#include <Monomials_fwd.hpp>

Detailed Description

template<class vec_t>
class mm::Monomials< vec_t >

A class representing Monomial basis.

This class satisfies the Basis function concept.

Usage example:

    Monomials<Vec2d> basis(2);  // {1, y, x, y^2, xy, x^2}
    std::cout << basis << std::endl;  // Monomials 2D: [0, 0; 0, 1; 1, 0; 0, 2; 1, 1; 2, 0], ...
    double value = basis.eval(2, {0.3, -2.4});  // value = 0.3
 
    basis = Monomials<Vec2d>::tensorBasis(1);  // tensor basis {1, x, y, xy}
    basis = Monomials<Vec2d>({{1, 0}, {2, 3}});  // from powers
    // Evaluate derivative or any other supported operator.
    double der = basis.evalOp(1, {0.3, -2.4}, Derivative<2>({2, 1}));  // der = 2*3*(-2.4)^2

Examples: test/end2end/poisson_explicit.cpp.

Definition at line 38 of file Monomials_fwd.hpp.

Public Member Functions
	Monomials ()=default
	Construct empty monomial basis with `size = 0`. More...

	Monomials (int order)
	Construct a basis of all monomials of combined order lower or equal to `order`. More...

	Monomials (const std::vector< std::vector< int >> &powers)
	Construct monomial basis from monomials with specific powers. More...

int	size () const
	Return number of monomials in this basis. More...

const Eigen::Matrix< int, dim, Eigen::Dynamic > &	powers () const
	Get saved monomial powers. More...

scalar_t	eval (int index, const vector_t &point, const std::vector< vector_t > &={}) const
	Evaluates `index`-th monomial' at `point`. More...

template<typename operator_t >
scalar_t	evalOp (int index, const vector_t &point, operator_t op, const std::vector< vector_t > &support={}, scalar_t scale=1) const
	Apply an operator at a given point. More...

scalar_t	evalOp (int index, const vector_t &point, Lap< dim > op, const std::vector< vector_t > &support={}, scalar_t scale=1) const
	Evaluate Laplacian. More...

scalar_t	evalOp (int index, const vector_t &point, Der1< dim > op, const std::vector< vector_t > &support={}, scalar_t scale=1) const
	Evaluate 1st derivative. More...

scalar_t	evalOp (int index, const vector_t &point, Der2< dim > op, const std::vector< vector_t > &support={}, scalar_t scale=1) const
	Evaluate 2nd derivative. More...

scalar_t	evalOp (int index, const vector_t &point, Derivative< dim > op, const std::vector< vector_t > &support={}, scalar_t scale=1) const
	Evaluate general derivative. More...

scalar_t	evalAt0 (int index, const std::vector< vector_t > &={}) const
	Evaluate `index`-th monomial at zero. More...

template<typename operator_t >
scalar_t	evalOpAt0 (int index, const operator_t&op, const std::vector< vector_t > &support={}, scalar_t scale=1) const
	Evaluate an operator on `index`-th monomial at zero. More...

scalar_t	evalOpAt0 (int index, const Lap< dim > &lap, const std::vector< vector_t > &={}, scalar_t scale=1) const
	Evaluate Laplacian `lap` at zero. More...

scalar_t	evalOpAt0 (int index, const Der1< dim > &der1, const std::vector< vector_t > &={}, scalar_t scale=1) const
	Evaluate first derivative operator `der1` at zero. More...

scalar_t	evalOpAt0 (int index, const Der2< dim > &der2, const std::vector< vector_t > &={}, scalar_t scale=1) const
	Evaluate second derivative operator `der2` at zero. More...

scalar_t	evalOpAt0 (int index, const Derivative< dim > &der, const std::vector< vector_t > &={}, scalar_t scale=1) const
	Evaluate general derivative operator at zero. More...

Static Public Member Functions
static Monomials< vec_t >	tensorBasis (int order)
	Construct a tensor basis of monomials ranging from 0 up to `order` (inclusive) in each dimension. More...

Public Types
enum	{ dim = vec_t::dim }
	Store dimension of the domain. More...

typedef vec_t	vector_t
	Vector type. More...

typedef vector_t::scalar_t	scalar_t
	Scalar type. More...

Friends
template<class V >
std::ostream &	operator<< (std::ostream &os, const Monomials< V > &m)
	Output basic info about given Monomial basis. More...

Private Member Functions
void	setFromPowers (const std::vector< std::vector< int >> &powers)
	Constructs basis from a vector of powers. More...

Static Private Member Functions
static std::vector< std::vector< int > >	generatePowers (int max_order, int dim)
	Generate powers for `dim`-d monomials up to `max_order`. More...

Private Attributes
Eigen::Matrix< int, dim, Eigen::Dynamic >	powers_
	A vector describing monomials with the powers of every coordinate. More...

Member Enumeration Documentation

◆ anonymous enum

template<class vec_t >

anonymous enum

Store dimension of the domain.

Enumerator
dim	Dimensionality of the function domain.

Definition at line 43 of file Monomials_fwd.hpp.

Constructor & Destructor Documentation

◆ Monomials() [1/3]

template<class vec_t >

mm::Monomials< vec_t >::Monomials ( )

default

Construct empty monomial basis with size = 0.

◆ Monomials() [2/3]

template<class vec_t >

mm::Monomials< vec_t >::Monomials ( int order )

Construct a basis of all monomials of combined order lower or equal to order.

Parameters

order Maximal combined order of monomials to be used. If order is -1 then empty basis is constructed.

Example: If you call this with an order = 2 parameter and Vec2d template parameter, your basis will consist of \( \{1, x, y, x^2, xy, y^2\} \) (not necessarily in that order).

Definition at line 18 of file Monomials.hpp.

◆ Monomials() [3/3]

template<class vec_t >

mm::Monomials< vec_t >::Monomials ( const std::vector< std::vector< int >> & powers )

inline

Construct monomial basis from monomials with specific powers.

Parameters

powers List of lists of size dim, each representing powers of a monomial. For example {{1, 2}, {0, 3}, {2, 0}} in 2D represents monomials \(\{x y^2, y^3, x^2\}\).

Definition at line 75 of file Monomials_fwd.hpp.

Member Function Documentation

◆ eval()

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::eval	(	int	index,
		const vector_t &	point,
		const std::vector< vector_t > &	= `{}`
	)		const

Evaluates index-th monomial' at point.

Parameters

index	A number in `[0, size())` specifying the index of a monomial to evaluate.
point	Point in which to evaluate the monomial.

Returns: Value of requested monomial at given point.

Exceptions

Assertion fails if index is out of range or if an invalid derivative is requested.

Definition at line 67 of file Monomials.hpp.

◆ evalAt0()

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalAt0	(	int	index,
		const std::vector< vector_t > &	= `{}`
	)		const

Evaluate index-th monomial at zero.

Definition at line 173 of file Monomials.hpp.

◆ evalOp() [1/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOp	(	int	index,
		const vector_t &	point,
		Der1< dim >	op,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate 1st derivative.

See also: evalOp

Definition at line 99 of file Monomials.hpp.

◆ evalOp() [2/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOp	(	int	index,
		const vector_t &	point,
		Der2< dim >	op,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate 2nd derivative.

See also: evalOp

Definition at line 115 of file Monomials.hpp.

◆ evalOp() [3/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOp	(	int	index,
		const vector_t &	point,
		Derivative< dim >	op,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate general derivative.

See also: evalOp

Definition at line 149 of file Monomials.hpp.

◆ evalOp() [4/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOp	(	int	index,
		const vector_t &	point,
		Lap< dim >	op,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate Laplacian.

See also: evalOp

Definition at line 79 of file Monomials.hpp.

◆ evalOp() [5/5]

template<class vec_t >

template<typename operator_t >

scalar_t mm::Monomials< vec_t >::evalOp	(	int	index,
		const vector_t &	point,
		operator_t	op,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

Apply an operator at a given point.

For \(\frac{d}{dx}\), the function computes \( \frac{d}{dx} p_i(x_s), x_s = \frac{x - c}{s} \).

Parameters

index	A number in `[0, size())` specifying the index of a monomial to evaluate.
point	Translated and scaled point \(x_s\) at which the basis function is evaluated.
op	The differential operator.
support	Translated and scaled values of nodes in the support domain. Unused.
scale	The scaling factor \(s\).

Returns: Value of the operator applied to index-th basis function at point point.

Exceptions

Assertion fails if index is out of range.

See also: Operator, Lap, Der1, Der2

◆ evalOpAt0() [1/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOpAt0	(	int	index,
		const Der1< dim > &	der1,
		const std::vector< vector_t > &	= `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate first derivative operator der1 at zero.

See also: evalOpAt0

Definition at line 207 of file Monomials.hpp.

◆ evalOpAt0() [2/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOpAt0	(	int	index,
		const Der2< dim > &	der2,
		const std::vector< vector_t > &	= `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate second derivative operator der2 at zero.

See also: evalOpAt0

Definition at line 228 of file Monomials.hpp.

◆ evalOpAt0() [3/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOpAt0	(	int	index,
		const Derivative< dim > &	der,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate general derivative operator at zero.

See also: evalOpAt0

Definition at line 262 of file Monomials.hpp.

◆ evalOpAt0() [4/5]

template<class vec_t >

vec_t::scalar_t mm::Monomials< vec_t >::evalOpAt0	(	int	index,
		const Lap< dim > &	lap,
		const std::vector< vector_t > &	= `{}`,
		scalar_t	scale = `1`
	)		const

Evaluate Laplacian lap at zero.

See also: evalOpAt0

Definition at line 189 of file Monomials.hpp.

◆ evalOpAt0() [5/5]

template<class vec_t >

template<typename operator_t >

scalar_t mm::Monomials< vec_t >::evalOpAt0	(	int	index,
		const operator_t&	op,
		const std::vector< vector_t > &	support = `{}`,
		scalar_t	scale = `1`
	)		const

inline

Evaluate an operator on index-th monomial at zero.

See also: eval

Definition at line 136 of file Monomials_fwd.hpp.

◆ generatePowers()

template<class vec_t >

std::vector< std::vector< int > > mm::Monomials< vec_t >::generatePowers	(	int	max_order,
		int	dim
	)

staticprivate

Generate powers for dim-d monomials up to max_order.

Definition at line 24 of file Monomials.hpp.

◆ powers()

template<class vec_t >

const Eigen::Matrix<int, dim, Eigen::Dynamic>& mm::Monomials< vec_t >::powers ( ) const

inline

Get saved monomial powers.

Definition at line 90 of file Monomials_fwd.hpp.

◆ setFromPowers()

template<class vec_t >

void mm::Monomials< vec_t >::setFromPowers ( const std::vector< std::vector< int >> & powers )

private

Constructs basis from a vector of powers.

Definition at line 54 of file Monomials.hpp.

◆ size()

template<class vec_t >

int mm::Monomials< vec_t >::size ( ) const

inline

Return number of monomials in this basis.

Definition at line 87 of file Monomials_fwd.hpp.

◆ tensorBasis()

template<class vec_t >

Monomials< vec_t > mm::Monomials< vec_t >::tensorBasis ( int order )

static

Construct a tensor basis of monomials ranging from 0 up to order (inclusive) in each dimension.

If order is -1 then empty basis is constructed.

Example: tensorBasis(2) in 2D constructs the set \(\{1, x, x^2, y, yx, yx^2, y^2, y^2x, y^2x^2\}\).

Examples: test/approximations/Monomials_test.cpp, test/approximations/WLS_test.cpp, test/end2end/cantilever_beam_implicit.cpp, and test/end2end/poisson_implicit.cpp.

Definition at line 41 of file Monomials.hpp.

Friends And Related Function Documentation

◆ operator<<

template<class vec_t >

template<class V >

std::ostream& operator<<	(	std::ostream &	os,
		const Monomials< V > &	m
	)

friend

Output basic info about given Monomial basis.

Definition at line 269 of file Monomials.hpp.

Member Data Documentation

◆ powers_

template<class vec_t >

Eigen::Matrix<int, dim, Eigen::Dynamic> mm::Monomials< vec_t >::powers_

private

A vector describing monomials with the powers of every coordinate.

Definition at line 47 of file Monomials_fwd.hpp.

The documentation for this class was generated from the following files:

include/medusa/bits/approximations/Monomials_fwd.hpp
include/medusa/bits/approximations/Monomials.hpp

Detailed Description

template<class vec_t> class mm::Monomials< vec_t >

Public Member Functions

Static Public Member Functions

Public Types

Friends

Private Member Functions

Static Private Member Functions

Private Attributes

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ Monomials() [1/3]

◆ Monomials() [2/3]

◆ Monomials() [3/3]

Member Function Documentation

◆ eval()

◆ evalAt0()

◆ evalOp() [1/5]

◆ evalOp() [2/5]

◆ evalOp() [3/5]

◆ evalOp() [4/5]

◆ evalOp() [5/5]

◆ evalOpAt0() [1/5]

◆ evalOpAt0() [2/5]

◆ evalOpAt0() [3/5]

◆ evalOpAt0() [4/5]

◆ evalOpAt0() [5/5]

◆ generatePowers()

◆ powers()

◆ setFromPowers()

◆ size()

◆ tensorBasis()

Friends And Related Function Documentation

◆ operator<<

Member Data Documentation

◆ powers_

template<class vec_t>
class mm::Monomials< vec_t >