tpl_multi_polynomial.H File Reference

Sparse multivariate polynomial over an arbitrary coefficient ring. More...

#include <sstream>
#include <string>
#include <type_traits>
#include <limits>
#include <cmath>
#include <utility>
#include <iomanip>
#include <future>
#include <thread>
#include <fstream>
#include <numeric>
#include <ah-errors.H>
#include <ahSort.H>
#include <tpl_array.H>
#include <tpl_dynBinHeap.H>
#include <tpl_dynMapTree.H>
#include <htlist.H>
#include <ah-parallel.H>
#include <tpl_polynomial.H>

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Classes
struct	Aleph::Lex_Order
	Lexicographic monomial order. More...
struct	Aleph::Grlex_Order
	Graded lexicographic monomial order. More...
struct	Aleph::Grevlex_Order
	Graded reverse lexicographic monomial order (default). More...
class	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >
	Sparse multivariate polynomial. More...
struct	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::Pair_Candidate
struct	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::Pair_Candidate_Less
struct	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::FactorTerm
	Factorization term: a factor polynomial with its multiplicity. More...
struct	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::Uni_Linear_Factor_Data
	Primitive linear factor data \(a x + b\) for a chosen main variable. More...
struct	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::Uni_Linear_Factor_Data_Less
	Deterministic ordering of primitive univariate linear factors. More...
struct	Aleph::Gen_MultiPolynomial< Coefficient, MonomOrder >::Uni_Factor_Signature_Less
	Deterministic ordering of primitive univariate factors by normalized signature. More...
struct	Aleph::PolyFitAnalysis< C >
	Structure holding residual analysis for polynomial fits. More...

Namespaces
namespace	Aleph
	Main namespace for Aleph-w library functions.
namespace	Aleph::multi_poly_detail
	Internal helpers for multivariate polynomial operations.

Typedefs
using	Aleph::MultiPolynomial = Gen_MultiPolynomial< double, Grevlex_Order >
	Multivariate polynomial with double coefficients, grevlex.
using	Aleph::MultiPolynomial_Lex = Gen_MultiPolynomial< double, Lex_Order >
	Multivariate polynomial with lex ordering.
using	Aleph::MultiPolynomial_Grlex = Gen_MultiPolynomial< double, Grlex_Order >
	Multivariate polynomial with grlex ordering.
using	Aleph::IntMultiPolynomial = Gen_MultiPolynomial< long long, Grevlex_Order >
	Multivariate polynomial with integer (long long) coefficients, grevlex.
using	Aleph::IntMultiPolynomial_Lex = Gen_MultiPolynomial< long long, Lex_Order >
	Multivariate polynomial with integer (long long) coefficients, lex.

Functions
size_t	Aleph::multi_poly_detail::total_degree (const Array< size_t > &idx) noexcept
	Total degree of a multi-index (sum of exponents).
bool	Aleph::multi_poly_detail::is_zero_index (const Array< size_t > &idx) noexcept
	Test whether a multi-index is the zero vector.
Array< size_t >	Aleph::multi_poly_detail::add_indices (const Array< size_t > &a, const Array< size_t > &b)
	Component-wise addition of two multi-indices.
Array< size_t >	Aleph::multi_poly_detail::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	Aleph::multi_poly_detail::int_power (const T &base, size_t exp)
	Integer power by repeated squaring.
Array< size_t >	Aleph::multi_poly_detail::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	Aleph::multi_poly_detail::falling_factorial (const size_t k, const size_t n)
	Falling factorial k(k-1)(k-2)...(k-n+1).
Array< size_t >	Aleph::multi_poly_detail::flat_to_multi_index (size_t flat, const Array< size_t > &sizes)
	Convert flat index to multi-index given sizes per dimension.
size_t	Aleph::multi_poly_detail::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	Aleph::multi_poly_detail::eval_monomial (const Array< Coefficient > &pt, const Array< size_t > &alpha)
	Evaluate monomial at a point (helper for fitting/interpolation).
bool	Aleph::multi_poly_detail::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 >	Aleph::multi_poly_detail::lcm_indices (const Array< size_t > &a, const Array< size_t > &b, const size_t nvars)
	Component-wise max = monomial LCM.
Array< size_t >	Aleph::multi_poly_detail::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).
template<typename C , class M >
PolyFitAnalysis< C >	Aleph::analyze_fit (const Gen_MultiPolynomial< C, M > &poly, const Array< std::pair< Array< C >, C > > &data)
	Compute residual analysis for a polynomial fit.
template<typename C , class M >
Gen_MultiPolynomial< C, M >	Aleph::operator* (const C &s, const Gen_MultiPolynomial< C, M > &p)
	Scalar * polynomial (commutative).
template<typename C , class M >
Gen_MultiPolynomial< C, M >	Aleph::operator+ (const C &s, const Gen_MultiPolynomial< C, M > &p)
	Scalar + polynomial (commutative).
template<typename C , class M >
Gen_MultiPolynomial< C, M >	Aleph::operator- (const C &s, const Gen_MultiPolynomial< C, M > &p)
	Scalar - polynomial.
template<typename C , class M >
std::ostream &	Aleph::operator<< (std::ostream &out, const Gen_MultiPolynomial< C, M > &p)
	Stream insertion.

Detailed Description

Sparse multivariate polynomial over an arbitrary coefficient ring.

Provides Gen_MultiPolynomial<Coefficient, MonomOrder>, a template class representing polynomials in \(n\) variables with a sparse map from multi-index (exponent vector) to coefficient. Three standard monomial orderings are included: lexicographic (Lex_Order), graded lexicographic (Grlex_Order), and graded reverse lexicographic (Grevlex_Order, the default).

Layer 1 of the multivariate polynomial subsystem — covers representation, arithmetic, evaluation, and pretty-printing.

Author

Leandro Rabindranath Leon

Definition in file tpl_multi_polynomial.H.