An affine chart on a product of projective spaces. More...

#include <patch.hpp>

Public Member Functions
	Patch ()
	Default constructor.
	Patch (Patch const &other)
	Custom copy constructor, ensuring the max-precision coefficients are copied in highest precision.
Patch &	operator= (Patch const &other)=default
	Copy assignment.
	Patch (std::vector< unsigned > const &sizes)
	Constructor making a random complex patch on a space whose structure is described by the input argument.
	Patch (std::vector< unsigned > const &sizes, std::vector< Vec< complex_mp > > const &coefficients)
	Construct a patch from given coefficients – for a loader restoring an archived patch exactly (the canonical-encoding reader), never for making a fresh one (use Random).
unsigned	Precision () const
	Get the current precision of the patch.
template<typename T>
void	SyncPrecision (Vec< T > const &x) const
	Materialize the working coefficients at the precision of the point being evaluated.
void	Precision (unsigned new_precision) const
	Set the precision of the patch.
template<typename Derived, typename T>
void	EvalInPlace (Eigen::MatrixBase< Derived > &function_values, Vec< T > const &x) const
	Evaluate the patch at a point, in place.
template<typename T>
Vec< T >	Eval (Vec< T > const &x) const
	Evaluate the patch at a point.
template<typename Derived, typename T>
void	JacobianInPlace (Eigen::MatrixBase< Derived > &jacobian, Vec< T > const &x) const
	Evaluate the Jacobian matrix, in place.
template<typename T>
Mat< T >	Jacobian (Vec< T > const &x) const
	Evaluate the Jacobian matrix, in place.
template<typename T>
void	RescalePointToFitInPlace (Vec< T > &x) const
	Rescale a point so that it satisfies the patch equations herein contained.
template<typename T>
Vec< T >	RescalePoint (Vec< T > const &x) const
	Rescale a point so that it satisfies the patch equations herein contained.
unsigned	NumVariableGroups () const
	Get the number of variable groups.
unsigned	NumVariables () const
	Get the number of variables in the patch.
std::vector< Vec< complex_mp > > const &	Coefficients () const
	The patch's highest-precision coefficients, one vector per variable group.
std::vector< unsigned > const &	VariableGroupSizes () const
	The number of variables in each variable group (homogenizing variable included).
bool	operator== (Patch const &rhs) const
	Check whether two patches are the same.
bool	operator!= (Patch const &rhs) const
	Check whether two patches are different.

Static Public Member Functions
static Patch	Random (std::vector< unsigned > const &sizes)
	Construct a random complex patch on a space.
static Patch	RandomReal (std::vector< unsigned > const &sizes)
	Construct random REAL patch on a space.

Friends
class	boost::serialization::access
std::ostream &	operator<< (std::ostream &out, Patch const &p)
	Stream-insertion for a Patch, summarizing its variable groups and coefficients.

Detailed Description

An affine chart on a product of projective spaces.

This class provides an adjustable-precision patch on a working space for solving a polynomial system using homotopy continuation.

The constant values for the patch are unity.

Example code, not using a bertini::System to generate the patch.

std::vector<unsigned> s{2,3};
 
Patch p(s);
 
Vec<complex_mp> v(5);
v << complex_mp(1),  complex_mp(1),  complex_mp(1),  complex_mp(1),  complex_mp(1);
 
auto v_rescaled = p.RescalePoint(v);
 
auto f = p.Eval(v_rescaled);

At the end of the above example, the patch evaluates to 0 in the container $f$ .

Other ways of patching

Patches can also be applied to a system, which should first be homogenized.

Constructor & Destructor Documentation

◆ Patch() [1/4]

bertini::Patch::Patch ( )

inline

Default constructor.

Makes an empty patch.

◆ Patch() [2/4]

bertini::Patch::Patch ( Patch const & other )

inline

Custom copy constructor, ensuring the max-precision coefficients are copied in highest precision.

◆ Patch() [3/4]

bertini::Patch::Patch ( std::vector< unsigned > const & sizes )

inline

Constructor making a random complex patch on a space whose structure is described by the input argument.

The sizes input give the number of total variables, including homogenizing variables, for the product of spaces forming the total space to be patched. The patch has no idea whether the underlying space is projective or affine – that is handled somewhere else, likely in a bertini::System.

The initial precision of the patch is set to current default precision, and the precision of the highest precision coefficients are set to the current default as well.

Parameters

sizes The sizes of the variable groups, including homogenizing variables if present.

◆ Patch() [4/4]

bertini::Patch::Patch	(	std::vector< unsigned > const &	sizes,
		std::vector< Vec< complex_mp > > const &	coefficients )

inline

Construct a patch from given coefficients – for a loader restoring an archived patch exactly (the canonical-encoding reader), never for making a fresh one (use Random).

Parameters

sizes	The sizes of the variable groups, including homogenizing variables.
coefficients	One coefficient vector per group, each of that group's size; every entry is kept at its own precision.

Member Function Documentation

◆ Coefficients()

std::vector< Vec< complex_mp > > const & bertini::Patch::Coefficients ( ) const

inline

The patch's highest-precision coefficients, one vector per variable group.

Coefficient Coefficients()[ii](jj) multiplies the jj-th variable of group ii (in the system's variable ordering). The patch function for group ii is sum_jj Coefficients()[ii](jj) * x_jj - 1, so these are exactly the constant entries of the patch's (symbolic) Jacobian rows.

◆ Eval()

template<typename T>

Vec< T > bertini::Patch::Eval ( Vec< T > const & x ) const

inline

Evaluate the patch at a point.

Returns: The values of the patch at the point.

Parameters

x	The current space point at which to evaluate.

◆ EvalInPlace()

template<typename Derived, typename T>

void bertini::Patch::EvalInPlace	(	Eigen::MatrixBase< Derived > &	function_values,
		Vec< T > const &	x ) const

inline

Evaluate the patch at a point, in place.

Parameters

function_values	The vector to populate. Must be at least as long as the number of variable groups.
x	The current space point at which to evaluate.

◆ Jacobian()

template<typename T>

Mat< T > bertini::Patch::Jacobian ( Vec< T > const & x ) const

inline

Evaluate the Jacobian matrix, in place.

Parameters

x	Point at which to evaluate. Not technically needed, because the Jacobian is simply the matrix of coefficients.

Returns: Jacobian matrix, of size (NumVariableGroups x NumVariables).

◆ JacobianInPlace()

template<typename Derived, typename T>

void bertini::Patch::JacobianInPlace	(	Eigen::MatrixBase< Derived > &	jacobian,
		Vec< T > const &	x ) const

inline

Evaluate the Jacobian matrix, in place.

Parameters

jacobian	Matrix to populate with the Jacobian. Must be large enough (NumVariableGroups x NumVariables).
x	Point at which to evaluate. Not technically needed, because the Jacobian is simply the matrix of coefficients.

◆ NumVariableGroups()

unsigned bertini::Patch::NumVariableGroups ( ) const

inline

Get the number of variable groups.

Returns: The number of variable groups in the problem.

◆ NumVariables()

unsigned bertini::Patch::NumVariables ( ) const

inline

Get the number of variables in the patch.

◆ operator!=()

bool bertini::Patch::operator!= ( Patch const & rhs ) const

inline

Check whether two patches are different.

◆ operator=()

Patch & bertini::Patch::operator= ( Patch const & other )

default

Copy assignment.

Explicitly defaulted (memberwise), matching the previously-implicit behavior. Note this differs from the custom copy constructor, which re-downsamples at current default precision.

◆ operator==()

bool bertini::Patch::operator== ( Patch const & rhs ) const

inline

Check whether two patches are the same.

Returns: true If have the same variable structure and coefficients. Otherwise, false.

◆ Precision() [1/2]

unsigned bertini::Patch::Precision ( ) const

inline

Get the current precision of the patch.

Returns: The current precision, in digits.

◆ Precision() [2/2]

void bertini::Patch::Precision ( unsigned new_precision ) const

inline

Set the precision of the patch.

Copies the patch coefficients into correct precision for subsequent precision.

Parameters

new_precision The precision to change to.

◆ Random()

Patch bertini::Patch::Random ( std::vector< unsigned > const & sizes )

inlinestatic

Construct a random complex patch on a space.

◆ RandomReal()

Patch bertini::Patch::RandomReal ( std::vector< unsigned > const & sizes )

inlinestatic

Construct random REAL patch on a space.

◆ RescalePoint()

template<typename T>

Vec< T > bertini::Patch::RescalePoint ( Vec< T > const & x ) const

inline

Rescale a point so that it satisfies the patch equations herein contained.

Returns: Point rescaled to fit the patch. cannot be the zero vector – zero is a degenerate point in projective space.

Parameters

x	The point to rescale.

Template Parameters

T	The number type.

◆ RescalePointToFitInPlace()

template<typename T>

void bertini::Patch::RescalePointToFitInPlace ( Vec< T > & x ) const

inline

Rescale a point so that it satisfies the patch equations herein contained.

The point must be as long as there are total variables.

This function modifies the vector in place, so ensure you have a copy stored somewhere else if you feel it necessary to still have the pre-rescaled vector around.

Parameters

x	The point to rescale

Template Parameters

T	The number type.

◆ SyncPrecision()

template<typename T>

void bertini::Patch::SyncPrecision ( Vec< T > const & x ) const

inline

Materialize the working coefficients at the precision of the point being evaluated.

Every evaluable type self-aligns this way, so no caller and no owning System has to fan a precision out beforehand – ADR-0057. Precision() short-circuits when already there, so the steady state is one integer compare. No-op for double.

◆ VariableGroupSizes()

std::vector< unsigned > const & bertini::Patch::VariableGroupSizes ( ) const

inline

The number of variables in each variable group (homogenizing variable included).

◆ boost::serialization::access

friend class boost::serialization::access

friend

◆ operator<<

std::ostream & operator<<	(	std::ostream &	out,
		Patch const &	p )

friend

Stream-insertion for a Patch, summarizing its variable groups and coefficients.

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

Example code, not using a bertini::System to generate the patch.

Other ways of patching

Constructor & Destructor Documentation

◆ Patch() [1/4]

◆ Patch() [2/4]

◆ Patch() [3/4]

◆ Patch() [4/4]

Member Function Documentation

◆ Coefficients()

◆ Eval()

◆ EvalInPlace()

◆ Jacobian()

◆ JacobianInPlace()

◆ NumVariableGroups()

◆ NumVariables()

◆ operator!=()

◆ operator=()

◆ operator==()

◆ Precision() [1/2]

◆ Precision() [2/2]

◆ Random()

◆ RandomReal()

◆ RescalePoint()

◆ RescalePointToFitInPlace()

◆ SyncPrecision()

◆ VariableGroupSizes()

Friends And Related Symbol Documentation

◆ boost::serialization::access

◆ operator<<