theoretica::autodiff Namespace Reference

Differential operators with automatic differentiation. More...

Typedefs
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 \(f: \mathbb{R}^N \rightarrow \mathbb{R}^M\).
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 $\vec x$.
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.
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.
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 \(f: \mathbb{R}^N \rightarrow \mathbb{R}\).
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 \(f: \mathbb{R}^N \rightarrow \mathbb{R}\).
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 $\vec x$.
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 $\vec x$.
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.

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.