real>::Identity(z.size(), z.size());
Tensor<Scalar, 2> IT = Eigen::TensorMap<Tensor<const Scalar, 2>>(I.data(), {z.size(), z.size()});
Eigen::array<Eigen::IndexPair<int>, 0> ei = {};
Eigen::array<Eigen::IndexPair<int>, 1> ip00 = {Eigen::IndexPair<int>(0, 0)};
Scalar z² = z.squaredNorm();
return - dddH + dddH.contract(zT.conjugate(), ip00).contract(zT, ei) / z²
+ IT.contract(ddHT.contract(zT.conjugate(), ip00), ei).shuffle((std::array<int, 3>){0, 2, 1}) / z²
+ ddHT.contract(zT.conjugate(), ip00).contract(IT, ei) / z²
- zT.conjugate().contract(ddHT.contract(zT.conjugate(), ip00), ei).contract(zT, ei) / pow(z², 2)
- zT.conjugate().contract(IT, ei) * (z.dot(dH) / pow(z², 2))
- IT.contract(zT.conjugate(), ei).shuffle((std::array<int, 3>){0, 2, 1}) * (z.dot(dH) / pow(z², 2))
- ddHT.contract(zT.conjugate(), ip00).contract(zT.conjugate(), ei).contract(zT, ei) / pow(z², 2)
+ zT.conjugate().contract(zT.conjugate(), ei).contract(zT, ei) * (2.0 * z.dot(dH) / pow(z², 3));
}
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