void Model::construct_preference_and_covariance(
Eigen::VectorXd &Eta, Eigen::MatrixXd &Omega, const Eigen::MatrixXd &C,
const Eigen::MatrixXd &M, const Eigen::VectorXd &W,
const Eigen::MatrixXd &S, const Parameter ¶meter) {
Eigen::VectorXd Mu = C * M * W;
Eta = Mu;
Eigen::MatrixXd Psi = W.asDiagonal();
Psi -= (W * W.transpose());
Eigen::MatrixXd I(n_alternatives, n_alternatives);
I.setIdentity();
Eigen::MatrixXd Phi = C * M * Psi * M.transpose() * C.transpose() +
parameter.sig2 * C * I * C.transpose();
Omega = Phi;
double rt;
Eigen::MatrixXd Si = I;
for (int i = 2; i <= parameter.stopping_time; i++) {
rt = 1.0 / (1 + exp(i - 202.0) / 25); // [Takao] Not sure what this is.
Si = S * Si;
Eta += rt * Si * Mu;
Omega += pow(rt, 2) * Si * Phi * Si.transpose();
}
}
void projectorFromSvd(const Eigen::MatrixXd& jac,
Eigen::JacobiSVD<Eigen::MatrixXd>& svd,
Eigen::VectorXd& svdSingular,
Eigen::MatrixXd& preResult,
Eigen::MatrixXd& result,
double epsilon=std::numeric_limits<double>::epsilon(),
double minTol=1e-8)
{
// we are force to compute the Full matrix because of
// the nullspace matrix computation
svd.compute(jac, Eigen::ComputeFullU | Eigen::ComputeFullV);
double tolerance =
epsilon*double(std::max(jac.cols(), jac.rows()))*
std::abs(svd.singularValues()[0]);
tolerance = std::max(tolerance, minTol);
svdSingular.setOnes();
for(int i = 0; i < svd.singularValues().rows(); ++i)
{
svdSingular[i] = svd.singularValues()[i] > tolerance ? 0. : 1.;
}
preResult.noalias() = svd.matrixV()*svdSingular.asDiagonal();
result.noalias() = preResult*svd.matrixV().adjoint();
}
void constMatrix::gradient_add( const Eigen::VectorXd & X, const Eigen::VectorXd & iV)
{
Eigen::VectorXd vtmp = Q * X;
double xtQx = vtmp.dot(iV.asDiagonal() * vtmp);
dtau += (d - xtQx)/ tau;
ddtau -= (d + xtQx)/ pow(tau, 2);
}
Eigen::VectorXd dphi_dq(
softabs_point& z,
interface_callbacks::writer::base_writer& info_writer,
interface_callbacks::writer::base_writer& error_writer) {
Eigen::VectorXd a
= z.softabs_lambda_inv.cwiseProduct(z.pseudo_j.diagonal());
Eigen::MatrixXd A = a.asDiagonal()
* z.eigen_deco.eigenvectors().transpose();
Eigen::MatrixXd B = z.eigen_deco.eigenvectors() * A;
stan::math::grad_tr_mat_times_hessian(
softabs_fun<Model>(this->model_, 0), z.q, B, a);
return - 0.5 * a + z.g;
}
Eigen::VectorXd dtau_dq(
softabs_point& z,
interface_callbacks::writer::base_writer& info_writer,
interface_callbacks::writer::base_writer& error_writer) {
Eigen::VectorXd a = z.softabs_lambda_inv.cwiseProduct(
z.eigen_deco.eigenvectors().transpose() * z.p);
Eigen::MatrixXd A = a.asDiagonal()
* z.eigen_deco.eigenvectors().transpose();
Eigen::MatrixXd B = z.pseudo_j.selfadjointView<Eigen::Lower>() * A;
Eigen::MatrixXd C = A.transpose() * B;
Eigen::VectorXd b(z.q.size());
stan::math::grad_tr_mat_times_hessian(
softabs_fun<Model>(this->model_, 0), z.q, C, b);
return 0.5 * b;
}
int main() {
int m = 40;
int n = 20;
Eigen::MatrixXd A = Eigen::MatrixXd(m,n);
Eigen::MatrixXd B = Eigen::MatrixXd(m,n);
double tolerance = 1e-14;
Eigen::MatrixXd U, V1, V2;
Eigen::VectorXd S;
int rank;
Compress_Cols(A, B, tolerance, U, S, V1, V2, rank);
std::cout << (A-U*S.asDiagonal()*V1.transpose()).cwiseAbs().maxCoeff() << std::endl;
std::cout << (B-U*S.asDiagonal()*V2.transpose()).cwiseAbs().maxCoeff() << std::endl;
}
double polyfit(size_t n, size_t deg, const double* xp, const double* yp,
const double* wp, double* pp)
{
ConstMappedVector x(xp, n);
Eigen::VectorXd y = ConstMappedVector(yp, n);
MappedVector p(pp, deg+1);
if (deg >= n) {
throw CanteraError("polyfit", "Polynomial degree ({}) must be less "
"than number of input data points ({})", deg, n);
}
// Construct A such that each row i of A has the elements
// 1, x[i], x[i]^2, x[i]^3 ... + x[i]^deg
Eigen::MatrixXd A(n, deg+1);
A.col(0).setConstant(1.0);
if (deg > 0) {
A.col(1) = x;
}
for (size_t i = 1; i < deg; i++) {
A.col(i+1) = A.col(i).array() * x.array();
}
if (wp != nullptr && wp[0] > 0) {
// For compatibility with old Fortran dpolft, input weights are the
// squares of the weight vector used in this algorithm
Eigen::VectorXd w = ConstMappedVector(wp, n).cwiseSqrt().eval();
// Multiply by the weights on both sides
A = w.asDiagonal() * A;
y.array() *= w.array();
}
// Solve W*A*p = W*y to find the polynomial coefficients
p = A.colPivHouseholderQr().solve(y);
// Evaluate the computed polynomial at the input x coordinates to compute
// the RMS error as the return value
return (A*p - y).eval().norm() / sqrt(n);
}
void pseudoInverse(const Eigen::MatrixXd& jac,
Eigen::JacobiSVD<Eigen::MatrixXd>& svd,
Eigen::VectorXd& svdSingular,
Eigen::MatrixXd& prePseudoInv,
Eigen::MatrixXd& result,
double epsilon=std::numeric_limits<double>::epsilon(),
double minTol=1e-8)
{
svd.compute(jac, Eigen::ComputeThinU | Eigen::ComputeThinV);
// singular values are sorted in decreasing order
// so the first one is the max one
double tolerance =
epsilon*double(std::max(jac.cols(), jac.rows()))*
std::abs(svd.singularValues()[0]);
tolerance = std::max(tolerance, minTol);
svdSingular = ((svd.singularValues().array().abs() > tolerance).
select(svd.singularValues().array().inverse(), 0.));
prePseudoInv.noalias() = svd.matrixV()*svdSingular.asDiagonal();
result.noalias() = prePseudoInv*svd.matrixU().adjoint();
}
void propa_2d()
{
trace.beginBlock("2d propagation");
typedef DiscreteExteriorCalculus<2, 2, EigenLinearAlgebraBackend> Calculus;
typedef DiscreteExteriorCalculusFactory<EigenLinearAlgebraBackend> CalculusFactory;
{
trace.beginBlock("solving time dependent equation");
const Calculus::Scalar cc = 8; // px/s
trace.info() << "cc = " << cc << endl;
const Z2i::Domain domain(Z2i::Point(0,0), Z2i::Point(29,29));
const Calculus calculus = CalculusFactory::createFromDigitalSet(generateDiskSet(domain), false);
//! [time_laplace]
const Calculus::DualIdentity0 laplace = calculus.laplace<DUAL>() + 1e-8 * calculus.identity<0, DUAL>();
//! [time_laplace]
trace.info() << "laplace = " << laplace << endl;
trace.beginBlock("finding eigen pairs");
//! [time_eigen]
typedef Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> EigenSolverMatrix;
const EigenSolverMatrix eigen_solver(laplace.myContainer);
const Eigen::VectorXd eigen_values = eigen_solver.eigenvalues();
const Eigen::MatrixXd eigen_vectors = eigen_solver.eigenvectors();
//! [time_eigen]
trace.info() << "eigen_values_range = " << eigen_values.minCoeff() << "/" << eigen_values.maxCoeff() << endl;
trace.endBlock();
//! [time_omega]
const Eigen::VectorXd angular_frequencies = cc * eigen_values.array().sqrt();
//! [time_omega]
Eigen::VectorXcd initial_wave = Eigen::VectorXcd::Zero(calculus.kFormLength(0, DUAL));
//set_initial_phase_null(calculus, initial_wave);
set_initial_phase_dir_yy(calculus, initial_wave);
{
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-1, 1));
board << Calculus::DualForm0(calculus, initial_wave.real());
board.saveSVG("propagation_time_wave_initial_coarse.svg");
}
//! [time_init_proj]
Eigen::VectorXcd initial_projections = eigen_vectors.transpose() * initial_wave;
//! [time_init_proj]
// low pass
//! [time_low_pass]
const Calculus::Scalar lambda_cutoff = 4.5;
const Calculus::Scalar angular_frequency_cutoff = 2*M_PI * cc / lambda_cutoff;
int cutted = 0;
for (int kk=0; kk<initial_projections.rows(); kk++)
{
const Calculus::Scalar angular_frequency = angular_frequencies(kk);
if (angular_frequency < angular_frequency_cutoff) continue;
initial_projections(kk) = 0;
cutted ++;
}
//! [time_low_pass]
trace.info() << "cutted = " << cutted << "/" << initial_projections.rows() << endl;
{
const Eigen::VectorXcd wave = eigen_vectors * initial_projections;
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-1, 1));
board << Calculus::DualForm0(calculus, wave.real());
board.saveSVG("propagation_time_wave_initial_smoothed.svg");
}
const int kk_max = 60;
trace.progressBar(0,kk_max);
for (int kk=0; kk<kk_max; kk++)
{
//! [time_solve_time]
const Calculus::Scalar time = kk/20.;
const Eigen::VectorXcd current_projections = (angular_frequencies * std::complex<double>(0,time)).array().exp() * initial_projections.array();
const Eigen::VectorXcd current_wave = eigen_vectors * current_projections;
//! [time_solve_time]
std::stringstream ss;
ss << "propagation_time_wave_solution_" << std::setfill('0') << std::setw(3) << kk << ".svg";
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-1, 1));
board << Calculus::DualForm0(calculus, current_wave.real());
board.saveSVG(ss.str().c_str());
trace.progressBar(kk+1,kk_max);
}
trace.info() << endl;
trace.endBlock();
}
{
trace.beginBlock("forced oscillations");
const Z2i::Domain domain(Z2i::Point(0,0), Z2i::Point(50,50));
const Calculus calculus = CalculusFactory::createFromDigitalSet(generateDiskSet(domain), false);
const Calculus::DualIdentity0 laplace = calculus.laplace<DUAL>();
trace.info() << "laplace = " << laplace << endl;
trace.beginBlock("finding eigen pairs");
typedef Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> EigenSolverMatrix;
const EigenSolverMatrix eigen_solver(laplace.myContainer);
const Eigen::VectorXd laplace_eigen_values = eigen_solver.eigenvalues();
const Eigen::MatrixXd laplace_eigen_vectors = eigen_solver.eigenvectors();
trace.info() << "eigen_values_range = " << laplace_eigen_values.minCoeff() << "/" << laplace_eigen_values.maxCoeff() << endl;
trace.endBlock();
Calculus::DualForm0 concentration(calculus);
{
const Z2i::Point point(25,25);
const Calculus::Cell cell = calculus.myKSpace.uSpel(point);
const Calculus::Index index = calculus.getCellIndex(cell);
concentration.myContainer(index) = 1;
}
{
Board2D board;
board << domain;
board << concentration;
board.saveSVG("propagation_forced_concentration.svg");
}
for (int ll=0; ll<6; ll++)
{
//! [forced_lambda]
const Calculus::Scalar lambda = 4*20.75/(1+2*ll);
//! [forced_lambda]
trace.info() << "lambda = " << lambda << endl;
//! [forced_dalembert_eigen]
const Eigen::VectorXd dalembert_eigen_values = (laplace_eigen_values.array() - (2*M_PI/lambda)*(2*M_PI/lambda)).array().inverse();
const Eigen::MatrixXd concentration_to_wave = laplace_eigen_vectors * dalembert_eigen_values.asDiagonal() * laplace_eigen_vectors.transpose();
//! [forced_dalembert_eigen]
//! [forced_wave]
Calculus::DualForm0 wave(calculus, concentration_to_wave * concentration.myContainer);
//! [forced_wave]
wave.myContainer /= wave.myContainer(calculus.getCellIndex(calculus.myKSpace.uSpel(Z2i::Point(25,25))));
{
trace.info() << "saving samples" << endl;
const Calculus::Properties properties = calculus.getProperties();
{
std::stringstream ss;
ss << "propagation_forced_samples_hv_" << ll << ".dat";
std::ofstream handle(ss.str().c_str());
for (int kk=0; kk<=50; kk++)
{
const Z2i::Point point_horizontal(kk,25);
const Z2i::Point point_vertical(25,kk);
const Calculus::Scalar xx = 2 * (kk/50. - .5);
handle << xx << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_horizontal) << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_vertical) << endl;
}
}
{
std::stringstream ss;
ss << "propagation_forced_samples_diag_" << ll << ".dat";
std::ofstream handle(ss.str().c_str());
for (int kk=0; kk<=50; kk++)
{
const Z2i::Point point_diag_pos(kk,kk);
const Z2i::Point point_diag_neg(kk,50-kk);
const Calculus::Scalar xx = 2 * sqrt(2) * (kk/50. - .5);
handle << xx << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_diag_pos) << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_diag_neg) << endl;
}
}
}
{
std::stringstream ss;
ss << "propagation_forced_wave_" << ll << ".svg";
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-.5, 1));
board << wave;
board.saveSVG(ss.str().c_str());
}
}
trace.endBlock();
}
trace.endBlock();
}