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();
}