math::Norm< N, Vector, Scalar > Struct Template Reference
[Concepts]

Concept Norm. More...

#include <vector_concepts.hpp>

List of all members.

Public Types
typedef associated_type	magnitude_type
	Associated type to represent real values in teh Field of scalar (with default).
typedef associated_type	result_type_norm
	Associated type for result of norm functor.
Public Member Functions
axiom	Positivity (N norm, Vector v, magnitude_type ref)
	Invariant: norm of vector is larger than zero.
axiom	PositiveHomogeneity (N norm, Vector v, Scalar a)
	Invariant: positive homogeneity with scalar.
axiom	TriangleInequality (N norm, Vector u, Vector v)
	Invariant: triangle inequality.

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Detailed Description

template<typename N, typename Vector, typename Scalar = typename Vector::value_type>
struct math::Norm< N, Vector, Scalar >

Concept Norm.

Semantic requirements of a norm

Parameters:

	N	Norm functor
	Vector	The the type of a vector or a collection
	Scalar	The scalar over which the vector field is defined

Refinement of:

• std::Callable1 <N, Vector>

Associated types:

• magnitude_type
• result_type_norm

Requires:

• VectorSpace <Vector, Scalar>;
• RealMagnitude < Scalar >;
• std::Convertible <magnitude_type, Scalar>;
• std::Convertible <result_type_norm, RealMagnitude<Scalar>::magnitude_type>;
• std::Convertible <result_type_norm, Scalar>;

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Member Typedef Documentation

template<typename N, typename Vector, typename Scalar = typename Vector::value_type>

typedef associated_type math::Norm< N, Vector, Scalar >::magnitude_type

Associated type to represent real values in teh Field of scalar (with default).

By default MagnitudeType<Scalar>::type

template<typename N, typename Vector, typename Scalar = typename Vector::value_type>

typedef associated_type math::Norm< N, Vector, Scalar >::result_type_norm

Associated type for result of norm functor.

Automatically detected

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Member Function Documentation

template<typename N, typename Vector, typename Scalar = typename Vector::value_type>

axiom math::Norm< N, Vector, Scalar >::PositiveHomogeneity	(	N	norm,
		Vector	v,
		Scalar	a
	)			[inline]

Invariant: positive homogeneity with scalar.

norm(a * v) == abs(a) * norm(v);

template<typename N, typename Vector, typename Scalar = typename Vector::value_type>

axiom math::Norm< N, Vector, Scalar >::Positivity	(	N	norm,
		Vector	v,
		magnitude_type	ref
	)			[inline]

Invariant: norm of vector is larger than zero.

norm(v) >= zero(ref);

template<typename N, typename Vector, typename Scalar = typename Vector::value_type>

axiom math::Norm< N, Vector, Scalar >::TriangleInequality	(	N	norm,
		Vector	u,
		Vector	v
	)			[inline]

Invariant: triangle inequality.

norm(u + v) <= norm(u) + norm(v);

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The documentation for this struct was generated from the following file:

• vector_concepts.hpp