ChebyshevPolynomial Class Reference

Detailed Description

Represents the Chebyshev polynomial of the first kind Tₙ(x).

One definition of Chebyshev polynomial of the first kind is Tₙ(cos(θ)) = cos(nθ) It can also be defined recursively as

T₀(x) = 1
T₁(x) = x
Tₙ₊₁(x) = 2xTₙ(x) − Tₙ₋₁(x)

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

Public Member Functions
	ChebyshevPolynomial (Variable var, int degree)
	Constructs a Chebyshev polynomial Tₙ(x) More...
const Variable &	var () const
	Getter for the variable. More...
int	degree () const
	Getter for the degree of the Chebyshev polynomial. More...
Polynomial	ToPolynomial () const
	Converts this Chebyshev polynomial to a polynomial with monomial basis. More...
double	Evaluate (double var_val) const
	Evaluates this Chebyshev polynomial at var_val. More...
bool	operator== (const ChebyshevPolynomial &other) const
	Checks if this and other represent the same Chebyshev polynomial. More...
bool	operator!= (const ChebyshevPolynomial &other) const
	Checks if this and other do not represent the same Chebyshev polynomial. More...
bool	operator< (const ChebyshevPolynomial &other) const
	Compare this to another Chebyshev polynomial, returns True if this is regarded as less than the other, otherwise returns false. More...
std::vector< std::pair< ChebyshevPolynomial, double > >	Differentiate () const
	Computes the differentiation of a Chebyshev polynomial dTₙ(x)/dx = nUₙ₋₁(x) where Uₙ₋₁(x) is a Chebyshev polynomial of the second kind. More...
Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	ChebyshevPolynomial (const ChebyshevPolynomial &)=default
ChebyshevPolynomial &	operator= (const ChebyshevPolynomial &)=default
	ChebyshevPolynomial (ChebyshevPolynomial &&)=default
ChebyshevPolynomial &	operator= (ChebyshevPolynomial &&)=default

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

Constructor & Destructor Documentation

ChebyshevPolynomial() [1/3]

ChebyshevPolynomial ( const ChebyshevPolynomial &  )

default

ChebyshevPolynomial() [2/3]

ChebyshevPolynomial ( ChebyshevPolynomial &&  )

default

ChebyshevPolynomial() [3/3]

ChebyshevPolynomial	(	Variable	var,
		int	degree
	)

Constructs a Chebyshev polynomial Tₙ(x)

Parameters

var	The variable x
degree	The Chebyshev polynomial is of degree n.

Precondition

degree >= 0.

Member Function Documentation

degree()

int degree

(

)

const

Getter for the degree of the Chebyshev polynomial.

Differentiate()

std::vector<std::pair<ChebyshevPolynomial, double> > Differentiate

(

)

const

Computes the differentiation of a Chebyshev polynomial dTₙ(x)/dx = nUₙ₋₁(x) where Uₙ₋₁(x) is a Chebyshev polynomial of the second kind.

Uₙ₋₁(x) can be written as a summation of Chebyshev polynomials of the first kind with lower degrees.

• If n is even dTₙ(x)/dx = 2n ∑ⱼ Tⱼ(x), j is odd and j <= n-1
• If n is odd dTₙ(x)/dx = 2n ∑ⱼ Tⱼ(x) - n, j is even and j <= n-1
• A special case is that dT₀(x)/dx = 0.

  Return values

  chebyshev_coeff_pairs.

  sum(chebyshev_coeff_pairs[j].first * chebyshev_coeff_pairs[j].second) is the differentiation dTₙ(x)/dx. If n is even, then chebyshev_coeff_pairs[j] = (T₂ⱼ₋₁(x), 2n). If n is odd, then chebyshev_coeff_pairs[j] = (T₂ⱼ(x), 2n) for j >= 1, and chebyshev_coeff_pairs[0] = (T₀(x), n). For the special case when degree() == 0, we return an empty vector.

Evaluate()

double Evaluate

(

double

var_val

)

const

Evaluates this Chebyshev polynomial at var_val.

operator!=()

bool operator!=

(

const ChebyshevPolynomial &

other

)

const

Checks if this and other do not represent the same Chebyshev polynomial.

operator<()

bool operator<

(

const ChebyshevPolynomial &

other

)

const

Compare this to another Chebyshev polynomial, returns True if this is regarded as less than the other, otherwise returns false.

If this.var() < other.var(), return True. If this.var() > other.var(), return False. If this.var() == other.var(), then return this.degree() < other.degree().

A special case is when this.degree() == 0 or other.degree() == 0. In this case the variable doesn't matter, and we return this.degree() < other.degree().

operator=() [1/2]

ChebyshevPolynomial& operator= ( ChebyshevPolynomial &&  )

default

operator=() [2/2]

ChebyshevPolynomial& operator= ( const ChebyshevPolynomial &  )

default

operator==()

bool operator==

(

const ChebyshevPolynomial &

other

)

const

Checks if this and other represent the same Chebyshev polynomial.

Two Chebyshev polynomials are equal iff their variable and degree are the same, or they both have degree 0.

Note

T₀(x) = T₀(y) = 1

ToPolynomial()

Polynomial ToPolynomial

(

)

const

Converts this Chebyshev polynomial to a polynomial with monomial basis.

var()

const Variable& var

(

)

const

Getter for the variable.

Friends And Related Function Documentation

hash_append

void hash_append ( HashAlgorithm &  hasher, const ChebyshevPolynomial &  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_polynomial.h