Euclidean Space (L-2)

Functions
template<typename InputIterator1 , typename InputIterator2 , typename T = value_type_i<InputIterator1>>
T	aaa::euclidean::dot (InputIterator1 first_left, InputIterator1 last_left, InputIterator2 first_right, T init=T{})
template<typename Container1 , typename Container2 , typename T = value_type<Container1>>
T	aaa::euclidean::dot (const Container1 &a, Container2 &b, T init=T{})
template<typename InputIterator , typename T = value_type_i<InputIterator>>
T	aaa::euclidean::squared_norm (InputIterator first, InputIterator last, T init=T{})
template<typename Container , typename T = value_type<Container>>
T	aaa::euclidean::squared_norm (const Container &a, T init=T{})
template<typename InputIterator , typename T = value_type_i<InputIterator>>
sqrt_type_t< T >	aaa::euclidean::norm (InputIterator first, InputIterator last, T init=T{})
template<typename Container , typename T = value_type<Container>>
sqrt_type_t< T >	aaa::euclidean::norm (const Container &a, T init=T{})
template<typename InputIterator1 , typename InputIterator2 , typename T = value_type_i<InputIterator1>>
T	aaa::euclidean::squared_distance (InputIterator1 first_left, InputIterator1 last_left, InputIterator2 first_right, T init=T{})
template<typename Container1 , typename Container2 , typename T = value_type<Container1>>
T	aaa::euclidean::squared_distance (const Container1 &left, const Container2 &right, T init=T{})
template<typename InputIterator1 , typename InputIterator2 , typename T = value_type_i<InputIterator1>>
sqrt_type_t< T >	aaa::euclidean::distance (InputIterator1 first_left, InputIterator1 last_left, InputIterator2 first_right, T init=T{})
template<typename Container1 , typename Container2 , typename T = value_type<Container1>>
sqrt_type_t< T >	aaa::euclidean::distance (const Container1 &left, const Container2 &right, T init=T{})

Detailed Description

Euclidean space is also known as L-2 space. It defines the the following functions:

• The dot product is also known as the scalar product or Euclidean inner product. The dot product of the vectors a and b is defined as: \( a \cdot b = \sum_i a_i b_i \)
• The Euclidean norm or length of a vector is defined as: \( \| a \|_2 = \sqrt{a \cdot a} = \sqrt{\sum_i a_i^2} \)
• The Euclidean distance between the vectors a and b is defined as: \( d_2(a, b) = \|a-b\|_2 = \sqrt{\sum_i (a_i-b_i)^2} \)

We represent mathematical vectors as either containers, or ranges of iterators. The functions in this module take one or two vectors as input and output a single scalar. This is sometimes refered to as a reduction or fold operation.

Function Documentation

template<typename InputIterator1 , typename InputIterator2 , typename T = value_type_i<InputIterator1>>

sqrt_type_t<T> aaa::euclidean::distance	(	InputIterator1	first_left,
		InputIterator1	last_left,
		InputIterator2	first_right,
		T	init = T{}
	)

The Euclidean distance of two vectors. Each vector is represented by a range of iterators. Returns a value of a floating point type following the same convention as std::sqrt.

template<typename Container1 , typename Container2 , typename T = value_type<Container1>>

sqrt_type_t<T> aaa::euclidean::distance	(	const Container1 &	left,
		const Container2 &	right,
		T	init = T{}
	)

The Euclidean distance of two vectors. Each vector is represented by a container. The two containers should have the same size. Returns a value of a floating point type following the same convention as std::sqrt.

template<typename InputIterator1 , typename InputIterator2 , typename T = value_type_i<InputIterator1>>

T aaa::euclidean::dot	(	InputIterator1	first_left,
		InputIterator1	last_left,
		InputIterator2	first_right,
		T	init = T{}
	)

The dot product of two vectors. Each vector is represented by a range of iterators.

template<typename Container1 , typename Container2 , typename T = value_type<Container1>>

T aaa::euclidean::dot	(	const Container1 &	a,
		Container2 &	b,
		T	init = T{}
	)

The dot product of two vectors. Each vector is represented by a container. The two containers should have the same size.

template<typename InputIterator , typename T = value_type_i<InputIterator>>

sqrt_type_t<T> aaa::euclidean::norm	(	InputIterator	first,
		InputIterator	last,
		T	init = T{}
	)

The Euclidean norm of a vector. The vector is represented by a range of iterators. Returns a value of floating point type following the same convention as std::sqrt.

template<typename Container , typename T = value_type<Container>>

sqrt_type_t<T> aaa::euclidean::norm	(	const Container &	a,
		T	init = T{}
	)

The Euclidean norm of a vector. The vector is represented by a container. Returns a value of floating point type following the same convention as std::sqrt.

template<typename InputIterator1 , typename InputIterator2 , typename T = value_type_i<InputIterator1>>

T aaa::euclidean::squared_distance	(	InputIterator1	first_left,
		InputIterator1	last_left,
		InputIterator2	first_right,
		T	init = T{}
	)

The squared Euclidean distance of two vectors. Each vector is represented by a range of iterators.

template<typename Container1 , typename Container2 , typename T = value_type<Container1>>

T aaa::euclidean::squared_distance	(	const Container1 &	left,
		const Container2 &	right,
		T	init = T{}
	)

The squared Euclidean distance of two vectors. Each vector is represented by a container. The two containers should have the same size.

template<typename InputIterator , typename T = value_type_i<InputIterator>>

T aaa::euclidean::squared_norm	(	InputIterator	first,
		InputIterator	last,
		T	init = T{}
	)

The squared Euclidean norm of a vector. The vector is represented by a range of iterators.

template<typename Container , typename T = value_type<Container>>

T aaa::euclidean::squared_norm	(	const Container &	a,
		T	init = T{}
	)

The squared Euclidean norm of a vector. The vector is represented by a container.