Detailed Description

template<typename T = double>
class drake::Polynomial< T >

A scalar multi-variate polynomial, modeled after the msspoly in spotless.

Polynomial represents a list of additive Monomials, each one of which is a product of a constant coefficient (of T, which by default is double) and any number of distinct Terms (variables raised to positive integer powers).

Variables are identified by integer indices rather than symbolic names, but an automatic facility is provided to convert variable names up to four characters into unique integers, provided those variables are named using only lowercase letters and the "@#_." characters followed by a number. For example, valid names include "dx4" and "m_x".

Monomials which have the same variables and powers may be constructed but will be automatically combined: (3 * a * b * a) + (1.5 * b * a**2) will be reduced to (4.5 * b * a**2) internally after construction.

Polynomials can be added, subtracted, and multiplied. They may only be divided by scalars (of T) because Polynomials are not closed under division.

Template Parameters

T	The scalar type, which must be one of the default scalars.

#include <drake/common/polynomial.h>

Classes
class	Monomial
	An additive atom of a Polynomial: The product of any number of Terms and a coefficient. More...

struct	Product

class	Term
	An individual variable raised to an integer power; e.g. x**2. More...

Public Types
typedef unsigned int	VarType

typedef int	PowerType
	This should be 'unsigned int' but MSVC considers a call to std::pow(..., unsigned int) ambiguous because it won't cast unsigned int to int. More...

typedef Eigen::NumTraits< T >::Real	RealScalar

typedef std::complex< RealScalar >	RootType

typedef Eigen::Matrix< RootType, Eigen::Dynamic, 1 >	RootsType

Public Member Functions
	Polynomial (void)
	Construct the vacuous polynomial, "0". More...

	Polynomial (const T &scalar)
	Construct a Polynomial of a single constant. e.g. "5". More...

	Polynomial (const T coeff, const std::vector< Term > &terms)
	Construct a Polynomial consisting of a single Monomial, e.g. "5xy**3". More...

	Polynomial (typename std::vector< Monomial >::const_iterator start, typename std::vector< Monomial >::const_iterator finish)
	Construct a Polynomial from a sequence of Monomials. More...

template<typename U = T>
	Polynomial (const std::enable_if_t< std::is_same_v< U, double >, std::string > &varname)
	Constructs a polynomial consisting of a single Monomial of the variable named `varname1`. More...

	Polynomial (const std::string &varname, unsigned int num)
	Construct a polynomial consisting of a single Monomial of the variable named varname + num. More...

	Polynomial (const T &coeff, const VarType &v)
	Construct a single Monomial of the given coefficient and variable. More...

template<typename Derived >
	Polynomial (const Eigen::MatrixBase< Derived > &coefficients)
	A constructor for univariate polynomials: takes a vector of coefficients for the x0, x1, x2, x3... More...

int	GetNumberOfCoefficients () const
	Returns the number of unique Monomials (and thus the number of coefficients) in this Polynomial. More...

int	GetDegree () const
	Returns the highest degree of any Monomial in this Polynomial. More...

bool	IsAffine () const
	Returns true iff this is a sum of terms of degree 1, plus a constant. More...

VarType	GetSimpleVariable () const
	If the polynomial is "simple" – e.g. More...

const std::vector< Monomial > &	GetMonomials () const

VectorX< T >	GetCoefficients () const
	Returns the vector of the coefficients of the polynomial in the order of powers of the variable - the [i]th element is the coefficient of the variable raised to the ith power. More...

std::set< VarType >	GetVariables () const
	Returns a set of all of the variables present in this Polynomial. More...

template<typename U >
Product< T, U >::type	EvaluateUnivariate (const U &x, int derivative_order=0) const
	Evaluate a univariate Polynomial at a specific point. More...

template<typename U >
Product< T, U >::type	EvaluateMultivariate (const std::map< VarType, U > &var_values) const
	Evaluate a multivariate Polynomial at a specific point. More...

Polynomial	EvaluatePartial (const std::map< VarType, T > &var_values) const
	Substitute values for some but not necessarily all variables of a Polynomial. More...

void	Subs (const VarType &orig, const VarType &replacement)
	Replaces all instances of variable orig with replacement. More...

Polynomial	Substitute (const VarType &orig, const Polynomial &replacement) const
	Replaces all instances of variable orig with replacement. More...

Polynomial	Derivative (int derivative_order=1) const
	Takes the derivative of this (univariate) Polynomial. More...

Polynomial	Integral (const T &integration_constant=0.0) const
	Takes the integral of this (univariate, non-constant) Polynomial. More...

bool	is_univariate () const
	Returns true if this is a univariate polynomial. More...

bool	operator== (const Polynomial &other) const

Polynomial &	operator+= (const Polynomial &other)

Polynomial &	operator-= (const Polynomial &other)

Polynomial &	operator *= (const Polynomial &other)

Polynomial &	operator+= (const T &scalar)

Polynomial &	operator-= (const T &scalar)

Polynomial &	operator *= (const T &scalar)

Polynomial &	operator/= (const T &scalar)

const Polynomial	operator+ (const Polynomial &other) const

const Polynomial	operator- (const Polynomial &other) const

const Polynomial	operator- () const

const Polynomial	operator * (const Polynomial &other) const

const Polynomial	operator/ (const T &scalar) const

bool	operator< (const Polynomial &other) const
	A comparison to allow std::lexicographical_compare on this class; does not reflect any sort of mathematical total order. More...

RootsType	Roots () const
	Returns the roots of this (univariate) Polynomial. More...

boolean< T >	CoefficientsAlmostEqual (const Polynomial< T > &other, const RealScalar &tol=0.0, const ToleranceType &tol_type=ToleranceType::kAbsolute) const
	Checks if a Polynomial is approximately equal to this one. More...

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

Polynomial &	operator= (const Polynomial &)=default

	Polynomial (Polynomial &&)=default

Polynomial &	operator= (Polynomial &&)=default

Static Public Member Functions
static Polynomial< T >	FromExpression (const drake::symbolic::Expression &e)
	Constructs a Polynomial representing the symbolic expression `e`. More...


static bool	IsValidVariableName (const std::string name)
	Variable name/ID conversion facility. More...

static VarType	VariableNameToId (const std::string name, unsigned int m=1)

static std::string	IdToVariableName (const VarType id)

Friends
const Polynomial	operator+ (const Polynomial &p, const T &scalar)

const Polynomial	operator+ (const T &scalar, const Polynomial &p)

const Polynomial	operator- (const Polynomial &p, const T &scalar)

const Polynomial	operator- (const T &scalar, const Polynomial &p)

const Polynomial	operator * (const Polynomial &p, const T &scalar)

const Polynomial	operator * (const T &scalar, const Polynomial &p)

std::ostream &	operator<< (std::ostream &os, const Monomial &m)

std::ostream &	operator<< (std::ostream &os, const Polynomial &poly)

template<typename U >
Polynomial< U >	pow (const Polynomial< U > &p, typename Polynomial< U >::PowerType n)

Member Typedef Documentation

◆ PowerType

typedef int PowerType

This should be 'unsigned int' but MSVC considers a call to std::pow(..., unsigned int) ambiguous because it won't cast unsigned int to int.

◆ RealScalar

typedef Eigen::NumTraits<T>::Real RealScalar

◆ RootsType

typedef Eigen::Matrix<RootType, Eigen::Dynamic, 1> RootsType

◆ RootType

typedef std::complex<RealScalar> RootType

◆ VarType

typedef unsigned int VarType

Constructor & Destructor Documentation

◆ Polynomial() [1/10]

Polynomial ( const Polynomial< T > & )

default

◆ Polynomial() [2/10]

Polynomial ( Polynomial< T > && )

default

◆ Polynomial() [3/10]

Polynomial ( void )

Construct the vacuous polynomial, "0".

◆ Polynomial() [4/10]

Polynomial ( const T & scalar )

Construct a Polynomial of a single constant. e.g. "5".

◆ Polynomial() [5/10]

Polynomial	(	const T	coeff,
		const std::vector< Term > &	terms
	)

Construct a Polynomial consisting of a single Monomial, e.g. "5xy**3".

◆ Polynomial() [6/10]

Polynomial	(	typename std::vector< Monomial >::const_iterator	start,
		typename std::vector< Monomial >::const_iterator	finish
	)

Construct a Polynomial from a sequence of Monomials.

◆ Polynomial() [7/10]

Polynomial ( const std::enable_if_t< std::is_same_v< U, double >, std::string > & varname )

explicit

Constructs a polynomial consisting of a single Monomial of the variable named varname1.

Note

: This constructor is only provided for T = double. For the other cases, a user should use the constructor with two arguments below (taking std::string and unsigned int). If we provided this constructor for T = AutoDiffXd and T = symbolic::Expression, there would be compiler errors for Polynomial<T>(0) as the following candidates are ambiguous:

Polynomial(const T& scalar)
Polynomial(const std::string& varname, const unsigned int num = 1)

◆ Polynomial() [8/10]

Polynomial	(	const std::string &	varname,
		unsigned int	num
	)

Construct a polynomial consisting of a single Monomial of the variable named varname + num.

◆ Polynomial() [9/10]

Polynomial	(	const T &	coeff,
		const VarType &	v
	)

Construct a single Monomial of the given coefficient and variable.

◆ Polynomial() [10/10]

Polynomial ( const Eigen::MatrixBase< Derived > & coefficients )

explicit

A constructor for univariate polynomials: takes a vector of coefficients for the x**0, x**1, x**2, x**3...

Monomials. All terms are always added, even if a coefficient is zero.

Member Function Documentation

◆ CoefficientsAlmostEqual()

boolean<T> CoefficientsAlmostEqual	(	const Polynomial< T > &	other,
		const RealScalar &	tol = `0.0`,
		const ToleranceType &	tol_type = `ToleranceType::kAbsolute`
	)		const

Checks if a Polynomial is approximately equal to this one.

Checks that every coefficient of other is within tol of the corresponding coefficient of this Polynomial.

Note: When tol_type is kRelative, if any monomials appear in this or other but not both, then the method returns false (since the comparison is relative to a missing zero coefficient). Use kAbsolute if you want to ignore non-matching monomials with coefficients less than tol.

◆ Derivative()

Polynomial Derivative ( int derivative_order = 1 ) const

Takes the derivative of this (univariate) Polynomial.

Returns a new Polynomial that is the derivative of this one in its sole variable.

Exceptions

std::exception if this Polynomial is not univariate.

If derivative_order is given, takes the nth derivative of this Polynomial.

◆ EvaluateMultivariate()

Product<T, U>::type EvaluateMultivariate ( const std::map< VarType, U > & var_values ) const

Evaluate a multivariate Polynomial at a specific point.

Evaluates a Polynomial with the given values for each variable.

Exceptions

std::exception if the Polynomial contains variables for which values were not provided.

The provided values may be of any type which is std::is_arithmetic (supporting the std::pow, *, and + operations) and need not be CoefficientsType or RealScalar)

◆ EvaluatePartial()

Polynomial EvaluatePartial ( const std::map< VarType, T > & var_values ) const

Substitute values for some but not necessarily all variables of a Polynomial.

Analogous to EvaluateMultivariate, but: (1) Restricted to T, and (2) Need not map every variable in var_values.

Returns a Polynomial in which each variable in var_values has been replaced with its value and constants appropriately combined.

◆ EvaluateUnivariate()

Product<T, U>::type EvaluateUnivariate	(	const U &	x,
		int	derivative_order = `0`
	)		const

Evaluate a univariate Polynomial at a specific point.

Evaluates a univariate Polynomial at the given x.

Exceptions

std::exception if this Polynomial is not univariate.

x may be of any type supporting the ** and + operations (which can be different from both CoefficientsType and RealScalar).

This method may also be used for efficient evaluation of the derivatives of the univariate polynomial, evaluated at x. derivative_order = 0 (the default) returns the polynomial value without differentiation. derivative_order = 1 returns the first derivative, etc.

Precondition: derivative_order must be non-negative.

◆ FromExpression()

static Polynomial<T> FromExpression ( const drake::symbolic::Expression & e )

static

Constructs a Polynomial representing the symbolic expression e.

Note that the ID of a variable is preserved in this translation.

Exceptions

std::exception if e is not polynomial-convertible.

Precondition: e.is_polynomial() is true.

◆ GetCoefficients()

VectorX<T> GetCoefficients ( ) const

Returns the vector of the coefficients of the polynomial in the order of powers of the variable - the [i]th element is the coefficient of the variable raised to the ith power.

Exceptions

std::runtime_error if this Polynomial is not univariate.

◆ GetDegree()

int GetDegree ( ) const

Returns the highest degree of any Monomial in this Polynomial.

The degree of a multivariate Monomial is the product of the degrees of each of its terms.

◆ GetMonomials()

const std::vector<Monomial>& GetMonomials ( ) const

◆ GetNumberOfCoefficients()

int GetNumberOfCoefficients ( ) const

Returns the number of unique Monomials (and thus the number of coefficients) in this Polynomial.

◆ GetSimpleVariable()

VarType GetSimpleVariable ( ) const

If the polynomial is "simple" – e.g.

just a single term with coefficient 1 – then returns that variable; otherwise returns 0.

◆ GetVariables()

std::set<VarType> GetVariables ( ) const

Returns a set of all of the variables present in this Polynomial.

◆ IdToVariableName()

static std::string IdToVariableName ( const VarType id )

static

◆ Integral()

Polynomial Integral ( const T & integration_constant = 0.0 ) const

Takes the integral of this (univariate, non-constant) Polynomial.

Returns a new Polynomial that is the indefinite integral of this one in its sole variable.

Exceptions

std::exception if this Polynomial is not univariate, or if it has no variables.

If integration_constant is given, adds that constant as the constant term (zeroth-order coefficient) of the resulting Polynomial.

◆ is_univariate()

bool is_univariate ( ) const

Returns true if this is a univariate polynomial.

◆ IsAffine()

bool IsAffine ( ) const

Returns true iff this is a sum of terms of degree 1, plus a constant.

◆ IsValidVariableName()

static bool IsValidVariableName ( const std::string name )

static

Variable name/ID conversion facility.

◆ operator *()

const Polynomial operator * ( const Polynomial< T > & other ) const

◆ operator *=() [1/2]

Polynomial& operator *= ( const Polynomial< T > & other )

◆ operator *=() [2/2]

Polynomial& operator *= ( const T & scalar )

◆ operator+()

const Polynomial operator+ ( const Polynomial< T > & other ) const

◆ operator+=() [1/2]

Polynomial& operator+= ( const Polynomial< T > & other )

◆ operator+=() [2/2]

Polynomial& operator+= ( const T & scalar )

◆ operator-() [1/2]

const Polynomial operator- ( const Polynomial< T > & other ) const

◆ operator-() [2/2]

const Polynomial operator- ( ) const

◆ operator-=() [1/2]

Polynomial& operator-= ( const Polynomial< T > & other )

◆ operator-=() [2/2]

Polynomial& operator-= ( const T & scalar )

◆ operator/()

const Polynomial operator/ ( const T & scalar ) const

◆ operator/=()

Polynomial& operator/= ( const T & scalar )

◆ operator<()

bool operator< ( const Polynomial< T > & other ) const

A comparison to allow std::lexicographical_compare on this class; does not reflect any sort of mathematical total order.

◆ operator=() [1/2]

Polynomial& operator= ( Polynomial< T > && )

default

◆ operator=() [2/2]

Polynomial& operator= ( const Polynomial< T > & )

default

◆ operator==()

bool operator== ( const Polynomial< T > & other ) const

◆ Roots()

RootsType Roots ( ) const

Returns the roots of this (univariate) Polynomial.

Returns the roots of a univariate Polynomial as an Eigen column vector of complex numbers whose components are of the RealScalar type.

Exceptions

std::exception of this Polynomial is not univariate.

◆ Subs()

void Subs	(	const VarType &	orig,
		const VarType &	replacement
	)

Replaces all instances of variable orig with replacement.

◆ Substitute()

Polynomial Substitute	(	const VarType &	orig,
		const Polynomial< T > &	replacement
	)		const

Replaces all instances of variable orig with replacement.

◆ VariableNameToId()

static VarType VariableNameToId	(	const std::string	name,
		unsigned int	m = `1`
	)

static

Friends And Related Function Documentation

◆ operator * [1/2]

const Polynomial operator *	(	const Polynomial< T > &	p,
		const T &	scalar
	)

friend

◆ operator * [2/2]

const Polynomial operator *	(	const T &	scalar,
		const Polynomial< T > &	p
	)

friend

◆ operator+ [1/2]

const Polynomial operator+	(	const Polynomial< T > &	p,
		const T &	scalar
	)

friend

◆ operator+ [2/2]

const Polynomial operator+	(	const T &	scalar,
		const Polynomial< T > &	p
	)

friend

◆ operator- [1/2]

const Polynomial operator-	(	const Polynomial< T > &	p,
		const T &	scalar
	)

friend

◆ operator- [2/2]

const Polynomial operator-	(	const T &	scalar,
		const Polynomial< T > &	p
	)

friend

◆ operator<< [1/2]

std::ostream& operator<<	(	std::ostream &	os,
		const Monomial &	m
	)

friend

◆ operator<< [2/2]

std::ostream& operator<<	(	std::ostream &	os,
		const Polynomial< T > &	poly
	)

friend

◆ pow

Polynomial<U> pow	(	const Polynomial< U > &	p,
		typename Polynomial< U >::PowerType	n
	)

friend

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

drake/common/polynomial.h

Detailed Description

template<typename T = double> class drake::Polynomial< T >

Classes

Public Types

Public Member Functions

Static Public Member Functions

Friends

Member Typedef Documentation

◆ PowerType

◆ RealScalar

◆ RootsType

◆ RootType

◆ VarType

Constructor & Destructor Documentation

◆ Polynomial() [1/10]

◆ Polynomial() [2/10]

◆ Polynomial() [3/10]

◆ Polynomial() [4/10]

◆ Polynomial() [5/10]

◆ Polynomial() [6/10]

◆ Polynomial() [7/10]

◆ Polynomial() [8/10]

◆ Polynomial() [9/10]

◆ Polynomial() [10/10]

Member Function Documentation

◆ CoefficientsAlmostEqual()

◆ Derivative()

◆ EvaluateMultivariate()

◆ EvaluatePartial()

◆ EvaluateUnivariate()

◆ FromExpression()

◆ GetCoefficients()

◆ GetDegree()

◆ GetMonomials()

◆ GetNumberOfCoefficients()

◆ GetSimpleVariable()

◆ GetVariables()

◆ IdToVariableName()

◆ Integral()

◆ is_univariate()

◆ IsAffine()

◆ IsValidVariableName()

◆ operator *()

◆ operator *=() [1/2]

◆ operator *=() [2/2]

◆ operator+()

◆ operator+=() [1/2]

◆ operator+=() [2/2]

◆ operator-() [1/2]

◆ operator-() [2/2]

◆ operator-=() [1/2]

◆ operator-=() [2/2]

◆ operator/()

◆ operator/=()

◆ operator<()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ Roots()

◆ Subs()

◆ Substitute()

◆ VariableNameToId()

Friends And Related Function Documentation

◆ operator * [1/2]

◆ operator * [2/2]

◆ operator+ [1/2]

◆ operator+ [2/2]

◆ operator- [1/2]

◆ operator- [2/2]

◆ operator<< [1/2]

◆ operator<< [2/2]

◆ pow

template<typename T = double>
class drake::Polynomial< T >