Polynomial Class Reference

Represents symbolic polynomials. More...

#include <drake/common/symbolic_polynomial.h>

Public Types
using	MapType = std::unordered_map< Monomial, Expression >

Public Member Functions
	Polynomial ()=default
	Constructs a zero polynomial. More...
	Polynomial (std::nullptr_t)
	Constructs a default value. More...
	Polynomial (MapType init)
	Constructs a polynomial from a map, Monomial → Expression. More...
	Polynomial (const Monomial &m)
	Constructs a polynomial from a monomial m. More...
	Polynomial (const Expression &e)
	Constructs a polynomial from an expression e. More...
	Polynomial (const Expression &e, const Variables &indeterminates)
	Constructs a polynomial from an expression e by decomposing it with respect to indeterminates. More...
Variables	indeterminates () const
	Returns the indeterminates of this polynomial. More...
Variables	decision_variables () const
	Returns the decision variables of this polynomial. More...
int	Degree (const Variable &v) const
	Returns the highest degree of this polynomial in a variable v. More...
int	TotalDegree () const
	Returns the total degree of this polynomial. More...
const MapType &	monomial_to_coefficient_map () const
	Returns the mapping from a Monomial to its corresponding coefficient of this polynomial. More...
Expression	ToExpression () const
	Returns an equivalent symbolic expression of this polynomial. More...
Polynomial	Differentiate (const Variable &x) const
	Differentiates this polynomial with respect to the variable x. More...
template<typename Derived >
Eigen::Matrix< Polynomial, 1, Derived::RowsAtCompileTime >	Jacobian (const Eigen::MatrixBase< Derived > &vars) const
	Computes the Jacobian matrix J of the polynomial with respect to vars. More...
double	Evaluate (const Environment &env) const
	Evaluates this polynomial under a given environment env. More...
Polynomial	EvaluatePartial (const Environment &env) const
	Partially evaluates this polynomial using an environment env. More...
Polynomial	EvaluatePartial (const Variable &var, double c) const
	Partially evaluates this polynomial by substituting var with c. More...
Polynomial &	AddProduct (const Expression &coeff, const Monomial &m)
	Adds coeff * m to this polynomial. More...
Polynomial	RemoveTermsWithSmallCoefficients (double coefficient_tol) const
	Removes the terms whose absolute value of the coefficients are smaller than or equal to coefficient_tol For example, if the polynomial is 2x² + 3xy + 10⁻⁴x - 10⁻⁵, then after calling RemoveTermsWithSmallCoefficients(1e-3), the returned polynomial becomes 2x² + 3xy. More...
Polynomial &	operator+= (const Polynomial &p)
Polynomial &	operator+= (const Monomial &m)
Polynomial &	operator+= (double c)
Polynomial &	operator-= (const Polynomial &p)
Polynomial &	operator-= (const Monomial &m)
Polynomial &	operator-= (double c)
Polynomial &	operator*= (const Polynomial &p)
Polynomial &	operator*= (const Monomial &m)
Polynomial &	operator*= (double c)
bool	EqualTo (const Polynomial &p) const
	Returns true if this polynomial and p are structurally equal. More...
bool	EqualToAfterExpansion (const Polynomial &p) const
	Returns true if this polynomial and p are equal, after expanding the coefficients. More...
Formula	operator== (const Polynomial &p) const
	Returns a symbolic formula representing the condition where this polynomial and p are the same. More...
Formula	operator!= (const Polynomial &p) const
	Returns a symbolic formula representing the condition where this polynomial and p are not the same. More...
Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	Polynomial (const Polynomial &)=default
Polynomial &	operator= (const Polynomial &)=default
	Polynomial (Polynomial &&)=default
Polynomial &	operator= (Polynomial &&)=default

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

Detailed Description

Represents symbolic polynomials.

A symbolic polynomial keeps a mapping from a monomial of indeterminates to its coefficient in a symbolic expression.

A polynomial p has to satisfy an invariant such that p.decision_variables() ∩ p.indeterminates() = ∅. We have CheckInvariant() method to check the invariant.

Note that arithmetic operations (+,-,*) between a Polynomial and a Variable are not provided. The problem is that Variable class has no intrinsic information if a variable is a decision variable or an indeterminate while we need this information to perform arithmetic operations over Polynomials.

Member Typedef Documentation

MapType

using MapType = std::unordered_map<Monomial, Expression>

Constructor & Destructor Documentation

Polynomial() [1/8]

Polynomial ( )

default

Constructs a zero polynomial.

Polynomial() [2/8]

Polynomial ( const Polynomial &  )

default

Polynomial() [3/8]

Polynomial ( Polynomial &&  )

default

Polynomial() [4/8]

Polynomial ( std::nullptr_t  )

inlineexplicit

Constructs a default value.

This overload is used by Eigen when EIGEN_INITIALIZE_MATRICES_BY_ZERO is enabled.

Polynomial() [5/8]

Polynomial ( MapType  init )

explicit

Constructs a polynomial from a map, Monomial → Expression.

Polynomial() [6/8]

Polynomial

(

const Monomial &

m

)

Constructs a polynomial from a monomial m.

Note that all variables in m are considered as indeterminates.

Polynomial() [7/8]

Polynomial ( const Expression &  e )

explicit

Constructs a polynomial from an expression e.

Note that all variables in e are considered as indeterminates.

Exceptions

std::runtime_error

if e is not a polynomial.

Polynomial() [8/8]

Polynomial	(	const Expression &	e,
		const Variables &	indeterminates
	)

Constructs a polynomial from an expression e by decomposing it with respect to indeterminates.

Note

It collects the intersection of the variables appeared in e and the provided indeterminates.

Exceptions

std::runtime_error

if e is not a polynomial in indeterminates.

Member Function Documentation

AddProduct()

Polynomial & AddProduct	(	const Expression &	coeff,
		const Monomial &	m
	)

Adds coeff * m to this polynomial.

decision_variables()

Variables decision_variables

(

)

const

Returns the decision variables of this polynomial.

Degree()

int Degree

(

const Variable &

v

)

const

Returns the highest degree of this polynomial in a variable v.

Differentiate()

Polynomial Differentiate

(

const Variable &

x

)

const

Differentiates this polynomial with respect to the variable x.

Note that a variable x can be either a decision variable or an indeterminate.

EqualTo()

bool EqualTo

(

const Polynomial &

p

)

const

Returns true if this polynomial and p are structurally equal.

EqualToAfterExpansion()

bool EqualToAfterExpansion

(

const Polynomial &

p

)

const

Returns true if this polynomial and p are equal, after expanding the coefficients.

Evaluate()

double Evaluate

(

const Environment &

env

)

const

Evaluates this polynomial under a given environment env.

Exceptions

std::out_of_range

if there is a variable in this polynomial whose assignment is not provided by env.

EvaluatePartial() [1/2]

Polynomial EvaluatePartial

(

const Environment &

env

)

const

Partially evaluates this polynomial using an environment env.

Exceptions

std::runtime_error

if NaN is detected during evaluation.

EvaluatePartial() [2/2]

Polynomial EvaluatePartial	(	const Variable &	var,
		double	c
	)		const

Partially evaluates this polynomial by substituting var with c.

Exceptions

std::runtime_error

if NaN is detected at any point during evaluation.

indeterminates()

Variables indeterminates

(

)

const

Returns the indeterminates of this polynomial.

Jacobian()

Eigen::Matrix<Polynomial, 1, Derived::RowsAtCompileTime> Jacobian ( const Eigen::MatrixBase< Derived > &  vars ) const

inline

Computes the Jacobian matrix J of the polynomial with respect to vars.

J(0,i) contains ∂f/∂vars(i).

monomial_to_coefficient_map()

const Polynomial::MapType & monomial_to_coefficient_map

(

)

const

Returns the mapping from a Monomial to its corresponding coefficient of this polynomial.

operator!=()

Formula operator!=

(

const Polynomial &

p

)

const

Returns a symbolic formula representing the condition where this polynomial and p are not the same.

operator*=() [1/3]

Polynomial & operator*=

(

const Polynomial &

p

)

operator*=() [2/3]

Polynomial & operator*=

(

const Monomial &

m

)

operator*=() [3/3]

Polynomial & operator*=

(

double

c

)

operator+=() [1/3]

Polynomial & operator+=

(

const Polynomial &

p

)

operator+=() [2/3]

Polynomial & operator+=

(

const Monomial &

m

)

operator+=() [3/3]

Polynomial & operator+=

(

double

c

)

operator-=() [1/3]

Polynomial & operator-=

(

const Polynomial &

p

)

operator-=() [2/3]

Polynomial & operator-=

(

const Monomial &

m

)

operator-=() [3/3]

Polynomial & operator-=

(

double

c

)

operator=() [1/2]

Polynomial& operator= ( Polynomial &&  )

default

operator=() [2/2]

Polynomial& operator= ( const Polynomial &  )

default

operator==()

Formula operator==

(

const Polynomial &

p

)

const

Returns a symbolic formula representing the condition where this polynomial and p are the same.

RemoveTermsWithSmallCoefficients()

Polynomial RemoveTermsWithSmallCoefficients

(

double

coefficient_tol

)

const

Removes the terms whose absolute value of the coefficients are smaller than or equal to coefficient_tol For example, if the polynomial is 2x² + 3xy + 10⁻⁴x - 10⁻⁵, then after calling RemoveTermsWithSmallCoefficients(1e-3), the returned polynomial becomes 2x² + 3xy.

Parameters

coefficient_tol

A positive scalar.

Return values

polynomial_cleaned

A polynomial whose terms with small coefficients are removed.

ToExpression()

Expression ToExpression

(

)

const

Returns an equivalent symbolic expression of this polynomial.

TotalDegree()

int TotalDegree

(

)

const

Returns the total degree of this polynomial.

Friends And Related Function Documentation

hash_append

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

friend

Implements the hash_append generic hashing concept.

---

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

• drake/common/symbolic_polynomial.h
• drake/common/symbolic_polynomial.cc