bertini::Slice Class Reference

A linear slice of affine or projective space: a stack of linear forms, M [x ; 1]. More...

#include <slice.hpp>

Public Member Functions
	Slice ()=default
	An empty slice (zero forms, zero variables).
template<typename NumT>
void	Eval (Vec< NumT > &result, Vec< NumT > const &x) const
	Evaluate the slice's linear-form values, in-place.
template<typename NumT>
Vec< NumT >	Eval (Vec< NumT > const &x) const
	Evaluate the slice's linear-form values.
template<typename NumT>
void	Jacobian (Mat< NumT > &result, Vec< NumT > const &x) const
	The slice's Jacobian (its constant variable-coefficient matrix), in-place.
template<typename NumT>
Mat< NumT >	Jacobian (Vec< NumT > const &x) const
	The slice's Jacobian (its constant variable-coefficient matrix).
Mat< complex_mp > const &	Coefficients () const
	The augmented coefficient matrix: one row per linear form, (Dimension) x (NumVariables+1), the trailing column carrying each form's constant term.
blocks::LinearFormsBlock const &	AsLinearFormsBlock () const
	The underlying linear-forms block (eval / Jacobian / precision engine).
void	AddTo (System &s) const
	Add this slice's linear forms to a System as a LinearFormsBlock. (Defined in slice.cpp.).
System	AsSystem () const
	A standalone System whose functions are exactly this slice's linear forms (over the slice's variable group).
Slice	Concatenate (Slice const &other) const
	A new slice stacking this slice's linear forms on top of other's.
Slice	Head (unsigned m) const
	A new slice over the same variables built from the first m linear forms.
Slice	Tail (unsigned m) const
	A new slice over the same variables built from the last m linear forms.
Slice	Rows (std::vector< unsigned > const &indices) const
	A new slice over the same variables built from the chosen linear forms.
unsigned	Dimension () const
	The dimension of the slice – the number of linear forms.
unsigned	NumVariables () const
	The number of variables sliced.
VariableGroup const &	Variables () const
	The variables the slice is a function of.
bool	IsHomogeneous () const
	Whether the slice was authored without constant terms (passes through the origin).
unsigned	Precision () const
	Get the current working precision of the slice, in digits.
void	Precision (unsigned new_precision) const
	Set the working precision of the slice, in digits.

Static Public Member Functions
static Slice	FromCoefficients (VariableGroup const &v, Mat< complex_mp > const &augmented_coefficients, bool homogeneous=false)
	Build a slice directly from an augmented coefficient matrix.
static Slice	ThroughPoint (VariableGroup const &v, Mat< complex_mp > const &coefficients, Vec< complex_mp > const &point)
	Build an affine slice whose every linear form vanishes at a given point.
static Slice	RandomReal (VariableGroup const &v, unsigned dim, bool homogeneous=false, bool orthogonal=true, Vec< complex_mp > const *through_point=nullptr)
	Produce a random real slice on a variable group, slicing a given number of dimensions.
static Slice	RandomComplex (VariableGroup const &v, unsigned dim, bool homogeneous=false, bool orthogonal=true, Vec< complex_mp > const *through_point=nullptr)
	Generate a random complex slice.
static Slice	Make (VariableGroup const &v, unsigned dim, bool homogeneous, bool orthogonal, bool real, std::function< void(complex_mp &, unsigned)> gen, Vec< complex_mp > const *through_point=nullptr)
	Factory for generating slices.

Friends
class	boost::serialization::access
std::ostream &	operator<< (std::ostream &out, Slice const &s)
	Provides output streaming for Slice.

Detailed Description

A linear slice of affine or projective space: a stack of linear forms, M [x ; 1].

The slice is held as an augmented coefficient matrix (one row per linear form, the trailing column being that form's constant term) inside a LinearFormsBlock. A homogeneous slice simply has a zero constant column. Slices compose: Head / Tail / Rows return a new Slice over the same variables built from a subset of the linear forms.

Constructor & Destructor Documentation

Slice()

bertini::Slice::Slice ( )

default

An empty slice (zero forms, zero variables).

Member Function Documentation

AddTo()

void bertini::Slice::AddTo

(

System &

s

)

const

Add this slice's linear forms to a System as a LinearFormsBlock. (Defined in slice.cpp.).

AsLinearFormsBlock()

blocks::LinearFormsBlock const & bertini::Slice::AsLinearFormsBlock ( ) const

inline

The underlying linear-forms block (eval / Jacobian / precision engine).

AsSystem()

System bertini::Slice::AsSystem

(

)

const

A standalone System whose functions are exactly this slice's linear forms (over the slice's variable group).

Lets a slice be carried around and evaluated / tracked on its own. (Defined in slice.cpp.)

Coefficients()

Mat< complex_mp > const & bertini::Slice::Coefficients ( ) const

inline

The augmented coefficient matrix: one row per linear form, (Dimension) x (NumVariables+1), the trailing column carrying each form's constant term.

These rows are also factor rows for a ProductsOfLinearsBlock, so a slice composes directly into the product-of-linears form regeneration uses.

Concatenate()

Slice bertini::Slice::Concatenate ( Slice const & other ) const

inline

A new slice stacking this slice's linear forms on top of other's.

Both slices must be on the same number of variables. The result is homogeneous only if both operands are. This is how you build a higher-codimension slice from pieces (and the Python + operator).

Dimension()

unsigned bertini::Slice::Dimension ( ) const

inline

The dimension of the slice – the number of linear forms.

Eval() [1/2]

template<typename NumT>

void bertini::Slice::Eval ( Vec< NumT > & result, Vec< NumT > const & x ) const

inline

Evaluate the slice's linear-form values, in-place.

Eval() [2/2]

template<typename NumT>

Vec< NumT > bertini::Slice::Eval ( Vec< NumT > const & x ) const

inline

Evaluate the slice's linear-form values.

FromCoefficients()

Slice bertini::Slice::FromCoefficients ( VariableGroup const & v, Mat< complex_mp > const & augmented_coefficients, bool homogeneous = false )

inlinestatic

Build a slice directly from an augmented coefficient matrix.

Parameters

v	The variables the slice is a function of.
augmented_coefficients	One row per linear form, (number-of-forms) x (v.size()+1); the trailing column is each form's constant term (zero, for a homogeneous slice).
homogeneous	Whether the slice was authored without constant terms.

Head()

Slice bertini::Slice::Head ( unsigned m ) const

inline

A new slice over the same variables built from the first m linear forms.

IsHomogeneous()

bool bertini::Slice::IsHomogeneous ( ) const

inline

Whether the slice was authored without constant terms (passes through the origin).

Jacobian() [1/2]

template<typename NumT>

void bertini::Slice::Jacobian ( Mat< NumT > & result, Vec< NumT > const & x ) const

inline

The slice's Jacobian (its constant variable-coefficient matrix), in-place.

Jacobian() [2/2]

template<typename NumT>

Mat< NumT > bertini::Slice::Jacobian ( Vec< NumT > const & x ) const

inline

The slice's Jacobian (its constant variable-coefficient matrix).

Make()

Slice bertini::Slice::Make ( VariableGroup const & v, unsigned dim, bool homogeneous, bool orthogonal, bool real, std::function< void(complex_mp &, unsigned)> gen, Vec< complex_mp > const * through_point = nullptr )

inlinestatic

Factory for generating slices.

Generates the variable-coefficient block (optionally orthonormalized by a QR factorization) and the constant column, then assembles the augmented matrix the LinearFormsBlock holds.

Parameters

v	The variable group the slice is over.
dim	The number of linear forms (the slice's dimension).
homogeneous	Whether the slice is homogeneous (zero constant column).
orthogonal	Whether to orthonormalize the coefficient block via a QR factorization.
real	Whether the coefficients are real (a real orthonormal block on the orthogonal path, matching the real gen used on the non-orthogonal path and for the constant column).
gen	The scalar generator used for the non-orthogonal coefficients and the constant column.
through_point	If non-null, a point the slice must pass through (length v.size(); affine, or homogeneous when the slice is homogeneous); overrides the random constant column.

NumVariables()

unsigned bertini::Slice::NumVariables ( ) const

inline

The number of variables sliced.

Precision() [1/2]

unsigned bertini::Slice::Precision ( ) const

inline

Get the current working precision of the slice, in digits.

Precision() [2/2]

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

inline

Set the working precision of the slice, in digits.

RandomComplex()

Slice bertini::Slice::RandomComplex ( VariableGroup const & v, unsigned dim, bool homogeneous = false, bool orthogonal = true, Vec< complex_mp > const * through_point = nullptr )

inlinestatic

Generate a random complex slice.

Parameters

v	The variable group the slice is over.
dim	The number of linear forms (the slice's dimension).
homogeneous	Whether the slice is homogeneous (zero constant column).
orthogonal	Whether to orthonormalize the coefficient block via a QR factorization.
through_point	If non-null, a point the slice must pass through (length v.size(); affine, or homogeneous when homogeneous=true).

RandomReal()

Slice bertini::Slice::RandomReal ( VariableGroup const & v, unsigned dim, bool homogeneous = false, bool orthogonal = true, Vec< complex_mp > const * through_point = nullptr )

inlinestatic

Produce a random real slice on a variable group, slicing a given number of dimensions.

Parameters

v	The variable group the slice is over.
dim	The number of linear forms (the slice's dimension).
homogeneous	Whether the slice is homogeneous (zero constant column).
orthogonal	Whether to orthonormalize the coefficient block via a QR factorization.
through_point	If non-null, a point the slice must pass through (length v.size(); affine, or homogeneous when homogeneous=true).

Rows()

Slice bertini::Slice::Rows ( std::vector< unsigned > const & indices ) const

inline

A new slice over the same variables built from the chosen linear forms.

Tail()

Slice bertini::Slice::Tail ( unsigned m ) const

inline

A new slice over the same variables built from the last m linear forms.

ThroughPoint()

Slice bertini::Slice::ThroughPoint ( VariableGroup const & v, Mat< complex_mp > const & coefficients, Vec< complex_mp > const & point )

inlinestatic

Build an affine slice whose every linear form vanishes at a given point.

Given a bare (non-augmented) coefficient block A and a point p, this assembles the augmented matrix [ A | -A*p ], so each form A_i*x - A_i*p = 0 passes through p. The result is affine (non-homogeneous): a homogeneous form cannot pass through an arbitrary point via a constant term (it would need its rows orthogonal to p – see the random factories' homogeneous through-point path).

Parameters

v	The variables the slice is a function of.
coefficients	The bare coefficient block, (number-of-forms) x v.size() (NOT augmented).
point	The point the slice must pass through, length v.size().

Variables()

VariableGroup const & bertini::Slice::Variables ( ) const

inline

The variables the slice is a function of.

Friends And Related Symbol Documentation

boost::serialization::access

friend class boost::serialization::access

friend

operator<<

std::ostream & operator<< ( std::ostream & out, Slice const & s )

friend

Provides output streaming for Slice.