Numerical Algebraic Geometry: homotopy continuation PathTracker and supporting numeric types. More...

#include "engine-includes.hpp"
#include <algorithm>
#include <assert.h>
#include <map>
#include <math.h>
#include <memory>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <sys/types.h>
#include <utility>
#include <vector>
#include "interface/NAG.h"
#include "matrix.hpp"
#include "SLP.hpp"
#include "aring-CC.hpp"
#include "aring-glue.hpp"
#include "aring.hpp"
#include "buffer.hpp"
#include "error.h"
#include "hash.hpp"
#include "newdelete.hpp"
#include "ring.hpp"
#include "ringelem.hpp"

Classes
class	M2PointArray
	MutableEngineObject wrapper that owns a PointArray via unique_ptr. More...
class	PointArray
	Container of numerical points equipped with an $\varepsilon$ -tolerance and a random weight vector used to bucket approximately equal points. More...
class	complex
class	ComplexField
	Field-traits tag used as the template parameter of SLP<Field> to pick the complex element type. More...
class	SLP< Field >
class	StraightLineProgram
struct	Solution
	One numerical solution produced by a PathTracker run, with the full per-path diagnostic record. More...
class	PathTracker
	Numerical homotopy-continuation path tracker for systems of polynomial equations. More...

Macros
#define	CCC M2::ConcreteRing<M2::ARingCC>
#define	libPREFIX "/tmp/slpFN."
#define	slpCOMPILED 100
#define	slpPREDICTOR 101
#define	slpCORRECTOR 102
#define	slpEND 0
#define	slpCOPY 1
#define	slpMULTIsum 2
#define	slpPRODUCT 3
#define	ZERO_CONST -1
#define	ONE_CONST -2
#define	RUNGE_KUTTA 1
#define	TANGENT 2
#define	EULER 3
#define	PROJECTIVE_NEWTON 4
#define	MAX_NUM_SLPs 100
#define	CONST_OFFSET 0x1000
#define	SLP_HEADER_LEN 4
#define	MAX_NUM_PATH_TRACKERS 10

Functions
const CCC *	cast_to_CCC (const Ring *R)
ring_elem	from_doubles (const CCC *C, double re, double im)
gmp_CC	toBigComplex (const CCC *C, ring_elem a)
template<class Field>
void	zero_complex_array (int n, typename Field::element_type *a)
template<class Field>
void	copy_complex_array (int n, const typename Field::element_type a, typename Field::element_type b)
template<class Field>
Field::element_type *	make_copy_complex_array (int n, const typename Field::element_type *a)
template<class Field>
void	multiply_complex_array_scalar (int n, typename Field::element_type *a, const typename Field::element_type b)
template<class Field>
void	add_to_complex_array (int n, typename Field::element_type a, const typename Field::element_type b)
template<class Field>
void	negate_complex_array (int n, typename Field::element_type *a)
template<class Field>
double	norm2_complex_array (int n, typename Field::element_type *a)
template<typename R>
void	evaluateSLP (const SLProgram &slp, std::vector< typename R::ElementType > &values)
int	degree_ring_elem (const PolyRing *R, ring_elem re)
void	print_complex_matrix (int size, const double *A)

Detailed Description

Numerical Algebraic Geometry: homotopy continuation PathTracker and supporting numeric types.

Note: AI-generated documentation. Verify against the source before relying on it.

Complements the engine's symbolic Groebner-basis / resolution machinery by tracking solutions of a polynomial system F : C^n -> C^m over hardware-precision complex numbers. Declares PathTracker, the predictor-corrector homotopy continuation driver — it owns the homotopy H, start system S, target system T, their derivative SLPs (slpHxt, slpHxtH, slpSxS, slpTxT, ...), and the Bombieri-Weyl pairing <S, T> that gates the projective-Newton geodesic — along with Solution (a tracked path's terminal coordinates, status, condition number, step count) and the auxiliary complex / PointArray / M2PointArray value types. Witness sets and monodromy live in top-level M2 packages that call these path-tracking primitives, not in this header.

Polynomial evaluation at every continuation step goes through a straight-line program compiled once from the input system; the SLP machinery lives in SLP.hpp, SLP-defs.hpp, and SLP-imp.hpp. The complex class is a plain (double, double) pair with MPFR-to-double down-conversion in its gmp_CC constructor; the file is unusually liberal with STL containers compared to the rest of the engine because its values are not garbage-collected. Originally contributed by Anton Leykin; large portions are in the public domain.

See also: SLP.hpp; SLP-defs.hpp; SLP-imp.hpp; interface/NAG.h; aring-CC.hpp

Definition in file NAG.hpp.

Classes

Macros

Functions

Detailed Description