Function space meshes

Detailed Description

Mesh types for function spaces.

Classes
class	triqs::mesh::dlr
	Discrete Lehmann representation (DLR) mesh type. More...
class	triqs::mesh::legendre
	Legendre mesh type. More...

Functions
auto	triqs::mesh::evaluate (dlr const &m, auto const &f, double tau)
	Evaluate the DLR approximation of a function \( f \) at a given imaginary time point \( \tau \in [0, \beta] \).
auto	triqs::mesh::evaluate (dlr const &m, auto const &f, matsubara_freq const &iw)
	Evaluate the DLR approximation of a function \( f \) at a given Matsubara frequency \( i\omega_n \).
auto	triqs::mesh::evaluate (legendre const &m, auto const &f, double tau)
	Evaluate the generalized Fourier approximation of a function \( f \) at a given imaginary time point \(\tau \in [0, \beta] \) in terms of Legendre polynomials.

Function Documentation

evaluate() [1/3]

auto triqs::mesh::evaluate	(	dlr const &	m,
		auto const &	f,
		double	tau )

#include <triqs/mesh/dlr.hpp>

Evaluate the DLR approximation of a function \( f \) at a given imaginary time point \( \tau \in [0, \beta] \).

We calculate

\[ f(\tau) \approx \sum_{l=0}^{N-1} K(\tau, \omega_l) f_l \; , \]

where \( f_l \) are the DLR coefficients, \( \omega_l \) are the DLR frequencies and

\[ K(\tau, \omega_l) = \frac{e^{-\omega_l \tau}}{1 + e^{-\omega_l \beta}} \; , \]

are the DLR basis functions.

Parameters

m	triqs::mesh::dlr mesh.
f	Callable object \( f \) containing the DLR coefficients \( f_l \).
tau	Imaginary time point \( \tau \in [0, \beta] \) at which to approximate the function.

Returns

DLR approximation of \( f(\tau) \).

Definition at line 429 of file dlr.hpp.

evaluate() [2/3]

auto triqs::mesh::evaluate	(	dlr const &	m,
		auto const &	f,
		matsubara_freq const &	iw )

#include <triqs/mesh/dlr.hpp>

Evaluate the DLR approximation of a function \( f \) at a given Matsubara frequency \( i\omega_n \).

We calculate

\[ f(i\omega_n) \approx \sum_{l=0}^{N-1} K(i\omega_n, \omega_l) f_l \; , \]

where \( f_l \) are the DLR coefficients, \( \omega_l \) are the DLR frequencies and

\[ K(i\omega_n, \omega_l) = \frac{1}{i\omega_n + \omega_l} \; , \]

are the Fourier transformed DLR basis functions.

Parameters

m	triqs::mesh::dlr mesh.
f	Callable object \( f \) containing the DLR coefficients \( f_l \).
iw	Matsubara frequency \( i\omega_n \) at which to approximate the function.

Returns

DLR approximation of \( f(i\omega_n) \).

Definition at line 453 of file dlr.hpp.

evaluate() [3/3]

auto triqs::mesh::evaluate ( legendre const & m, auto const & f, double tau )

inline

#include <triqs/mesh/legendre.hpp>

Evaluate the generalized Fourier approximation of a function \( f \) at a given imaginary time point \(\tau \in [0, \beta] \) in terms of Legendre polynomials.

We calculate

\[ f(\tau) \approx \sum_{n=0}^{N-1} \frac{\sqrt{2n + 1}}{\beta} f_n P_n(x(\tau)) \; , \]

where \( P_n \) denotes the Legendre polynomial of degree \( n \), \( x(\tau) = 2\tau / \beta - 1 \) is a linear map from \( [0, \beta] \) to \( [-1, 1] \) and

\[ f_n = \sqrt{2n + 1} \int_0^\beta d\tau \, f(\tau) P_n(x(\tau)) \; , \]

are the expansion coefficients.

Parameters

m	triqs::mesh::legendre mesh.
f	Callable object \( f \) containing the expansion coefficients \( f_n \).
tau	Imaginary time point \( \tau \in [0, \beta] \) at which to approximate the function.

Returns

Generalized Fourier approximation of \( f(\tau) \).

Definition at line 304 of file legendre.hpp.