Differential operators with automatic differentiation. More...

Classes
struct	is_dual_type
	Type trait to check whether the given type is a multidual number. More...

struct	is_dual_type< dual >
	Type trait to check whether the given type is a multidual number. More...

struct	is_dual2_type
	Type trait to check whether the given type is a multidual number. More...

struct	is_dual2_type< dual2 >
	Type trait to check whether the given type is a multidual number. More...

struct	is_multidual_type
	Type trait to check whether the given type is a multidual number. More...

struct	is_multidual_type< multidual< N > >
	Type trait to check whether the given type is a multidual number. More...

Typedefs
template<typename Function >
using	is_dual_func = std::conditional_t< is_dual_type< typename _internal::return_type_or_void< Function, dual >::type >::value, std::true_type, std::false_type >
	Type trait to check whether the given function takes a dual number as its first argument.

template<typename Function , typename T = bool>
using	enable_dual_func = typename std::enable_if< is_dual_func< Function >::value, T >::type
	Enable a certain function overload if the given type is a function taking as first argument a dual number.

template<typename Function >
using	is_dual2_func = std::conditional_t< is_dual2_type< typename _internal::return_type_or_void< Function, dual2 >::type >::value, std::true_type, std::false_type >
	Type trait to check whether the given function takes a dual2 number as its first argument.

template<typename Function , typename T = bool>
using	enable_dual2_func = typename std::enable_if< is_dual2_func< Function >::value, T >::type
	Enable a certain function overload if the given type is a function taking as first argument a dual2 number.

template<typename Type , typename T = bool>
using	enable_multidual = typename std::enable_if< is_multidual_type< Type >::value, T >::type
	Enable a certain function overload if the given type is an instantiation of the multidual template class.

template<typename Function , typename T = bool>
using	enable_scalar_field = typename std::enable_if< is_multidual_type< return_type_t< Function > >::value, T >::type
	Enable a certain function overload if the given type is a Callable object corresponding to a multidual function representing a scalar field, that is, a function taking a dvec_t and returning a dreal_t.

template<typename Function , typename T = bool>
using	enable_vector_field = typename std::enable_if< is_multidual_type< vector_element_t< return_type_t< Function > > >::value, T >::type
	Enable a certain function overload if the given type is a Callable object corresponding to a multidual function representing a vector field, that is, a function taking a dvec_t and returning a dvec_t.

template<unsigned int N = 0>
using	dreal_t = multidual< N >
	Real type for multivariate automatic differentiation (read "differential real").

template<unsigned int N = 0>
using	dvec_t = vec< dreal_t< N >, N >
	Vector type for multivariate automatic differentiation (read "differential vector").

using	dreal = dreal_t< 0 >
	Real type for multivariate automatic differentiation with dynamically allocated vectors.

using	dvec = dvec_t< 0 >
	Vector type for multivariate automatic differentiation with dynamically allocated vectors.

using	dreal2 = dreal_t< 2 >
	Real type for multivariate automatic differentiation with two-dimensional statically allocated vectors.

using	dvec2 = dvec_t< 2 >
	Vector type for multivariate automatic differentiation with two-dimensional statically allocated vectors.

using	dreal3 = dreal_t< 3 >
	Real type for multivariate automatic differentiation with three-dimensional statically allocated vectors.

using	dvec3 = dvec_t< 3 >
	Vector type for multivariate automatic differentiation with three-dimensional statically allocated vectors.

using	dreal4 = dreal_t< 4 >
	Real type for multivariate automatic differentiation with four-dimensional statically allocated vectors.

using	dvec4 = dvec_t< 4 >
	Vector type for multivariate automatic differentiation with four-dimensional statically allocated vectors.

Functions
template<typename DualFunction = std::function<dual(dual)>, enable_dual_func< DualFunction > = true>
real	deriv (DualFunction f, real x)
	Compute the derivative of a function at the given point using univariate automatic differentiation. More...

template<typename DualFunction = std::function<dual(dual)>, enable_dual_func< DualFunction > = true>
auto	deriv (DualFunction f)
	Get a lambda function which computes the derivative of the given function at the given point, using automatic differentiation. More...

template<typename Dual2Function = std::function<dual2(dual2)>, enable_dual2_func< Dual2Function > = true>
real	deriv2 (Dual2Function f, real x)
	Compute the second derivative of a function at the given point using univariate automatic differentiation. More...

template<typename Dual2Function = std::function<dual2(dual2)>, enable_dual2_func< Dual2Function > = true>
auto	deriv2 (Dual2Function f)
	Get a lambda function which computes the second derivative of the given function at the given point, using automatic differentiation. More...

template<typename MultidualType , typename Vector = vec<real>>
auto	make_autodiff_arg (const Vector &x)
	Prepare a vector of multidual numbers in "canonical" form, where the i-th element of the vector has a dual part which is the i-th canonical vector. More...

template<typename Function , typename Vector = vec<real>, enable_scalar_field< Function > = true, enable_vector< Vector > = true>
auto	gradient (Function f, const Vector &x)
	Compute the gradient $\nabla f = \sum_i^n \vec e_i \frac{\partial}{\partial x_i} f(\vec x)$ for a given $\vec x$ of a scalar field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ using automatic differentiation. More...

template<typename Function , enable_scalar_field< Function > = true>
auto	gradient (Function f)
	Get a lambda function which computes the gradient $\nabla f = \sum_i^n \vec e_i \frac{\partial}{\partial x_i} f(\vec x)$ of a given scalar field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at $\vec x$ using automatic differentiation. More...

template<typename Function , typename Vector = vec<real>, enable_scalar_field< Function > = true, enable_vector< Vector > = true>
real	divergence (Function f, const Vector &x)
	Compute the divergence $\sum_i^n \frac{\partial}{\partial x_i} f(\vec x)$ for a given $\vec x$ of a scalar field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ using automatic differentiation. More...

template<typename Function , enable_scalar_field< Function > = true>
auto	divergence (Function f)
	Get a lambda function which computes the divergence of a given function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at a given $\vec x$ using automatic differentiation. More...

template<unsigned int N = 0, unsigned int M = 0>
mat< real, M, N >	jacobian (vec< multidual< N >, M >(*f)(vec< multidual< N >, N >), const vec< real, N > &x)
	Compute the jacobian of a vector field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}^M$. More...

template<unsigned int N = 0, unsigned int M = 0>
auto	jacobian (vec< multidual< N >, M >(*f)(vec< multidual< N >, N >))
	Get a lambda function which computes the jacobian of a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}^M$ for a given $\vec x$. More...

template<unsigned int N = 0>
vec< real, N >	curl (vec< multidual< N >, N >(*f)(vec< multidual< N >, N >), const vec< real, N > &x)
	Compute the curl for a given $\vec x$ of a vector field defined by $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ using automatic differentiation. More...

template<unsigned int N = 0>
auto	curl (vec< multidual< N >, N >(*f)(vec< multidual< N >, N >))
	Get a lambda function which computes the curl for a given $\vec x$ of a vector field defined by $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ using automatic differentiation. More...

template<unsigned int N = 0>
vec< real, N >	directional_derivative (multidual< N >(*f)(vec< multidual< N >, N >), const vec< real, N > &x, const vec< real, N > &v)
	Compute the directional derivative of a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$. More...

template<unsigned int N = 0>
auto	directional_derivative (multidual< N >(*f)(vec< multidual< N >, N >), const vec< real, N > &v)
	Get a lambda function which computes the directional derivative of a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$. More...

template<unsigned int N = 0>
real	laplacian (dual2(*f)(vec< dual2, N >), const vec< real, N > &x)
	Compute the Laplacian differential operator for a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at a given $\vec x$. More...

template<unsigned int N = 0>
auto	laplacian (dual2(*f)(vec< dual2, N >))
	Get a lambda function which computes the Laplacian differential operator for a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at a given $\vec x$. More...

template<unsigned int N = 0>
real	sturm_liouville (multidual< N >(f)(vec< multidual< N >, N >), multidual< N >(H)(vec< multidual< N >, N >), vec< real, N > eta)
	Compute the Sturm-Liouville operator on a generic function of the form $f: \mathbb{R}^{2N} \rightarrow \mathbb{R}$ with respect to a given Hamiltonian function of the form $H: \mathbb{R}^{2N} \rightarrow \mathbb{R}$ where the first N arguments are the coordinates in phase space and the last N arguments are the conjugate momenta, for a given point in phase space. More...

Detailed Description

Differential operators with automatic differentiation.

Function Documentation

◆ curl() [1/2]

template<unsigned int N = 0>

auto theoretica::autodiff::curl ( vec< multidual< N >, N >(*)(vec< multidual< N >, N >) f )

inline

Get a lambda function which computes the curl for a given $\vec x$ of a vector field defined by $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ using automatic differentiation.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.

Returns: A lambda function which computes the curl of f.

◆ curl() [2/2]

template<unsigned int N = 0>

vec<real, N> theoretica::autodiff::curl	(	vec< multidual< N >, N >(*)(vec< multidual< N >, N >)	f,
		const vec< real, N > &	x
	)

inline

Compute the curl for a given $\vec x$ of a vector field defined by $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ using automatic differentiation.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.
x	The point to compute the curl at.

Returns: The curl of f at x.

◆ deriv() [1/2]

template<typename DualFunction = std::function<dual(dual)>, enable_dual_func< DualFunction > = true>

auto theoretica::autodiff::deriv ( DualFunction f )

inline

Get a lambda function which computes the derivative of the given function at the given point, using automatic differentiation.

Parameters

f	The function to differentiate, with dual argument and return value.

Returns: A lambda function which computes the derivative of f using automatic differentiation.

◆ deriv() [2/2]

template<typename DualFunction = std::function<dual(dual)>, enable_dual_func< DualFunction > = true>

real theoretica::autodiff::deriv	(	DualFunction	f,
		real	x
	)

inline

Compute the derivative of a function at the given point using univariate automatic differentiation.

Parameters

f	The function to differentiate, with dual argument and return value.
x	The coordinate to compute the derivative at.

Returns: The derivative of f at x.

◆ deriv2() [1/2]

template<typename Dual2Function = std::function<dual2(dual2)>, enable_dual2_func< Dual2Function > = true>

auto theoretica::autodiff::deriv2 ( Dual2Function f )

inline

Get a lambda function which computes the second derivative of the given function at the given point, using automatic differentiation.

Parameters

f	The function to differentiate, with dual2 argument and return value.

Returns: A lambda function which computes the derivative of f using automatic differentiation.

◆ deriv2() [2/2]

template<typename Dual2Function = std::function<dual2(dual2)>, enable_dual2_func< Dual2Function > = true>

real theoretica::autodiff::deriv2	(	Dual2Function	f,
		real	x
	)

inline

Compute the second derivative of a function at the given point using univariate automatic differentiation.

Parameters

f	The function to differentiate, with dual2 argument and return value.
x	The coordinate to compute the derivative at.

Returns: The derivative of f at x.

◆ directional_derivative() [1/2]

template<unsigned int N = 0>

auto theoretica::autodiff::directional_derivative	(	multidual< N >(*)(vec< multidual< N >, N >)	f,
		const vec< real, N > &	v
	)

inline

Get a lambda function which computes the directional derivative of a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$.

Parameters

f	The function to partially differentiate
x	The point to compute the derivative at
v	The direction to compute the derivative on

Note: In most applications, the vector v should be a unit vector, but the function does not control whether the vector has unit length or not.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.
v	The direction of the derivative.

Returns: A lambda function which computes the directional derivative over v of f.

◆ directional_derivative() [2/2]

template<unsigned int N = 0>

vec<real, N> theoretica::autodiff::directional_derivative	(	multidual< N >(*)(vec< multidual< N >, N >)	f,
		const vec< real, N > &	x,
		const vec< real, N > &	v
	)

inline

Compute the directional derivative of a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$.

Parameters

f	The function to partially differentiate
x	The point to compute the derivative at
v	The direction to compute the derivative on

Note: In most applications, the vector v should be a unit vector, but the function does not control whether the vector has unit length or not.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.
x	The point to compute the directional derivative at.
v	The direction of the derivative.

Returns: The directional derivative over v of f at x.

◆ divergence() [1/2]

template<typename Function , enable_scalar_field< Function > = true>

auto theoretica::autodiff::divergence ( Function f )

inline

Get a lambda function which computes the divergence of a given function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at a given $\vec x$ using automatic differentiation.

The returned lambda function accepts a vec<real, N> argument. The argument function may be a function pointer or lambda function, with the type dvec_t as first argument and dreal_t return type.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.

Returns: A lambda function which computes the divergence of f at x.

◆ divergence() [2/2]

template<typename Function , typename Vector = vec<real>, enable_scalar_field< Function > = true, enable_vector< Vector > = true>

real theoretica::autodiff::divergence	(	Function	f,
		const Vector &	x
	)

inline

Compute the divergence $\sum_i^n \frac{\partial}{\partial x_i} f(\vec x)$ for a given $\vec x$ of a scalar field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ using automatic differentiation.

The argument function may be a function pointer or lambda function, with the type dvec_t as first argument and dreal_t return type.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.
x	The point to compute the divergence at.

Returns: The divergence of f at x.

◆ gradient() [1/2]

template<typename Function , enable_scalar_field< Function > = true>

auto theoretica::autodiff::gradient ( Function f )

inline

Get a lambda function which computes the gradient $\nabla f = \sum_i^n \vec e_i \frac{\partial}{\partial x_i} f(\vec x)$ of a given scalar field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at $\vec x$ using automatic differentiation.

The returned lambda function accepts a vec<real, N> argument. The argument function may be a function pointer or lambda function, with the type dvec_t as first argument and dreal_t return type.

Parameters

f	A function with a vector of multidual numbers as input and a multidual number as output.

Returns: A lambda function which computes the gradient of f.

◆ gradient() [2/2]

template<typename Function , typename Vector = vec<real>, enable_scalar_field< Function > = true, enable_vector< Vector > = true>

auto theoretica::autodiff::gradient	(	Function	f,
		const Vector &	x
	)

inline

Compute the gradient $\nabla f = \sum_i^n \vec e_i \frac{\partial}{\partial x_i} f(\vec x)$ for a given $\vec x$ of a scalar field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ using automatic differentiation.

The argument function may be a function pointer or lambda function, with the type dvec_t as first argument and dreal_t return type.

Parameters

f	A function with a vector of multidual numbers as input and a multidual number as output.
x	The point to compute the gradient at.

Returns: The gradient of f computed at x.

◆ jacobian() [1/2]

template<unsigned int N = 0, unsigned int M = 0>

auto theoretica::autodiff::jacobian ( vec< multidual< N >, M >(*)(vec< multidual< N >, N >) f )

inline

Get a lambda function which computes the jacobian of a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}^M$ for a given $\vec x$.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.

Returns: A lambda function which computes the Jacobian matrix of f.

◆ jacobian() [2/2]

template<unsigned int N = 0, unsigned int M = 0>

mat<real, M, N> theoretica::autodiff::jacobian	(	vec< multidual< N >, M >(*)(vec< multidual< N >, N >)	f,
		const vec< real, N > &	x
	)

inline

Compute the jacobian of a vector field of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}^M$.

Parameters

f	A function with a vector of multidual numbers as input and a vector of multidual numbers as output.
x	The point to compute the Jacobian at.

Returns: The Jacobian matrix of f at x.

◆ laplacian() [1/2]

template<unsigned int N = 0>

auto theoretica::autodiff::laplacian ( dual2(*)(vec< dual2, N >) f )

inline

Get a lambda function which computes the Laplacian differential operator for a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at a given $\vec x$.

Parameters

f	A function taking a vector of dual2 numbers and returning a dual2 number.

Returns: A lambda function which computes the Laplacian of f at x.

◆ laplacian() [2/2]

template<unsigned int N = 0>

real theoretica::autodiff::laplacian	(	dual2(*)(vec< dual2, N >)	f,
		const vec< real, N > &	x
	)

inline

Compute the Laplacian differential operator for a generic function of the form $f: \mathbb{R}^N \rightarrow \mathbb{R}$ at a given $\vec x$.

Parameters

f	A function taking a vector of dual2 numbers and returning a dual2 number.
x	The point to compute the Laplacian at.

Returns: The Laplacian of f at x.

◆ make_autodiff_arg()

template<typename MultidualType , typename Vector = vec<real>>

auto theoretica::autodiff::make_autodiff_arg ( const Vector & x )

inline

Prepare a vector of multidual numbers in "canonical" form, where the i-th element of the vector has a dual part which is the i-th canonical vector.

This form is used to evaluate a multidual function for automatic differentiation.

◆ sturm_liouville()

template<unsigned int N = 0>

real theoretica::autodiff::sturm_liouville	(	multidual< N >(*)(vec< multidual< N >, N >)	f,
		multidual< N >(*)(vec< multidual< N >, N >)	H,
		vec< real, N >	eta
	)

inline

Compute the Sturm-Liouville operator on a generic function of the form $f: \mathbb{R}^{2N} \rightarrow \mathbb{R}$ with respect to a given Hamiltonian function of the form $H: \mathbb{R}^{2N} \rightarrow \mathbb{R}$ where the first N arguments are the coordinates in phase space and the last N arguments are the conjugate momenta, for a given point in phase space.

Parameters

f	A function taking a vector of multidual numbers and returning a multidual number.
H	The Hamiltonian of the system.
eta	A vector containing N = 2K elements, where the first K elements are the coordinates and the last K elements are the conjugate momenta.

Classes

Typedefs

Functions

Detailed Description

Function Documentation

◆ curl() [1/2]

◆ curl() [2/2]

◆ deriv() [1/2]

◆ deriv() [2/2]

◆ deriv2() [1/2]

◆ deriv2() [2/2]

◆ directional_derivative() [1/2]

◆ directional_derivative() [2/2]

◆ divergence() [1/2]

◆ divergence() [2/2]

◆ gradient() [1/2]

◆ gradient() [2/2]

◆ jacobian() [1/2]

◆ jacobian() [2/2]

◆ laplacian() [1/2]

◆ laplacian() [2/2]

◆ make_autodiff_arg()

◆ sturm_liouville()