Aleph::multi_poly_detail Namespace Reference

Internal helpers for multivariate polynomial operations. More...

Functions
size_t	total_degree (const Array< size_t > &idx) noexcept
	Total degree of a multi-index (sum of exponents).
bool	is_zero_index (const Array< size_t > &idx) noexcept
	Test whether a multi-index is the zero vector.
Array< size_t >	add_indices (const Array< size_t > &a, const Array< size_t > &b)
	Component-wise addition of two multi-indices.
Array< size_t >	extend_index (const Array< size_t > &idx, size_t target)
	Pad a multi-index with trailing zeros to reach target size.
template<typename T >
T	int_power (const T &base, size_t exp)
	Integer power by repeated squaring.
Array< size_t >	decrement_index_at (const Array< size_t > &idx, const size_t var, const size_t n=1)
	Decrement exponent at a specific variable.
template<typename T >
T	falling_factorial (const size_t k, const size_t n)
	Falling factorial k(k-1)(k-2)...(k-n+1).
Array< size_t >	flat_to_multi_index (size_t flat, const Array< size_t > &sizes)
	Convert flat index to multi-index given sizes per dimension.
size_t	multi_to_flat_index (const Array< size_t > &midx, const Array< size_t > &sizes)
	Convert multi-index to flat index given sizes per dimension.
template<typename Coefficient >
Coefficient	eval_monomial (const Array< Coefficient > &pt, const Array< size_t > &alpha)
	Evaluate monomial at a point (helper for fitting/interpolation).
bool	divides_index (const Array< size_t > &alpha, const Array< size_t > &beta, size_t nvars) noexcept
	Test whether alpha divides beta component-wise (alpha(i) <= beta(i) all i).
Array< size_t >	lcm_indices (const Array< size_t > &a, const Array< size_t > &b, const size_t nvars)
	Component-wise max = monomial LCM.
Array< size_t >	sub_indices (const Array< size_t > &beta, const Array< size_t > &alpha, const size_t nvars)
	Component-wise subtraction beta - alpha (requires alpha divides beta).

Detailed Description

Internal helpers for multivariate polynomial operations.

Function Documentation

add_indices()

Array< size_t > Aleph::multi_poly_detail::add_indices ( const Array< size_t > &  a, const Array< size_t > &  b  )

inline

Component-wise addition of two multi-indices.

Pads the shorter one with zeros.

Definition at line 94 of file tpl_multi_polynomial.H.

References r, and Aleph::Array< T >::size().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::operator*().

decrement_index_at()

Array< size_t > Aleph::multi_poly_detail::decrement_index_at ( const Array< size_t > &  idx, const size_t  var, const size_t  n = 1  )

inline

Decrement exponent at a specific variable.

Definition at line 137 of file tpl_multi_polynomial.H.

References r.

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::partial().

divides_index()

bool Aleph::multi_poly_detail::divides_index ( const Array< size_t > &  alpha, const Array< size_t > &  beta, size_t  nvars  )

inlinenoexcept

Test whether alpha divides beta component-wise (alpha(i) <= beta(i) all i).

Definition at line 200 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::divmod(), Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::minimize_groebner_basis(), and Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::redundant_pair_by_chain_criterion().

eval_monomial()

template<typename Coefficient >

Coefficient Aleph::multi_poly_detail::eval_monomial	(	const Array< Coefficient > &	pt,
		const Array< size_t > &	alpha
	)

Evaluate monomial at a point (helper for fitting/interpolation).

Definition at line 190 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp(), int_power(), and Aleph::Array< T >::size().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::fit_parallel().

extend_index()

Array< size_t > Aleph::multi_poly_detail::extend_index ( const Array< size_t > &  idx, size_t  target  )

inline

Pad a multi-index with trailing zeros to reach target size.

Definition at line 106 of file tpl_multi_polynomial.H.

References r, and Aleph::Array< T >::size().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::norm(), Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::operator*(), and Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::promote().

falling_factorial()

template<typename T >

T Aleph::multi_poly_detail::falling_factorial	(	const size_t	k,
		const size_t	n
	)

Falling factorial k(k-1)(k-2)...(k-n+1).

Returns

k!/(k-n)! if k >= n, else 0 (when k < n or n==0 -> 1).

Definition at line 148 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp(), and k.

flat_to_multi_index()

Array< size_t > Aleph::multi_poly_detail::flat_to_multi_index ( size_t  flat, const Array< size_t > &  sizes  )

inline

Convert flat index to multi-index given sizes per dimension.

Assumes row-major order (rightmost dimension varies fastest).

Definition at line 163 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp(), r, and Aleph::Array< T >::size().

int_power()

template<typename T >

T Aleph::multi_poly_detail::int_power	(	const T &	base,
		size_t	exp
	)

Integer power by repeated squaring.

Definition at line 118 of file tpl_multi_polynomial.H.

References exp().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::_eval_monomial(), Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::_substitute_var(), Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::eval(), eval_monomial(), and Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::homomorphic_eval().

is_zero_index()

bool Aleph::multi_poly_detail::is_zero_index ( const Array< size_t > &  idx )

inlinenoexcept

Test whether a multi-index is the zero vector.

Definition at line 83 of file tpl_multi_polynomial.H.

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::is_constant(), and Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::to_str().

lcm_indices()

Array< size_t > Aleph::multi_poly_detail::lcm_indices ( const Array< size_t > &  a, const Array< size_t > &  b, const size_t  nvars  )

inline

Component-wise max = monomial LCM.

Definition at line 213 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp(), r, and Aleph::Array< T >::size().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::make_pair_candidate(), and Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::s_poly().

multi_to_flat_index()

size_t Aleph::multi_poly_detail::multi_to_flat_index ( const Array< size_t > &  midx, const Array< size_t > &  sizes  )

inline

Convert multi-index to flat index given sizes per dimension.

Definition at line 176 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp(), and Aleph::Array< T >::size().

sub_indices()

Array< size_t > Aleph::multi_poly_detail::sub_indices ( const Array< size_t > &  beta, const Array< size_t > &  alpha, const size_t  nvars  )

inline

Component-wise subtraction beta - alpha (requires alpha divides beta).

Definition at line 222 of file tpl_multi_polynomial.H.

References Aleph::divide_and_conquer_partition_dp(), r, and Aleph::Array< T >::size().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::divmod(), and Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::s_poly().

total_degree()

size_t Aleph::multi_poly_detail::total_degree ( const Array< size_t > &  idx )

inlinenoexcept

Total degree of a multi-index (sum of exponents).

Definition at line 74 of file tpl_multi_polynomial.H.

References Aleph::sum().

Referenced by Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::degree(), Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::make_pair_candidate(), Aleph::Grlex_Order::operator()(), and Aleph::Grevlex_Order::operator()().