gtsam 4.1.1 gtsam
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 >

## ◆ 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

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