Projective max-plus matrices

Defined in matrix.hpp.

This page describes the functionality for \(n \times n\) projective matrices over the max-plus semiring. Projective max-plus matrices belong to the quotient of the monoid of all max-plus matrices by the congruence where two matrices are related if they differ by a scalar multiple; see MaxPlusMat.

There are two types of such matrices those whose dimension is known at compile-time, and those where it is not. Both types can be accessed via the alias template ProjMaxPlusMat : if R or C has value 0, then the dimensions can be set at run time, otherwise R and C are the dimensions. The default value of R is 0, C is R, and Scalar is int.

Matrices in both these classes are modified when constructed to be in a normal form which is obtained by subtracting the maximum finite entry in the matrix from the every finite entry.

A ProjMaxPlusMat is neither a StaticMatrix nor a DynamicMatrix, but has the same interface as each of these types. Every instance of ProjMaxPlusMat wraps a MaxPlusMat.

Note

The types RowView and Row are the same as those in the wrapped matrix. This means that a Row object for a projective max-plus matrix is never normalised, because if they were they would be normalised according to their entries, and this might not correspond to the normalised entries of the matrix.

Please refer to the documentation of these class templates for more details. The only substantial difference in the interface of static and dynamic matrices is that the former can be default constructed and the latter should be constructed using the dimensions.

Example

// default construct an uninitialized 3 x 3 static matrix
ProjMaxPlusMat<3> m;
// construct an uninitialized 4 x 4 dynamic matrix
ProjMaxPlusMat<>  m(4, 4);

Typedefs
template<typename Scalar>
using	DynamicProjMaxPlusMat = detail::ProjMaxPlusMat<DynamicMaxPlusMat<Scalar>>
	Alias for dynamic projective max-plus matrices with run-time dimensions.
template<size_t R = 0, size_t C = R, typename Scalar = int>
using	ProjMaxPlusMat
	Alias template for projective max-plus matrices.
template<size_t R, size_t C, typename Scalar>
using	StaticProjMaxPlusMat = detail::ProjMaxPlusMat<StaticMaxPlusMat<R, C, Scalar>>
	Alias for static projective max-plus matrices with compile-time arithmetic and dimensions.

Variables
template<typename T>
static constexpr bool	IsProjMaxPlusMat = detail::IsProjMaxPlusMatHelper<T>::value
	Helper to check if a type is ProjMaxPlusMat.

Typedef Documentation

DynamicProjMaxPlusMat

template<typename Scalar>

using DynamicProjMaxPlusMat = detail::ProjMaxPlusMat<DynamicMaxPlusMat<Scalar>>

Defined in matrix.hpp.

Alias for the type of dynamic projective max-plus matrices where the dimensions of the matrices can be defined at run time.

Template Parameters

Scalar

the type of the entries in the matrix.

ProjMaxPlusMat

template<size_t R = 0, size_t C = R, typename Scalar = int>

using ProjMaxPlusMat

Initial value:

 std::conditional_t<R == 0 || C == 0,
                                            DynamicProjMaxPlusMat<Scalar>,
                                            StaticProjMaxPlusMat<R, C, Scalar>>

Defined in matrix.hpp.

Alias template for projective max-plus matrices.

Template Parameters

R	the number of rows. A value of 0 indicates that the value will be set at run time (default: 0).
C	the number of columns. A value of 0 indicates that the value will be set at run time (default: R).
Scalar	The type of the entries in the matrix (default: int).

StaticProjMaxPlusMat

template<size_t R, size_t C, typename Scalar>

using StaticProjMaxPlusMat = detail::ProjMaxPlusMat<StaticMaxPlusMat<R, C, Scalar>>

Defined in matrix.hpp.

Alias for static projective max-plus matrices whose arithmetic and dimensions are defined at compile-time.

Template Parameters

R	the number of rows.
C	the number of columns.
Scalar	the type of the entries in the matrix.

Variable Documentation

IsProjMaxPlusMat

template<typename T>

bool IsProjMaxPlusMat = detail::IsProjMaxPlusMatHelper<T>::value

staticconstexpr

Defined in matrix.hpp.

This variable has value true if the template parameter T is the same as ProjMaxPlusMat<R, C, Scalar> for some values of R, C, and Scalar; and false if it is not.

Template Parameters

T	the type to check.