Detailed Description

Numerical and mathematical helpers (Legendre, Chebyshev, Padé, real-or-complex, ...).

Classes
struct	triqs::utility::gmp_complex
	Lightweight complex number backed by GMP mpf_class floats, used during Padé coefficient computation. More...
class	triqs::utility::legendre_generator
	Recursive generation of Legendre polynomials \( P_l(x) \). More...
class	triqs::utility::pade_approximant
	Padé approximant of a complex function sampled at \( \{ z_i \} \) with values \( \{ u_i \} \). More...
class	triqs::utility::real_or_complex
	Type that can represent either a real or a complex number. More...
struct	triqs::utility::time_pt
	A point in imaginary time, \( \tau_n \in [0, \beta] \), stored on a very fine integer grid. More...
struct	triqs::utility::time_segment
	Represents the imaginary-time segment \( [0, \beta] \). More...

Functions
template<typename T1, typename T2> requires (std::is_arithmetic_v<T1> and std::is_arithmetic_v<T2>)
bool	triqs::utility::kronecker (T1 i, T2 j)
	Kronecker delta \( \delta_{ij} \).
std::complex< double >	triqs::utility::legendre_T (int n, int l)
	Get the quantity \( T_{nl} \) from Eq.(E2) in the paper https://doi.org/10.1103/PhysRevB.84.075145.
double	triqs::utility::legendre_t (int l, int p)
	Get the quantity \( t_l^{(p)} \) from Eq.(E8) in the paper https://doi.org/10.1103/PhysRevB.84.075145.
double	triqs::utility::mod_cyl_bessel_i (int n, double x)
	Get the modified spherical bessel function \( i_n(x) \) of the first kind of order \( n \) evaluated at \( x \).
double	operator* (time_pt const &tp1, time_pt const &tp2)
	Multiplication between two time points.
double	operator/ (time_pt const &tp1, time_pt const &tp2)
	Division between two time points.
nda::vector< double >	triqs::utility::chebyshev_points (long N)
	Compute Chebyshev points of the first kind on the interval \( [-1, 1] \).
nda::vector< double >	triqs::utility::chebyshev_barycentric_weights (long N)
	Compute barycentric weights for Chebyshev points of the first kind.
double	triqs::utility::to_standard_interval (double x, double a, double b)
	Scale a value from interval \( [a, b] \) to the standard interval \( [-1, 1] \).
double	triqs::utility::from_standard_interval (double x, double a, double b)
	Scale a value from the standard interval \( [-1, 1] \) to interval \( [a, b] \).
template<std::integral I>
bool	triqs::utility::is_zero (I const &x)
	Exact zero check for integral values.
template<std::floating_point T>
bool	triqs::utility::is_zero (T const &x, T tolerance=100 *std::numeric_limits< T >::epsilon())
	Zero check for floating-point values.
template<typename T>
bool	triqs::utility::is_zero (std::complex< T > const &z, T tolerance=100 *std::numeric_limits< T >::epsilon())
	Zero check for complex values.
template<std::integral I>
I	triqs::utility::conj (I const &x)
	Complex conjugate of an integral value.
template<std::floating_point T>
T	triqs::utility::conj (T const &x)
	Complex conjugate of a floating-point value.
template<typename Z> requires (triqs::is_complex<Z>::value)
Z	triqs::utility::conj (Z const &z)
	Complex conjugate of a complex value.
template<std::integral I>
I	triqs::utility::real (I const &x)
	Real part of an integral value.
template<std::floating_point T>
T	triqs::utility::real (T const &x)
	Real part of a floating-point value.
template<typename Z> requires (triqs::is_complex<Z>::value)
Z	triqs::utility::real (Z const &z)
	Real part of a complex value.
template<std::integral I>
I	triqs::utility::imag (I const &x)
	Imaginary part of an integral value.
template<std::floating_point T>
T	triqs::utility::imag (T const &x)
	Imaginary part of a floating-point value.
template<typename Z> requires (triqs::is_complex<Z>::value)
Z	triqs::utility::imag (Z const &z)
	Imaginary part of a complex value.

Function Documentation

◆ chebyshev_barycentric_weights()

nda::vector< double > triqs::utility::chebyshev_barycentric_weights ( long N )

inline

#include <triqs/utility/chebyshev.hpp>

Compute barycentric weights for Chebyshev points of the first kind.

Weights are computed as:

\[ w_i = (-1)^i \sin\left(\frac{(2i + 1) \pi}{2N}\right) \]

These weights are used in the barycentric interpolation formula. Weights are returned in the same order as chebyshev_points().

Parameters

N	Number of Chebyshev points.

Returns: Vector of \( N \) barycentric weights.

Definition at line 73 of file chebyshev.hpp.

◆ chebyshev_points()

nda::vector< double > triqs::utility::chebyshev_points ( long N )

inline

#include <triqs/utility/chebyshev.hpp>

Compute Chebyshev points of the first kind on the interval \( [-1, 1] \).

Points are computed as:

\[ x_i = \cos\left(\frac{(2i + 1) \pi}{2N}\right) \quad \text{for } i = 0, \ldots, N-1 \]

These are the roots of the Chebyshev polynomial \( T_N(x) \). Points are returned in ascending order.

Parameters

N	Number of Chebyshev points.

Returns: Vector of \( N \) Chebyshev points in ascending order on \( [-1, 1] \).

Definition at line 50 of file chebyshev.hpp.

◆ conj() [1/3]

template<std::integral I>

I triqs::utility::conj ( I const & x )

#include <triqs/utility/numeric_ops.hpp>

Complex conjugate of an integral value.

Template Parameters

I	Integral type.

Parameters

x	Value to conjugate.

Returns: Value \( x \) unchanged.

Definition at line 80 of file numeric_ops.hpp.

◆ conj() [2/3]

template<std::floating_point T>

T triqs::utility::conj ( T const & x )

#include <triqs/utility/numeric_ops.hpp>

Complex conjugate of a floating-point value.

Template Parameters

T	Floating-point type.

Parameters

x	Value to conjugate.

Returns: Value \( x \) unchanged.

Definition at line 89 of file numeric_ops.hpp.

◆ conj() [3/3]

template<typename Z>
requires (triqs::is_complex<Z>::value)

Z triqs::utility::conj ( Z const & z )

#include <triqs/utility/numeric_ops.hpp>

Complex conjugate of a complex value.

Template Parameters

Z	Complex type.

Parameters

z	Value to conjugate.

Returns: \( z^* \).

Definition at line 100 of file numeric_ops.hpp.

◆ from_standard_interval()

double triqs::utility::from_standard_interval	(	double	x,
		double	a,
		double	b )

inline

#include <triqs/utility/chebyshev.hpp>

Scale a value from the standard interval \( [-1, 1] \) to interval \( [a, b] \).

Parameters

x	Value in \( [-1, 1] \).
a	Lower bound of the target interval.
b	Upper bound of the target interval.

Returns: Scaled value in \( [a, b] \).

Definition at line 102 of file chebyshev.hpp.

◆ imag() [1/3]

template<std::integral I>

I triqs::utility::imag ( I const & x )

#include <triqs/utility/numeric_ops.hpp>

Imaginary part of an integral value.

Template Parameters

I	Integral type.

Parameters

x	Value to inspect.

Returns: Zero.

Definition at line 142 of file numeric_ops.hpp.

◆ imag() [2/3]

template<std::floating_point T>

T triqs::utility::imag ( T const & x )

#include <triqs/utility/numeric_ops.hpp>

Imaginary part of a floating-point value.

Template Parameters

T	Floating-point type.

Parameters

x	Value to inspect.

Returns: Zero.

Definition at line 151 of file numeric_ops.hpp.

◆ imag() [3/3]

template<typename Z>
requires (triqs::is_complex<Z>::value)

Z triqs::utility::imag ( Z const & z )

#include <triqs/utility/numeric_ops.hpp>

Imaginary part of a complex value.

Template Parameters

Z	Complex type.

Parameters

z	Value to inspect.

Returns: Imaginary part of \( z \), i.e. \( \Im(z) \).

Definition at line 162 of file numeric_ops.hpp.

◆ is_zero() [1/3]

template<std::integral I>

bool triqs::utility::is_zero ( I const & x )

#include <triqs/utility/numeric_ops.hpp>

Exact zero check for integral values.

Template Parameters

I	Integral type.

Parameters

x	Value to test.

Returns: True if the given integral value is exactly zero, false otherwise.

Definition at line 49 of file numeric_ops.hpp.

◆ is_zero() [2/3]

template<typename T>

bool triqs::utility::is_zero	(	std::complex< T > const &	z,
		T	tolerance = 100 * std::numeric_limits<T>::epsilon() )

#include <triqs/utility/numeric_ops.hpp>

Zero check for complex values.

Template Parameters

T	Value type.

Parameters

z	Value to test.
tolerance	Absolute tolerance \( \epsilon \).

Returns: True if \( |\Re(z)| < \epsilon \) and \( |\Im(z)| < \epsilon \), false otherwise.

Definition at line 69 of file numeric_ops.hpp.

◆ is_zero() [3/3]

template<std::floating_point T>

bool triqs::utility::is_zero	(	T const &	x,
		T	tolerance = 100 * std::numeric_limits<T>::epsilon() )

#include <triqs/utility/numeric_ops.hpp>

Zero check for floating-point values.

Template Parameters

T	Floating-point type.

Parameters

x	Value to test.
tolerance	Absolute tolerance \( \epsilon \).

Returns: True if \( |x| < \epsilon \), false otherwise.

Definition at line 59 of file numeric_ops.hpp.

◆ kronecker()

template<typename T1, typename T2>
requires (std::is_arithmetic_v<T1> and std::is_arithmetic_v<T2>)

bool triqs::utility::kronecker	(	T1	i,
		T2	j )

inline

#include <triqs/utility/kronecker.hpp>

Kronecker delta \( \delta_{ij} \).

Template Parameters

T1	Arithmetic type of the first index.
T2	Arithmetic type of the second index.

Parameters

i	Index \( i \).
j	Index \( j \).

Returns: True if the given indices are equal, false otherwise.

Definition at line 45 of file kronecker.hpp.

◆ legendre_T()

std::complex< double > triqs::utility::legendre_T	(	int	n,
		int	l )

#include <triqs/utility/legendre.cpp>

Get the quantity \( T_{nl} \) from Eq.(E2) in the paper https://doi.org/10.1103/PhysRevB.84.075145.

Parameters

n	Matsubara index \( n \).
l	Legendre polynomial order \( l \).

Returns: \( T_{nl} = (-1)^n i^{l+1} \sqrt{2l + 1} j_l(\bar{\nu_n} / 2) \).

Definition at line 37 of file legendre.cpp.

◆ legendre_t()

double triqs::utility::legendre_t	(	int	l,
		int	p )

#include <triqs/utility/legendre.cpp>

Get the quantity \( t_l^{(p)} \) from Eq.(E8) in the paper https://doi.org/10.1103/PhysRevB.84.075145.

Parameters

l	Legendre polynomial order \( l \).
p	Index \( p \).

Returns: \( t_l^{(p)} = (-1)^p 2 \sqrt{2l + 1} \frac{(l + p - 1)!}{(p - 1)! (l - p + 1)!} \delta_{p+l,\text{odd}} \).

Definition at line 53 of file legendre.cpp.

◆ mod_cyl_bessel_i()

double triqs::utility::mod_cyl_bessel_i	(	int	n,
		double	x )

#include <triqs/utility/legendre.cpp>

Get the modified spherical bessel function \( i_n(x) \) of the first kind of order \( n \) evaluated at \( x \).

Parameters

n	Order \( n \) of the modified spherical Bessel function.
x	Value \( x \) at which to evaluate the function.

Returns: \( i_n(x) = \sqrt{\frac{\pi}{2x}} I_{n+1/2}(x) \).

Definition at line 68 of file legendre.cpp.

◆ operator*()

double operator*	(	time_pt const &	tp1,
		time_pt const &	tp2 )

Multiplication between two time points.

Parameters

tp1	Left-hand side time point \( \tau_l \).
tp2	Right-hand side time point \( \tau_r \).

Returns: Product of their floating-point values.

Definition at line 224 of file time_pt.hpp.

◆ operator/()

double operator/	(	time_pt const &	tp1,
		time_pt const &	tp2 )

Division between two time points.

Parameters

tp1	Left-hand side time point \( \tau_l \).
tp2	Right-hand side time point \( \tau_r \).

Returns: Ratio of their floating-point values.

Definition at line 234 of file time_pt.hpp.

◆ real() [1/3]

template<std::integral I>

I triqs::utility::real ( I const & x )

#include <triqs/utility/numeric_ops.hpp>

Real part of an integral value.

Template Parameters

I	Integral type.

Parameters

x	Value to inspect.

Returns: Value \( x \) unchanged.

Definition at line 111 of file numeric_ops.hpp.

◆ real() [2/3]

template<std::floating_point T>

T triqs::utility::real ( T const & x )

#include <triqs/utility/numeric_ops.hpp>

Real part of a floating-point value.

Template Parameters

T	Floating-point type.

Parameters

x	Value to inspect.

Returns: Value \( x \) unchanged.

Definition at line 120 of file numeric_ops.hpp.

◆ real() [3/3]

template<typename Z>
requires (triqs::is_complex<Z>::value)

Z triqs::utility::real ( Z const & z )

#include <triqs/utility/numeric_ops.hpp>

Real part of a complex value.

Template Parameters

Z	Complex type.

Parameters

z	Value to inspect.

Returns: Real part of \( z \), i.e. \( \Re(z) \).

Definition at line 131 of file numeric_ops.hpp.

◆ to_standard_interval()

double triqs::utility::to_standard_interval	(	double	x,
		double	a,
		double	b )

inline

#include <triqs/utility/chebyshev.hpp>

Scale a value from interval \( [a, b] \) to the standard interval \( [-1, 1] \).

Parameters

x	Value in \( [a, b] \).
a	Lower bound of the original interval.
b	Upper bound of the original interval.

Returns: Scaled value in \( [-1, 1] \).

Definition at line 92 of file chebyshev.hpp.

Detailed Description

Classes

Functions

Function Documentation

◆ chebyshev_barycentric_weights()

◆ chebyshev_points()

◆ conj() [1/3]

◆ conj() [2/3]

◆ conj() [3/3]

◆ from_standard_interval()

◆ imag() [1/3]

◆ imag() [2/3]

◆ imag() [3/3]

◆ is_zero() [1/3]

◆ is_zero() [2/3]

◆ is_zero() [3/3]

◆ kronecker()

◆ legendre_T()

◆ legendre_t()

◆ mod_cyl_bessel_i()

◆ operator*()

◆ operator/()

◆ real() [1/3]

◆ real() [2/3]

◆ real() [3/3]

◆ to_standard_interval()