Matrices over the natural numbers mod (t = t + p)

Defined in matrix.hpp.

This page describes the functionality for \(n \times n\) matrices over the finite quotient of the usual semiring of natural number by the congruence \(t = t + p\) for arbitrary \(n\), \(t\), and \(p\). The value \(t\) is referred to as the threshold and \(p\) is called the period. The matrices of this type are referred to by the acronym ntp matrices, for "natural threshold period". The NTPSemiring has elements \(\{0, 1, ..., t, t + 1, ..., t + p - 1\}\) where \(t\), and \(p\) are the threshold and period, respectively; addition and multiplication in the NTPSemiring is defined below.

There are three types of such matrices where:

1. the dimension is known at compile-time;
2. the dimension is to be defined a run time but the arithmetic operations are known at compile-time (i.e. the values of \(t\) and \(p\) are known at compile time)
3. both the dimension and the arithmetic operations (i.e. \(t\) and \(p\)) are to be defined a run time.

All three of these types can be accessed via the alias template NTPMat<T, P, R, C, Scalar> if T and P have value 0, then the threshold and period can be set at run time, and if R or C is 0, then the dimension can be set at run time. The default values of T, P, and R are 0, and the default value of C is R.

The alias NTPMat is one of StaticMatrix, DynamicMatrix (compile-time arithmetic), or DynamicMatrix (run-time arithmetic); 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

// construct an uninitialized 3 x 3 static matrix with threshold
// 11, period 2
NTPMat<11, 2, 3> m;
// construct an uninitialized 4 x 4 dynamic matrix with threshold 11,
// period 2
NTPMat<11, 2> m(4, 4);
// construct an ntp semiring with threshold 11, period 2
NTPSemiring sr(11, 2);
// construct an uninitialized 5 x 5 dynamic matrix with threshold 11,
// period 2
NTPMat<> m(sr, 5, 5);

Classes
struct	NTPPlus< T, P, Scalar >
	Function object for addition in ntp semirings. More...
struct	NTPProd< T, P, Scalar >
	Function object for multiplication in an ntp semirings. More...
class	NTPSemiring< Scalar >
	Class representing an ntp semiring. More...

Typedefs
template<size_t T, size_t P, typename Scalar>
using	DynamicNTPMatWithoutSemiring
	Alias for ntp matrices with static threshold and period.
template<typename Scalar>
using	DynamicNTPMatWithSemiring = DynamicMatrix<NTPSemiring<Scalar>, Scalar>
	Alias for ntp matrices with dynamic threshold and period.
template<size_t T = 0, size_t P = 0, size_t R = 0, size_t C = R, typename Scalar = size_t>
using	NTPMat
	Alias template for ntp matrices.
template<size_t T, size_t P, size_t R, size_t C, typename Scalar>
using	StaticNTPMat
	Alias for ntp matrices with static threshold and period, and dimensions.

Functions
template<typename Mat>
std::enable_if_t< IsNTPMat< Mat > >	throw_if_bad_entry (Mat const &m)
	Check that the entries in an ntp matrix are valid.
template<typename Mat>
std::enable_if_t< IsNTPMat< Mat > >	throw_if_bad_entry (Mat const &m, typename Mat::scalar_type val)
	Check that an entry in an ntp matrix is valid.

Variables
template<typename U>
static constexpr bool	IsNTPMat = detail::IsNTPMatHelper<U>::value
	Helper to check if a type is NTPMat.

Typedef Documentation

DynamicNTPMatWithoutSemiring

template<size_t T, size_t P, typename Scalar>

using DynamicNTPMatWithoutSemiring

Initial value:

 DynamicMatrix<NTPPlus<T, P, Scalar>,
                                                     NTPProd<T, P, Scalar>,
                                                     IntegerZero<Scalar>,
                                                     IntegerOne<Scalar>,
                                                     Scalar>

Defined in matrix.hpp.

Alias for the type of dynamic ntp matrices where the dimension is defined at run time, but the threshold and period are defined at compile-time.

Template Parameters

T	the threshold.
P	the period.
Scalar	the type of the entries in the matrix.

DynamicNTPMatWithSemiring

template<typename Scalar>

using DynamicNTPMatWithSemiring = DynamicMatrix<NTPSemiring<Scalar>, Scalar>

Defined in matrix.hpp.

Alias for ntp matrices with dimensions, threshold, and period defined at runtime.

Template Parameters

Scalar

the type of the entries in the matrix.

NTPMat

template<size_t T = 0, size_t P = 0, size_t R = 0, size_t C = R, typename Scalar = size_t>

using NTPMat

Initial value:

 std::conditional_t<
      R == 0 || C == 0,
      std::conditional_t<T == 0 && P == 0,
                         DynamicNTPMatWithSemiring<Scalar>,
                         DynamicNTPMatWithoutSemiring<T, P, Scalar>>,
      StaticNTPMat<T, P, R, C, Scalar>>

Defined in matrix.hpp.

Alias template for ntp matrices.

Template Parameters

T	the threshold. If both T and P are 0, this indicates that the value will be set at run time (default: 0).
P	the period. If both T and P are 0, this indicates that the value will be set at run time (default: 0).
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: size_t).

StaticNTPMat

template<size_t T, size_t P, size_t R, size_t C, typename Scalar>

using StaticNTPMat

Initial value:

 StaticMatrix<NTPPlus<T, P, Scalar>,
                                    NTPProd<T, P, Scalar>,
                                    IntegerZero<Scalar>,
                                    IntegerOne<Scalar>,
                                    R,
                                    C,
                                    Scalar>

Defined in matrix.hpp.

Alias for static ntp matrices where the threshold, period, and dimensions are defined at compile-time.

Template Parameters

T	the threshold.
P	the period.
R	the number of rows.
C	the number of columns.
Scalar	the type of the entries in the matrix (default: int).

Function Documentation

throw_if_bad_entry() [1/2]

template<typename Mat>

std::enable_if_t< IsNTPMat< Mat > > throw_if_bad_entry

(

Mat const &

m

)

Defined in matrix.hpp.

This function can be used to check that a matrix contains values in the underlying semiring.

Template Parameters

Mat	the type of the matrix m (must satisfy IsNTPMat<Mat>).

Parameters

m	the matrix to check.

Exceptions

LibsemigroupsException	if any entry in the matrix is not in the set \(\{0, 1, \ldots, t + p - 1\}\) where \(t\) is the threshold, and \(p\) the period, of the matrix.
LibsemigroupsException	if the underlying semiring is not defined (only applies to matrices with run time arithmetic).

throw_if_bad_entry() [2/2]

template<typename Mat>

std::enable_if_t< IsNTPMat< Mat > > throw_if_bad_entry	(	Mat const &	m,
		typename Mat::scalar_type	val )

Defined in matrix.hpp.

This function can be used to check that a matrix contains values in the underlying semiring.

Template Parameters

Mat	the type of the 1st argument (must satisfy IsNTPMat<Mat>).

Parameters

m	the matrix.
val	the entry to check.

Exceptions

LibsemigroupsException	if the entry val is not in the set \(\{0, 1, \ldots, t + p - 1\}\) where \(t\) is the threshold, and \(p\) the period, of the matrix.
LibsemigroupsException	if the underlying semiring is not defined (only applies to matrices with run time arithmetic).

Variable Documentation

IsNTPMat

template<typename U>

bool IsNTPMat = detail::IsNTPMatHelper<U>::value

staticconstexpr

Defined in matrix.hpp.

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

Template Parameters

U	the type to check.