void pade (gf_view<refreq> &gr, gf_view<imfreq> const &gw, int n_points, double freq_offset) { // make sure the GFs have the same structure //assert(gw.shape() == gr.shape()); // copy the tail. it doesn't need to conform to the pade approximant gr.singularity() = gw.singularity(); auto sh = gw.data().shape().front_pop(); int N1 = sh[0], N2 = sh[1]; for (int n1=0; n1<N1; n1++) { for (int n2=0; n2<N2; n2++) { arrays::vector<dcomplex> z_in(n_points); // complex points arrays::vector<dcomplex> u_in(n_points); // values at these points arrays::vector<dcomplex> a(n_points); // corresponding Pade coefficients for (int i=0; i < n_points; ++i) z_in(i) = gw.mesh()[i]; for (int i=0; i < n_points; ++i) u_in(i) = gw.on_mesh(i)(n1,n2); triqs::utility::pade_approximant PA(z_in,u_in); gr() = 0.0; for (auto om : gr.mesh()) { dcomplex e = om + dcomplex(0.0,1.0)*freq_offset; gr[om](n1,n2) = PA(e); } } } }
void inverse(gf_view<imtime,scalar_valued> gt, gf_view<imfreq,scalar_valued> const gw){ using namespace impl_local_matsubara; static bool Green_Function_Are_Complex_in_time = false; // If the Green function are NOT complex, then one use the symmetry property // fold the sum and get a factor 2 auto ta = gw(freq_infty()); //TO BE MODIFIED AFTER SCALAR IMPLEMENTATION TODO dcomplex d= ta(1)(0,0), A= ta.get_or_zero(2)(0,0), B = ta.get_or_zero(3)(0,0); double b1, b2, b3; dcomplex a1, a2, a3; double beta=gw.domain().beta; size_t L= gt.mesh().size() - ( gt.mesh().kind() == full_bins ? 1 : 0); //L can be different from gt.mesh().size() (depending on the mesh kind) and is given to the FFT algorithm dcomplex iomega = dcomplex(0.0,1.0) * std::acos(-1) / beta; dcomplex iomega2 = -iomega * 2 * gt.mesh().delta() * (gt.mesh().kind() == half_bins ? 0.5 : 0.0) ; double fact = (Green_Function_Are_Complex_in_time ? 1 : 2)/beta; g_in.resize( gw.mesh().size()); g_out.resize(gt.mesh().size()); if (gw.domain().statistic == Fermion){ b1 = 0; b2 =1; b3 =-1; a1 = d-B; a2 = (A+B)/2; a3 = (B-A)/2; } else { b1 = -0.5; b2 =-1; b3 =1; a1=4*(d-B)/3; a2=B-(d+A)/2; a3=d/6+A/2+B/3; } g_in() = 0; for (auto & w: gw.mesh()) { g_in[ w.index() ] = fact * exp(w.index()*iomega2) * ( gw[w] - (a1/(w-b1) + a2/(w-b2) + a3/(w-b3)) ); } // for bosons GF(w=0) is divided by 2 to avoid counting it twice if (gw.domain().statistic == Boson && !Green_Function_Are_Complex_in_time ) g_in(0) *= 0.5; details::fourier_base(g_in, g_out, L, false); // CORRECT FOR COMPLEX G(tau) !!! typedef double gt_result_type; //typedef typename gf<imtime>::mesh_type::gf_result_type gt_result_type; if (gw.domain().statistic == Fermion){ for (auto & t : gt.mesh()){ gt[t] = convert_green<gt_result_type> ( g_out( t.index() == L ? 0 : t.index() ) * exp(-iomega*t) + oneFermion(a1,b1,t,beta) + oneFermion(a2,b2,t,beta)+ oneFermion(a3,b3,t,beta) ); } } else { for (auto & t : gt.mesh()) gt[t] = convert_green<gt_result_type> ( g_out( t.index() == L ? 0 : t.index() ) + oneBoson(a1,b1,t,beta) + oneBoson(a2,b2,t,beta) + oneBoson(a3,b3,t,beta) ); } if (gt.mesh().kind() == full_bins) gt.on_mesh(L) = -gt.on_mesh(0)-convert_green<gt_result_type>(ta(1)(0,0)); // set tail gt.singularity() = gw.singularity(); }
// Impose a discontinuity G(\tau=0)-G(\tau=\beta) void enforce_discontinuity(gf_view<legendre> & gl, arrays::array_view<double,2> disc) { double norm = 0.0; arrays::vector<double> t(gl.data().shape()[0]); for (int i=0; i<t.size(); ++i) { t(i) = triqs::utility::legendre_t(i,1) / gl.domain().beta; norm += t(i)*t(i); } arrays::array<dcomplex, 2> corr(disc.shape()); corr() = 0; for (auto const &l : gl.mesh()) corr += t(l.index()) * gl[l]; auto _ = arrays::range{}; for (auto const& l : gl.mesh()) gl.data()(l.index(), _, _) += (disc - corr) * t(l.index()) / norm; }
auto curry_impl(gf_view<cartesian_product<Ms...>, Target, Singularity, Evaluator, IsConst> g) { // pick up the meshed corresponding to the curryed variables auto meshes_tuple = triqs::tuple::filter<pos...>(g.mesh().components()); using var_t = cart_prod<triqs::tuple::filter_t<std::tuple<Ms...>, pos...>>; auto m = triqs::tuple::apply_construct<gf_mesh<var_t>>(meshes_tuple); auto l = [g](auto&&... x) { return partial_eval_linear_index<pos...>(g, std::make_tuple(x...)); }; return make_gf_view_lambda_valued<var_t>(m, l); };
// Impose a discontinuity G(\tau=0)-G(\tau=\beta) void enforce_discontinuity(gf_view<legendre> & gl, tqa::array_view<double,2> disc) { double norm = 0.0; tqa::vector<double> t(gl.data().shape()[0]); for (int i=0; i<t.size(); ++i) { t(i) = triqs::utility::legendre_t(i,1) / gl.domain().beta; norm += t(i)*t(i); } tqa::array<double,2> corr(disc.shape()); corr() = 0; for (auto l : gl.mesh()) { corr += t(l.index()) * gl[l]; } tqa::range R; for (auto l : gl.mesh()) { gl.data()(l.index(),R,R) += (disc - corr) * t(l.index()) / norm; } }
void fit_tail(gf_view<imfreq> gf, tail_view known_moments, int n_moments, int n_min, int n_max, bool replace_by_fit ) { if (get_target_shape(gf) != known_moments.shape()) TRIQS_RUNTIME_ERROR << "shape of tail does not match shape of gf"; gf.singularity() = fit_tail_impl(gf, known_moments, n_moments, n_min, n_max); if (replace_by_fit) { // replace data in the fitting range by the values from the fitted tail int i = 0; for (auto iw : gf.mesh()) { // (arrays::range(n_min,n_max+1)) { if (i >= n_min) gf[iw] = evaluate(gf.singularity(),iw); i++; } } }
// compute a tail from the Legendre GF // this is Eq. 8 of our paper local::tail_view get_tail(gf_view<legendre> const & gl, int size = 10, int omin = -1) { auto sh = gl.data().shape().front_pop(); local::tail t(sh, size, omin); t.data() = 0.0; for (int p=1; p<=t.order_max(); p++) for (auto l : gl.mesh()) t(p) += (triqs::utility::legendre_t(l.index(),p)/pow(gl.domain().beta,p)) * gl[l]; return t; }
void legendre_matsubara_direct(gf_view<imtime> gt, gf_const_view<legendre> gl) { gt() = 0.0; legendre_generator L; for (auto t : gt.mesh()) { L.reset(2 * t / gt.domain().beta - 1); for (auto l : gl.mesh()) { gt[t] += sqrt(2 * l.index() + 1) / gt.domain().beta * gl[l] * L.next(); } } gt.singularity() = get_tail(gl, gt.singularity().size(), gt.singularity().order_min()); }
void legendre_matsubara_direct(gf_view<imfreq> gw, gf_const_view<legendre> gl) { gw() = 0.0; triqs::arrays::range R; // Use the transformation matrix for (auto om : gw.mesh()) { for (auto l : gl.mesh()) { gw[om] += legendre_T(om.index(), l.index()) * gl[l]; } } gw.singularity() = get_tail(gl, gw.singularity().size(), gw.singularity().order_min()); }
tqa::matrix<double> density( gf_view<legendre> const & gl) { auto sh = gl.data().shape().front_pop(); tqa::matrix<double> res(sh); res() = 0.0; for (auto l : gl.mesh()) { res -= sqrt(2*l.index()+1) * gl[l]; } res /= gl.domain().beta; return res; }
static auto invoke(gf_view<cartesian_product<Ms...>, Target, Singularity, Evaluator, IsConst> g, XTuple const& x_tuple) { using var_t = cart_prod<triqs::tuple::filter_out_t<std::tuple<Ms...>, pos...>>; // meshes of the returned gf_view : just drop the mesh of the evaluated variables auto meshes_tuple_partial = triqs::tuple::filter_out<pos...>(g.mesh().components()); // The mesh of the resulting function auto m = triqs::tuple::apply_construct<gf_mesh<var_t>>(meshes_tuple_partial); // rebuild a tuple of the size sizeof...(Ms), containing the linear indices and range at the position of evaluated variables. auto arr_args = triqs::tuple::inverse_filter<sizeof...(Ms), pos...>(x_tuple, arrays::range()); // from it, we make a slice of the array of g, corresponding to the data of the returned gf_view auto arr2 = triqs::tuple::apply(g.data(), std::tuple_cat(arr_args, std::make_tuple(arrays::ellipsis{}))); // We now also partial_eval the singularity auto singv = partial_eval_linear_index<pos...>(g.singularity(), x_tuple); using r_sing_t = typename decltype(singv)::regular_type; // finally, we build the view on this data. using r_t = gf_view<var_t, Target, r_sing_t, void, IsConst>; return r_t{m, arr2, singv, {}, {}}; }
void legendre_matsubara_inverse(gf_view<legendre> gl, gf_const_view<imtime> gt) { gl() = 0.0; legendre_generator L; auto N = gt.mesh().size() - 1; double coef; // Do the integral over imaginary time for (auto t : gt.mesh()) { if (t.index()==0 || t.index()==N) coef = 0.5; else coef = 1.0; L.reset(2 * t / gt.domain().beta - 1); for (auto l : gl.mesh()) { gl[l] += coef * sqrt(2 * l.index() + 1) * L.next() * gt[t]; } } gl.data() *= gt.mesh().delta(); }
//------------------------------------------------------- // For Imaginary Matsubara Frequency functions // ------------------------------------------------------ tqa::matrix<double> density( gf_view<imfreq> const & G) { dcomplex I(0,1); auto sh = G.data().shape().front_pop(); auto Beta = G.domain().beta; local::tail_view t = G(freq_infty()); if (!t.is_decreasing_at_infinity()) TRIQS_RUNTIME_ERROR<<" density computation : Green Function is not as 1/omega or less !!!"; const size_t N1=sh[0], N2 = sh[1]; tqa::array<dcomplex,2> dens_part(sh), dens_tail(sh), dens(sh); tqa::matrix<double> res(sh); dens_part()=0;dens()=0;dens_tail()=0; for (size_t n1=0; n1<N1;n1++) for (size_t n2=0; n2<N2;n2++) { dcomplex d= t(1)(n1,n2) , A=t(2)(n1,n2),B = t(3)(n1,n2) ; double b1 = 0,b2 =1, b3 =-1; dcomplex a1 = d-B, a2 = (A+B)/2, a3 = (B-A)/2; for (auto & w : G.mesh()) dens_part(n1,n2)+= G[w](n1,n2) - (a1/(w - b1) + a2 / (w-b2) + a3/(w-b3)); dens_part(n1,n2) = dens_part(n1,n2)/Beta; dens_tail(n1,n2) = d + F(a1,b1,Beta) + F(a2,b2,Beta)+ F(a3,b3,Beta); // If the Green function are NOT complex, then one use the symmetry property // fold the sum and get a factor 2 //double fact = (Green_Function_Are_Complex_in_time ? 1 : 2); //dens_part(n1,n2) = dens_part(n1,n2)*(fact/Beta) + (d + F(a1,b1,Beta) + F(a2,b2,Beta)+ F(a3,b3,Beta)); //if (!Green_Function_Are_Complex_in_time) dens_part = 0+real(dens_part); } for (size_t n1=0; n1<N1;n1++) for (size_t n2=0; n2<N2;n2++) { dens_part(n1,n2) = dens_part(n1,n2) + real(dens_part(n2,n1)) - I * imag(dens_part(n2,n1)) + dens_tail(n1,n2); // ?? STRANGE ?? dens_part(n2,n1) = real(dens_part(n1,n2)) - I * imag(dens_part(n1,n2)); } for (size_t n1=0; n1<N1;n1++) for (size_t n2=0; n2<N2;n2++) { res(n1,n2) = real(dens_part(n1,n2)); } return res; }
template <typename T> size_t n_blocks(gf_view<block_index, T> const &g) { return g.mesh().size(); }
tail fit_tail_impl(gf_view<imfreq> gf, const tail_view known_moments, int n_moments, int n_min, int n_max) { // precondition : check that n_max is not too large n_max = std::min(n_max, int(gf.mesh().size()-1)); tail res(get_target_shape(gf)); if (known_moments.size()) for (int i = known_moments.order_min(); i <= known_moments.order_max(); i++) res(i) = known_moments(i); // if known_moments.size()==0, the lowest order to be obtained from the fit is determined by order_min in known_moments // if known_moments.size()==0, the lowest order is the one following order_max in known_moments int n_unknown_moments = n_moments - known_moments.size(); if (n_unknown_moments < 1) return known_moments; // get the number of even unknown moments: it is n_unknown_moments/2+1 if the first // moment is even and n_moments is odd; n_unknown_moments/2 otherwise int omin = known_moments.size() == 0 ? known_moments.order_min() : known_moments.order_max() + 1; // smallest unknown moment int omin_even = omin % 2 == 0 ? omin : omin + 1; int omin_odd = omin % 2 != 0 ? omin : omin + 1; int size_even = n_unknown_moments / 2; if (n_unknown_moments % 2 != 0 && omin % 2 == 0) size_even += 1; int size_odd = n_unknown_moments - size_even; int size1 = n_max - n_min + 1; // size2 is the number of moments arrays::matrix<double> A(size1, std::max(size_even, size_odd), FORTRAN_LAYOUT); arrays::matrix<double> B(size1, 1, FORTRAN_LAYOUT); arrays::vector<double> S(std::max(size_even, size_odd)); const double rcond = 0.0; int rank; for (int i = 0; i < get_target_shape(gf)[0]; i++) { for (int j = 0; j < get_target_shape(gf)[1]; j++) { // fit the odd moments S.resize(size_odd); A.resize(size1,size_odd); //when resizing, gelss segfaults for (int k = 0; k < size1; k++) { auto n = n_min + k; auto iw = std::complex<double>(gf.mesh().index_to_point(n)); B(k, 0) = imag(gf.data()(gf.mesh().index_to_linear(n), i, j)); // subtract known tail if present if (known_moments.size() > 0) B(k, 0) -= imag(evaluate(slice_target(known_moments, arrays::range(i, i + 1), arrays::range(j, j + 1)), iw)(0, 0)); for (int l = 0; l < size_odd; l++) { int order = omin_odd + 2 * l; A(k, l) = imag(pow(iw, -1.0 * order)); // set design matrix for odd moments } } arrays::lapack::gelss(A, B, S, rcond, rank); for (int m = 0; m < size_odd; m++) { res(omin_odd + 2 * m)(i, j) = B(m, 0); } // fit the even moments S.resize(size_even); A.resize(size1,size_even); //when resizing, gelss segfaults for (int k = 0; k < size1; k++) { auto n = n_min + k; auto iw = std::complex<double>(gf.mesh().index_to_point(n)); B(k, 0) = real(gf.data()(gf.mesh().index_to_linear(n), i, j)); // subtract known tail if present if (known_moments.size() > 0) B(k, 0) -= real(evaluate(slice_target(known_moments, arrays::range(i, i + 1), arrays::range(j, j + 1)), iw)(0, 0)); for (int l = 0; l < size_even; l++) { int order = omin_even + 2 * l; A(k, l) = real(pow(iw, -1.0 * order)); // set design matrix for odd moments } } arrays::lapack::gelss(A, B, S, rcond, rank); for (int m = 0; m < size_even; m++) { res(omin_even + 2 * m)(i, j) = B(m, 0); } } } res.mask()()=n_moments; return res; // return tail }
void fit_tail(gf_view<block_index, gf<imfreq>> block_gf, tail_view known_moments, int n_moments, int n_min, int n_max, bool replace_by_fit ) { // for(auto &gf : block_gf) fit_tail(gf, known_moments, n_moments, n_min, n_max, replace_by_fit); for (int i = 0; i < block_gf.mesh().size(); i++) fit_tail(block_gf[i], known_moments, n_moments, n_min, n_max, replace_by_fit); }
template <typename M, typename T, typename S, typename E> gf<M, real_target_t<T>, S> real(gf_view<M, T, S, E> g) { return {g.mesh(), real(g.data()), g.singularity(), g.symmetry(), {}, {}}; }
void direct (gf_view<imfreq,scalar_valued> gw, gf_view<imtime,scalar_valued> const gt) { using namespace impl_local_matsubara; auto ta = gt(freq_infty()); //TO BE MODIFIED AFTER SCALAR IMPLEMENTATION TODO dcomplex d= ta(1)(0,0), A= ta.get_or_zero(2)(0,0), B = ta.get_or_zero(3)(0,0); double b1=0, b2=0, b3=0; dcomplex a1, a2, a3; double beta=gt.mesh().domain().beta; auto L = ( gt.mesh().kind() == full_bins ? gt.mesh().size()-1 : gt.mesh().size()); double fact= beta/ gt.mesh().size(); dcomplex iomega = dcomplex(0.0,1.0) * std::acos(-1) / beta; dcomplex iomega2 = iomega * 2 * gt.mesh().delta() * ( gt.mesh().kind() == half_bins ? 0.5 : 0.0); g_in.resize(gt.mesh().size()); g_out.resize(gw.mesh().size()); if (gw.domain().statistic == Fermion){ b1 = 0; b2 =1; b3 =-1; a1 = d-B; a2 = (A+B)/2; a3 = (B-A)/2; } else { b1 = -0.5; b2 =-1; b3 =1; a1 = 4*(d-B)/3; a2 = B-(d+A)/2; a3 = d/6+A/2+B/3; } if (gw.domain().statistic == Fermion){ for (auto & t : gt.mesh()) g_in[t.index()] = fact * exp(iomega*t) * ( gt[t] - ( oneFermion(a1,b1,t,beta) + oneFermion(a2,b2,t,beta)+ oneFermion(a3,b3,t,beta) ) ); } else { for (auto & t : gt.mesh()) g_in[t.index()] = fact * ( gt[t] - ( oneBoson(a1,b1,t,beta) + oneBoson(a2,b2,t,beta) + oneBoson(a3,b3,t,beta) ) ); } details::fourier_base(g_in, g_out, L, true); for (auto & w : gw.mesh()) { gw[w] = g_out( w.index() ) * exp(iomega2*w.index() ) + a1/(w-b1) + a2/(w-b2) + a3/(w-b3); } gw.singularity() = gt.singularity();// set tail }
std14::enable_if_t<!arrays::is_scalar<RHS>::value> triqs_gf_view_assign_delegation(gf_view<M, T, S, E> g, RHS const &rhs) { if (!(g.mesh() == rhs.mesh())) TRIQS_RUNTIME_ERROR << "Gf Assignment in View : incompatible mesh" << g.mesh() << " vs " << rhs.mesh(); for (auto const &w : g.mesh()) g[w] = rhs[w]; g.singularity() = rhs.singularity(); }