math::InnerProduct< I, Vector, Scalar > Struct Template Reference

Concept InnerProduct. More...

#include <vector_concepts.hpp>

Inheritance diagram for math::InnerProduct< I, Vector, Scalar >:

Collaboration diagram for math::InnerProduct< I, Vector, Scalar >:

List of all members.

Public Member Functions
axiom	ConjugateSymmetry (I inner, Vector v, Vector w)
	The arguments can be changed and the result is then the complex conjugate.
axiom	SequiLinearity (I inner, Scalar a, Scalar b, Vector u, Vector v, Vector w)
	The inner product is linear in the second argument and conjugate linear in the first one.
axiom	NonNegativity (I inner, Vector v, MagnitudeType< Scalar >::type magnitude)
	The inner product of a vector with itself is not negative.
axiom	NonDegeneracy (I inner, Vector v, Vector w, Scalar s)
	Non-degeneracy not representable with axiom.
Public Attributes
associated_type	magnitude_type
	Associated type: the real magnitude type of the scalar.

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Detailed Description

template<typename I, typename Vector, typename Scalar = typename Vector::value_type>
struct math::InnerProduct< I, Vector, Scalar >

Concept InnerProduct.

Semantic requirements of a inner product

Parameters:

I	The inner product functor
Vector	The the type of a vector or a collection
Scalar	The scalar over which the vector field is defined

Refinement of:

• std::Callable2 <I, Vector, Vector>

Associated types:

• magnitude_type

Requires:

• std::Convertible<std::Callable2 <I, Vector, Vector>::result_type, Scalar> ; result of inner product convertible to scalar to be used in expressions
• HasConjugate < Scalar >
• RealMagnitude < Scalar > ; the scalar value needs a real magnitude type

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Member Function Documentation

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

axiom math::InnerProduct< I, Vector, Scalar >::ConjugateSymmetry	(	I	inner,
		Vector	v,
		Vector	w
	)		[inline]

The arguments can be changed and the result is then the complex conjugate.

inner(v, w) == conj(inner(w, v));

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

axiom math::InnerProduct< I, Vector, Scalar >::NonDegeneracy	(	I	inner,
		Vector	v,
		Vector	w,
		Scalar	s
	)		[inline]

Non-degeneracy not representable with axiom.

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

axiom math::InnerProduct< I, Vector, Scalar >::NonNegativity	(	I	inner,
		Vector	v,
		MagnitudeType< Scalar >::type	magnitude
	)		[inline]

The inner product of a vector with itself is not negative.

inner(v, v) == conj(inner(v, v)) implies inner(v, v) is representable as real

magnitude_type(inner(v, v)) >= zero(magnitude);

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

axiom math::InnerProduct< I, Vector, Scalar >::SequiLinearity	(	I	inner,
		Scalar	a,
		Scalar	b,
		Vector	u,
		Vector	v,
		Vector	w
	)		[inline]

The inner product is linear in the second argument and conjugate linear in the first one.

The equalities are partly redundant with ConjugateSymmetry

inner(v, b * w) == b * inner(v, w);

inner(u, v + w) == inner(u, v) + inner(u, w);

inner(a * v, w) == conj(a) * inner(v, w);

inner(u + v, w) == inner(u, w) + inner(v, w);

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Member Data Documentation

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

associated_type math::InnerProduct< I, Vector, Scalar >::magnitude_type

Associated type: the real magnitude type of the scalar.

By default RealMagnitude<Scalar>::type

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

• vector_concepts.hpp