gtsam::Chebyshev1Basis Struct Reference

Detailed Description

Basis of Chebyshev polynomials of the first kind https://en.wikipedia.org/wiki/Chebyshev_polynomials#First_kind These are typically denoted with the symbol T_n, where n is the degree.

The parameter N is the number of coefficients, i.e., N = n+1.

Inheritance diagram for gtsam::Chebyshev1Basis:

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 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< Chebyshev1Basis >
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 Attributes
Parameters	parameters_

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

Member Function Documentation

CalculateWeights()

Weights gtsam::Chebyshev1Basis::CalculateWeights ( size_t  N, double  x, double  a = -1, double  b = 1  )

static

Evaluate Chebyshev Weights on [-1,1] at 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::Chebyshev1Basis::DerivativeWeights ( size_t  N, double  x, double  a = -1, double  b = 1  )

static

Evaluate Chebyshev derivative at x.

The derivative weights are pre-multiplied to the polynomial Parameters.

From Wikipedia we have D[T_n(x),x] = n*U_{n-1}(x) I.e. the derivative fo a first kind cheb is just a second kind cheb So, we define a second kind basis here of order N-1 Note that it has one less weight.

The Parameters pertain to 1st kind chebs up to order N-1 But of course the first one (order 0) is constant, so omit that weight.

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