gtsam::Chebyshev2Basis Struct Reference

Detailed Description

Basis of Chebyshev polynomials of the second kind.

https://en.wikipedia.org/wiki/Chebyshev_polynomials#Second_kind These are typically denoted with the symbol U_n, where n is the degree. The parameter N is the number of coefficients, i.e., N = n+1. In contrast to the templates in Chebyshev2, the classes below specify basis functions, weighted combinations of which are used to approximate functions. In this sense, they are like the sines and cosines of the Fourier basis.

Inheritance diagram for gtsam::Chebyshev2Basis:

Static Public Member Functions
static Weights	CalculateWeights (size_t N, double x, double a=-1, double b=1)
	Evaluate Chebyshev Weights on [-1,1] at any x up to order N-1 (N values). More...
static Weights	DerivativeWeights (size_t N, double x, double a=-1, double b=1)
	Evaluate Chebyshev derivative at x. More...
Static Public Member Functions inherited from gtsam::Basis< Chebyshev2Basis >
static Matrix	WeightMatrix (size_t N, const Vector &X)
	Calculate weights for all x in vector X. More...
static Matrix	WeightMatrix (size_t N, const Vector &X, double a, double b)
	Calculate weights for all x in vector X, with interval [a,b]. More...
static double	Derivative (double x, const Vector &p, OptionalJacobian< -1, -1 > H=boost::none)

Public Types
using	Parameters = Eigen::Matrix< double, -1, 1 >

Member Function Documentation

CalculateWeights()

Weights gtsam::Chebyshev2Basis::CalculateWeights ( size_t  N, double  x, double  a = -1, double  b = 1  )

static

Evaluate Chebyshev Weights on [-1,1] at any x up to order N-1 (N values).

Parameters

N	Degree of the polynomial.
x	Point to evaluate polynomial at.
a	Lower limit of polynomial (default=-1).
b	Upper limit of polynomial (default=1).

DerivativeWeights()

Weights gtsam::Chebyshev2Basis::DerivativeWeights ( size_t  N, double  x, double  a = -1, double  b = 1  )

static

Evaluate Chebyshev derivative at x.

Parameters

N	Degree of the polynomial.
x	Point to evaluate polynomial at.
a	Lower limit of polynomial (default=-1).
b	Upper limit of polynomial (default=1).

Returns

Weights

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

• /Users/dellaert/git/github/gtsam/basis/Chebyshev.h
• /Users/dellaert/git/github/gtsam/basis/Chebyshev.cpp