Detailed Description

MonomialBasisElement represents a monomial, a product of powers of variables with non-negative integer exponents.

Note that it doesn't not include the coefficient part of a monomial. So x, x³y, xy²z are all valid MonomialBasisElement instances, but 1+x or 2xy²z are not. TODO(hongkai.dai): deprecate Monomial class and replace Monomial class with MonomialBasisElement class. For more information regarding the motivation of this class, please see Drake github issue #13602 and #13803.

#include <drake/common/symbolic/monomial_basis_element.h>

Public Member Functions
	MonomialBasisElement ()
	Constructs a monomial equal to 1. More...

	MonomialBasisElement (std::nullptr_t)
	Constructs a default value. More...

	MonomialBasisElement (const std::map< Variable, int > &var_to_degree_map)
	Constructs a MonomialBasisElement from variable to degree map. More...

	MonomialBasisElement (const Expression &e)
	Converts an expression to a monomial if the expression is written as ∏ᵢpow(xᵢ, kᵢ), otherwise throws a runtime error. More...

	MonomialBasisElement (const Eigen::Ref< const VectorX< Variable >> &vars, const Eigen::Ref< const Eigen::VectorXi > &degrees)
	Constructs a Monomial from a vector of variables `vars` and their corresponding integer degrees `degrees`. More...

	MonomialBasisElement (const Variable &var)
	Constructs a monomial basis element with only one variable, and the degree is 1. More...

	MonomialBasisElement (const Variable &var, int degree)
	Constructs a monomial basis element with only one variable, and the degree of that variable is given by `degree`. More...

std::pair< double, MonomialBasisElement >	EvaluatePartial (const Environment &env) const
	Partially evaluates using a given environment `env`. More...

MonomialBasisElement &	pow_in_place (int p)
	Returns this monomial raised to `p`. More...

bool	operator< (const MonomialBasisElement &other) const
	Compares two MonomialBasisElement in lexicographic order. More...

std::map< MonomialBasisElement, double >	Differentiate (const Variable &var) const
	Differentiates this MonomialBasisElement. More...

std::map< MonomialBasisElement, double >	Integrate (const Variable &var) const
	Integrates this MonomialBasisElement on a variable. More...

void	MergeBasisElementInPlace (const MonomialBasisElement &other)
	Merges this basis element with another basis element `other` by merging their var_to_degree_map. More...

std::map< ChebyshevBasisElement, double >	ToChebyshevBasis () const
	Converts this monomial to Chebyshev polynomial basis. More...

template<typename BasisElement >
std::map< BasisElement, double >	ToBasis () const
	Converts this monomial to a weighted sum of basis elements of type BasisElement. More...

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	MonomialBasisElement (const MonomialBasisElement &)=default

MonomialBasisElement &	operator= (const MonomialBasisElement &)=default

	MonomialBasisElement (MonomialBasisElement &&)=default

MonomialBasisElement &	operator= (MonomialBasisElement &&)=default

Public Member Functions inherited from PolynomialBasisElement
	PolynomialBasisElement ()=default
	Constructs a polynomial basis with empty var_to_degree map. More...

	PolynomialBasisElement (const std::map< Variable, int > &var_to_degree_map)
	Constructs a polynomial basis given the variable and the degree of that variable. More...

	PolynomialBasisElement (const Eigen::Ref< const VectorX< Variable >> &vars, const Eigen::Ref< const Eigen::VectorXi > &degrees)
	Constructs a polynomial basis, such that it contains the variable-to-degree map vars(i)→degrees(i). More...

virtual	~PolynomialBasisElement ()=default

const std::map< Variable, int > &	var_to_degree_map () const

const std::map< Variable, int > &	get_powers () const
	Returns variable to degree map. More...

int	total_degree () const
	Returns the total degree of a polynomial basis. More...

int	degree (const Variable &v) const
	Returns the degree of this PolynomialBasisElement in a variable `v`. More...

Variables	GetVariables () const

double	Evaluate (const Environment &env) const
	Evaluates under a given environment `env`. More...

bool	operator== (const PolynomialBasisElement &other) const

bool	operator!= (const PolynomialBasisElement &other) const

Expression	ToExpression () const

	PolynomialBasisElement (const PolynomialBasisElement &)=default

PolynomialBasisElement &	operator= (const PolynomialBasisElement &)=default

	PolynomialBasisElement (PolynomialBasisElement &&)=default

PolynomialBasisElement &	operator= (PolynomialBasisElement &&)=default

Friends
template<class HashAlgorithm >
void	hash_append (HashAlgorithm &hasher, const MonomialBasisElement &item) noexcept
	Implements the hash_append generic hashing concept. More...

Additional Inherited Members
Protected Member Functions inherited from PolynomialBasisElement
bool	lexicographical_compare (const PolynomialBasisElement &other) const
	Compares two PolynomialBasisElement using lexicographical order. More...

virtual bool	EqualTo (const PolynomialBasisElement &other) const

void	DoEvaluatePartial (const Environment &e, double coeff, std::map< Variable, int > new_basis_element) const

int *	get_mutable_total_degree ()

std::map< Variable, int > *	get_mutable_var_to_degree_map ()

void	DoMergeBasisElementInPlace (const PolynomialBasisElement &other)
	Merge this basis element with another basis element by merging their var_to_degree_map. More...

Constructor & Destructor Documentation

◆ MonomialBasisElement() [1/9]

MonomialBasisElement ( const MonomialBasisElement & )

default

◆ MonomialBasisElement() [2/9]

MonomialBasisElement ( MonomialBasisElement && )

default

◆ MonomialBasisElement() [3/9]

MonomialBasisElement ( )

Constructs a monomial equal to 1.

Namely the toal degree is zero.

◆ MonomialBasisElement() [4/9]

MonomialBasisElement ( std::nullptr_t )

explicit

Constructs a default value.

This overload is used by Eigen when EIGEN_INITIALIZE_MATRICES_BY_ZERO is enabled.

◆ MonomialBasisElement() [5/9]

MonomialBasisElement ( const std::map< Variable, int > & var_to_degree_map )

explicit

Constructs a MonomialBasisElement from variable to degree map.

◆ MonomialBasisElement() [6/9]

MonomialBasisElement ( const Expression & e )

explicit

Converts an expression to a monomial if the expression is written as ∏ᵢpow(xᵢ, kᵢ), otherwise throws a runtime error.

Precondition: is_polynomial(e) should be true.

◆ MonomialBasisElement() [7/9]

MonomialBasisElement	(	const Eigen::Ref< const VectorX< Variable >> &	vars,
		const Eigen::Ref< const Eigen::VectorXi > &	degrees
	)

Constructs a Monomial from a vector of variables vars and their corresponding integer degrees degrees.

For example, MonomialBasisElement([x, y, z], [2, 0, 1]) constructs a MonomialBasisElement x²z.

Precondition: The size of vars should be the same as the size of degrees.

Exceptions

std::exception if degrees includes a negative integer.

◆ MonomialBasisElement() [8/9]

MonomialBasisElement ( const Variable & var )

explicit

Constructs a monomial basis element with only one variable, and the degree is 1.

◆ MonomialBasisElement() [9/9]

MonomialBasisElement	(	const Variable &	var,
		int	degree
	)

Constructs a monomial basis element with only one variable, and the degree of that variable is given by degree.

Member Function Documentation

◆ Differentiate()

std::map<MonomialBasisElement, double> Differentiate ( const Variable & var ) const

Differentiates this MonomialBasisElement.

Since dxⁿ/dx = nxⁿ⁻¹, we return the map from the MonomialBasisElement to its coefficient. So if this MonomialBasisElement is x³y², then differentiate with x will return (x²y² → 3) as dx³y²/dx = 3x²y² If var is not a variable in MonomialBasisElement, then returns an empty map.

◆ EvaluatePartial()

std::pair<double, MonomialBasisElement> EvaluatePartial ( const Environment & env ) const

Partially evaluates using a given environment env.

The evaluation result is of type pair<double, MonomialBasisElement>. The first component (: double) represents the coefficient part while the second component represents the remaining parts of the MonomialBasisElement which was not evaluated.

Example 1. Evaluate with a fully-specified environment (x³*y²).EvaluatePartial({{x, 2}, {y, 3}}) = (2³ * 3² = 8 * 9 = 72, MonomialBasisElement{} = 1).

Example 2. Evaluate with a partial environment (x³*y²).EvaluatePartial({{x, 2}}) = (2³ = 8, y²).

◆ Integrate()

std::map<MonomialBasisElement, double> Integrate ( const Variable & var ) const

Integrates this MonomialBasisElement on a variable.

Since ∫ xⁿ dx = 1 / (n+1) xⁿ⁺¹, we return the map from the MonomialBasisElement to its coefficient in the integration result. So if this MonomialBasisElement is x³y², then we return (x⁴y² → 1/4) as ∫ x³y²dx = 1/4 x⁴y². If var is not a variable in this MonomialBasisElement, for example ∫ x³y²dz = x³y²z, then we return (x³y²z → 1)

◆ MergeBasisElementInPlace()

void MergeBasisElementInPlace ( const MonomialBasisElement & other )

Merges this basis element with another basis element other by merging their var_to_degree_map.

This is equivalent to multiplying this monomial basis element in place with monomial basis element other.

◆ operator<()

bool operator< ( const MonomialBasisElement & other ) const

Compares two MonomialBasisElement in lexicographic order.

◆ operator=() [1/2]

MonomialBasisElement& operator= ( MonomialBasisElement && )

default

◆ operator=() [2/2]

MonomialBasisElement& operator= ( const MonomialBasisElement & )

default

◆ pow_in_place()

MonomialBasisElement& pow_in_place ( int p )

Returns this monomial raised to p.

Exceptions

std::exception if p is negative.

◆ ToBasis()

std::map<BasisElement, double> ToBasis ( ) const

Converts this monomial to a weighted sum of basis elements of type BasisElement.

We return the map from each BasisElement to its coefficient. For example, if BasisElement=ChebyshevBasisElement, then when this = x²y³, it returns {[T₂(x)T₃(y)⇒1/8], [T₂(x)T₁(y)⇒3/8], [T₀(x)T₃(y)⇒1/8], [T₀(x)T₁(y)⇒3/8]}.

Note: Currently we only support

Template Parameters

BasisElement being MonomialBasisElement and ChebyshevBasisElement.

◆ ToChebyshevBasis()

std::map<ChebyshevBasisElement, double> ToChebyshevBasis ( ) const

Converts this monomial to Chebyshev polynomial basis.

For example,

For x², it returns 0.5T₂(x) + 0.5T₀(x).
For x²y³, it returns 1/8T₂(x)T₃(y) + 3/8T₂(x)T₁(y) + 1/8T₀(x)T₃(y) + 3/8T₀(x)T₁(y).

We return the map from each ChebyshevBasisElement to its coefficient. For example, when this = x², it returns {[T₂(x)⇒0.5], [T₀(x)⇒0.5]}. When this = x²y³, it returns {[T₂(x)T₃(y)⇒1/8], [T₂(x)T₁(y)⇒3/8], [T₀(x)T₃(y)⇒1/8], [T₀(x)T₁(y)⇒3/8]}.

Friends And Related Function Documentation

◆ hash_append

void hash_append	(	HashAlgorithm &	hasher,
		const MonomialBasisElement &	item
	)

friend

Implements the hash_append generic hashing concept.

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

drake/common/symbolic/monomial_basis_element.h

Detailed Description

Public Member Functions

Friends

Additional Inherited Members

Constructor & Destructor Documentation

◆ MonomialBasisElement() [1/9]

◆ MonomialBasisElement() [2/9]

◆ MonomialBasisElement() [3/9]

◆ MonomialBasisElement() [4/9]

◆ MonomialBasisElement() [5/9]

◆ MonomialBasisElement() [6/9]

◆ MonomialBasisElement() [7/9]

◆ MonomialBasisElement() [8/9]

◆ MonomialBasisElement() [9/9]

Member Function Documentation

◆ Differentiate()

◆ EvaluatePartial()

◆ Integrate()

◆ MergeBasisElementInPlace()

◆ operator<()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ pow_in_place()

◆ ToBasis()

◆ ToChebyshevBasis()

Friends And Related Function Documentation

◆ hash_append