DiFfRG::LoopIntegrals Namespace Reference

Functions
template<typename NT , int d, typename FUN >
NT	integrate (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	angle_integrate (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const QGauss< 1 > &cos_quadrature)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	two_angle_integrate (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const QGauss< 1 > &cos_quadrature)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	three_angle_integrate (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const QGauss< 1 > &cos_quadrature)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	spatial_integrate_and_integrate (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const QGauss< 1 > &m_quadrature, const double m_extent)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	spatial_angle_integrate_and_integrate (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const QGauss< 1 > &cos_quadrature, const QGauss< 1 > &m_quadrature, const double m_extent)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	spatial_integrate_and_sum (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const int m_order, const QGauss< 1 > &m_quadrature, const double m_extent, const double T)
	Performs the integral.
template<typename NT , int d, typename FUN >
NT	spatial_angle_integrate_and_sum (const FUN &fun, const QGauss< 1 > &x_quadrature, const double x_extent, const double k, const QGauss< 1 > &cos_quadrature, const int m_order, const QGauss< 1 > &m_quadrature, const double m_extent, const double T)
	Performs the integral.
template<typename NT , typename FUN >
NT	sum (const FUN &fun, const int m_order, const QGauss< 1 > &m_quadrature, const double m_extent, const double T)

Function Documentation

angle_integrate()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::angle_integrate	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const QGauss< 1 > &	cos_quadrature )

Performs the integral.

\[ \int d\Omega_{d-1} \int_{-1}^1 d\cos \int_0^\infty dq f(q^2, cos) q^{d-1} \]

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
cos_quadrature	Quadrature rule for the integral over \(\cos\).

Returns

NT Result of the integral.

integrate()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::integrate	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k )

Performs the integral.

\[ \int d\Omega_{d} \int_0^\infty dq f(q^2) q^{d-1} \]

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.

Returns

NT Result of the integral.

spatial_angle_integrate_and_integrate()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::spatial_angle_integrate_and_integrate	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const QGauss< 1 > &	cos_quadrature,
		const QGauss< 1 > &	m_quadrature,
		const double	m_extent )

Performs the integral.

\[ \int_{-\infty}^\infty dq_0\int d\Omega_{d-2} \int_{-1}^1 d\cos \int_0^\infty dq f(q^2, cos, q0) q^{d-2} \]

with \(q_0\) being the zeroth momentum mode.

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
cos_quadrature	Quadrature rule for the integral over \(\cos\).
m_quadrature	Quadrature rule for the integral over \(q0\), specifically for the domain [0,m_extent).
m_extent	Extent of the integral over \(q0\).

Returns

NT Result of the integral.

spatial_angle_integrate_and_sum()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::spatial_angle_integrate_and_sum	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const QGauss< 1 > &	cos_quadrature,
		const int	m_order,
		const QGauss< 1 > &	m_quadrature,
		const double	m_extent,
		const double	T )

Performs the integral.

\[ \sum_{q0}\int d\Omega_{d-2} \int_{-1}^1 d\cos \int_0^\infty dq f(q^2, cos, q0) q^{d-2} \]

with \(q0 \,\in 2\pi\mathbb{Z}\) being a Matsubara frequency. The Matsubara sum is performed only for the first \(|n| <= m_order\) summands, the rest is approximated by an integral.

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
cos_quadrature	Quadrature rule for the integral over \(\cos\).
m_order	Number of Matsubara frequencies to be summed over.
m_quadrature	Quadrature rule for the integral over \(q0\).
m_extent	Extent of the integral over \(q0\).

Returns

NT Result of the integral.

spatial_integrate_and_integrate()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::spatial_integrate_and_integrate	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const QGauss< 1 > &	m_quadrature,
		const double	m_extent )

Performs the integral.

\[ \int_{-\infty}^\infty dq_0\int d\Omega_{d-1} \int_0^\infty dq f(q^2, cos, q0) q^{d-2} \]

with \(q_0\) being the zeroth momentum mode.

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
m_quadrature	Quadrature rule for the integral over \(q0\), specifically for the domain [0,m_extent).
m_extent	Extent of the integral over \(q0\).

Returns

NT Result of the integral.

spatial_integrate_and_sum()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::spatial_integrate_and_sum	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const int	m_order,
		const QGauss< 1 > &	m_quadrature,
		const double	m_extent,
		const double	T )

Performs the integral.

\[ \sum_{q0}\int d\Omega_{d-1} \int_0^\infty dq f(q^2, q0) q^{d-2} \]

with \(q0 \,\in 2\pi\mathbb{Z}\) being a Matsubara frequency. The Matsubara sum is performed only for the first \(|n| <= m_order\) summands, the rest is approximated by an integral.

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
m_order	Number of Matsubara frequencies to be summed over.
m_quadrature	Quadrature rule for the integral over \(q0\).
m_extent	Extent of the integral over \(q0\).

Returns

NT Result of the integral.

sum()

template<typename NT , typename FUN >

NT DiFfRG::LoopIntegrals::sum	(	const FUN &	fun,
		const int	m_order,
		const QGauss< 1 > &	m_quadrature,
		const double	m_extent,
		const double	T )

three_angle_integrate()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::three_angle_integrate	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const QGauss< 1 > &	cos_quadrature )

Performs the integral.

\[ \int d\Omega_{d-2} \int_{-1}^1 d\cos \int_0^{2\pi}d\phi \int_0^\infty dq f(q^2, cos, phi) q^{d-1} \]

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
cos_quadrature	Quadrature rule for the integral over \(\cos\).

Returns

NT Result of the integral.

two_angle_integrate()

template<typename NT , int d, typename FUN >

NT DiFfRG::LoopIntegrals::two_angle_integrate	(	const FUN &	fun,
		const QGauss< 1 > &	x_quadrature,
		const double	x_extent,
		const double	k,
		const QGauss< 1 > &	cos_quadrature )

Performs the integral.

\[ \int d\Omega_{d-2} \int_{-1}^1 d\cos \int_0^{2\pi}d\phi \int_0^\infty dq f(q^2, cos, phi) q^{d-1} \]

Template Parameters

NT	Number type used throughout.
d	Dimension of the integral.
FUN	Function type of the integrand.

Parameters

fun	Integrand, should be a callable with signature NT(double q^2).
x_quadrature	Quadrature rule for the integral over \(q^2\), \(q^2 = x * k^2\).
x_extent	Extent of the integral over \(q^2\), \(q^2 = x * k^2\).
k	Momentum scale as defined above.
cos_quadrature	Quadrature rule for the integral over \(\cos\).

Returns

NT Result of the integral.