ChebyshevBasisElement Class Reference

Detailed Description

ChebyshevBasisElement represents an element of Chebyshev polynomial basis, written as the product of Chebyshev polynomials, in the form Tₚ₀(x₀)Tₚ₁(x₁)...Tₚₙ(xₙ), where each Tₚᵢ(xᵢ) is a (univariate) Chebyshev polynomial of degree pᵢ.

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

Public Member Functions
	ChebyshevBasisElement ()
	Constructs a ChebyshevBasisElement equals to 1.
	ChebyshevBasisElement (const std::map< Variable, int > &var_to_degree_map)
	ChebyshevBasisElement (const Variable &var)
	Constructs a Chebyshev polynomial T₁(var).
	ChebyshevBasisElement (const Variable &var, int degree)
	Constructs a Chebyshev polynomial Tₙ(var) where n = degree.
	ChebyshevBasisElement (std::nullptr_t)
	Constructs a default value 1.
	ChebyshevBasisElement (const Eigen::Ref< const VectorX< Variable > > &vars, const Eigen::Ref< const Eigen::VectorXi > &degrees)
	~ChebyshevBasisElement () override
bool	operator< (const ChebyshevBasisElement &other) const
	Compares two ChebyshevBasisElement in lexicographic order.
std::map< ChebyshevBasisElement, double >	Differentiate (const Variable &var) const
	Differentiates the ChebyshevBasisElement with respect to a variable.
std::map< ChebyshevBasisElement, double >	Integrate (const Variable &var) const
	Integrates a ChebyshevBasisElement for a variable.
void	MergeBasisElementInPlace (const ChebyshevBasisElement &other)
	Merges this Chebyshev basis element with another Chebyshev basis element other by merging their var_to_degree_map.
std::pair< double, ChebyshevBasisElement >	EvaluatePartial (const Environment &env) const
	Partially evaluates using a given environment env.
Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	ChebyshevBasisElement (const ChebyshevBasisElement &)=default
ChebyshevBasisElement &	operator= (const ChebyshevBasisElement &)=default
	ChebyshevBasisElement (ChebyshevBasisElement &&)=default
ChebyshevBasisElement &	operator= (ChebyshevBasisElement &&)=default
Public Member Functions inherited from PolynomialBasisElement
	PolynomialBasisElement ()=default
	Constructs a polynomial basis with empty var_to_degree map.
	PolynomialBasisElement (const std::map< Variable, int > &var_to_degree_map)
	Constructs a polynomial basis given the variable and the degree of that variable.
	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).
virtual	~PolynomialBasisElement ()
const std::map< Variable, int > &	var_to_degree_map () const
const std::map< Variable, int > &	get_powers () const
	Returns variable to degree map.
int	total_degree () const
	Returns the total degree of a polynomial basis.
int	degree (const Variable &v) const
	Returns the degree of this PolynomialBasisElement in a variable v.
Variables	GetVariables () const
double	Evaluate (const Environment &env) const
	Evaluates under a given environment env.
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 ChebyshevBasisElement &item) noexcept
	Implements the hash_append generic hashing concept.

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

Constructor & Destructor Documentation

ChebyshevBasisElement() [1/8]

ChebyshevBasisElement ( const ChebyshevBasisElement & )

default

ChebyshevBasisElement() [2/8]

ChebyshevBasisElement ( ChebyshevBasisElement && )

default

ChebyshevBasisElement() [3/8]

ChebyshevBasisElement

(

)

Constructs a ChebyshevBasisElement equals to 1.

ChebyshevBasisElement() [4/8]

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

explicit

ChebyshevBasisElement() [5/8]

ChebyshevBasisElement ( const Variable & var )

explicit

Constructs a Chebyshev polynomial T₁(var).

ChebyshevBasisElement() [6/8]

ChebyshevBasisElement	(	const Variable &	var,
		int	degree )

Constructs a Chebyshev polynomial Tₙ(var) where n = degree.

ChebyshevBasisElement() [7/8]

ChebyshevBasisElement ( std::nullptr_t )

explicit

Constructs a default value 1.

This overload is used by Eigen when EIGEN_INITIALIZE_MATRICES_BY_ZERO is enabled.

ChebyshevBasisElement() [8/8]

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

~ChebyshevBasisElement()

~ChebyshevBasisElement ( )

override

Member Function Documentation

Differentiate()

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

nodiscard

Differentiates the ChebyshevBasisElement with respect to a variable.

We use the fact that

• If n is even dTₙ(x)/dx = 2n ∑ⱼ Tⱼ(x), j is odd and 1 <= j <= n-1
• If n is odd dTₙ(x)/dx = 2n ∑ⱼ Tⱼ(x) - n, j is even and 0 <= j <= n-1 We return result, a map from ChebyshevBasisElement to double, such that sum(result.key() * result[key]) is the differentiation of this w.r.t the variable. For example if n is even, dTₙ(x)Tₘ(y)/dx = 2n∑ⱼ Tⱼ(x)Tₘ(y), j is odd and 1 <= j <= n-1, then the returned result is {T₁(x)Tₘ(y), 2n}, {T₃(x)Tₘ(y), 2n}, ..., {T₂ₙ₋₁(x)Tₘ(y), 2n}. A special case is that var is not a variable in this, then we return an empty map.

  Parameters

  var	A variable to differentiate with.

EvaluatePartial()

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

nodiscard

Partially evaluates using a given environment env.

The evaluation result is of type pair<double, ChebyshevBasisElement>. The first component (: double) represents the coefficient part while the second component represents the remaining parts of the ChebyshevBasisElement which was not evaluated, the product of the first and the second component is the result of the partial evaluation. For example, if this ChebyshevBasisElement is T₂(x)T₃(y)T₁(z), and env stores x→ 3, y→ 2, then the partial evaluation is T₂(3)*T₃(2)*T₁(z) = 17 * 26 * T₁(z) = 442*T₁(z), then we return the pair (442, T₁(z)).

Integrate()

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

nodiscard

Integrates a ChebyshevBasisElement for a variable.

We use the fact that ∫ Tₙ(x)dx = 1/(2n+2)Tₙ₊₁(x) − 1/(2n−2)Tₙ₋₁(x) A special case is ∫ T₀(x)dx = T₁(x)

Parameters

var	The variable to integrate. If var is not a variable in this ChebyshevBasisElement, then the integration result is *this * T₁(var).

Return values

result

sum(key * result[key]) is the integration result. For example, ∫ T₂(x)T₃(y)dx = 1/6*T₃(x)T₃(y) − 1/2 * T₁(x)T₃(y), then the result is the map containing {T₃(x)T₃(y), 1/6} and {T₁(x)T₃(y), -1/2}.

MergeBasisElementInPlace()

void MergeBasisElementInPlace

(

const ChebyshevBasisElement &

other

)

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

After merging, the degree of each variable is raised to the sum of the degree in each basis element (if a variable does not show up in either one of the basis element, we regard its degree to be 0). For example, merging T₁(x)T₃(y) and T₂(x)T₄(z) gets T₃(x)T₃(y)T₄(z).

operator<()

bool operator<

(

const ChebyshevBasisElement &

other

)

const

Compares two ChebyshevBasisElement in lexicographic order.

operator=() [1/2]

ChebyshevBasisElement & operator= ( ChebyshevBasisElement && )

default

operator=() [2/2]

ChebyshevBasisElement & operator= ( const ChebyshevBasisElement & )

default

Friends And Related Symbol Documentation

hash_append

template<class HashAlgorithm>

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

friend

Implements the hash_append generic hashing concept.

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

• drake/common/symbolic/chebyshev_basis_element.h