#include <Chebyshev2.h>

Inheritance diagram for gtsam::Chebyshev2:

Public Types
using	Base = Basis< Chebyshev2 >

using	DiffMatrix = Eigen::Matrix< double, -1, -1 >

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

Static Public Member Functions
static Weights	CalculateWeights (size_t N, double x, double a=-1, double b=1)

static Weights	DerivativeWeights (size_t N, double x, double a=-1, double b=1)

static DiffMatrix	DifferentiationMatrix (size_t N)

static DiffMatrix	DifferentiationMatrix (size_t N, double a, double b)
	Compute D = differentiation matrix, for interval [a,b]. More...

static Weights	DoubleIntegrationWeights (size_t N)

static Weights	DoubleIntegrationWeights (size_t N, double a, double b)
	Calculate Double Clenshaw-Curtis integration weights, for interval [a,b]. More...

static Matrix	IntegrationMatrix (size_t N)

static Matrix	IntegrationMatrix (size_t N, double a, double b)
	IntegrationMatrix returns the (N+1)×(N+1) matrix P for interval [a,b]. More...

static Weights	IntegrationWeights (size_t N)

static Weights	IntegrationWeights (size_t N, double a, double b)
	Calculate Clenshaw-Curtis integration weights, for interval [a,b]. More...

template<size_t M>
static Matrix	matrix (std::function< Eigen::Matrix< double, M, 1 >(double)> f, size_t N, double a=-1, double b=1)
	Create matrix of values at Chebyshev points given vector-valued function. More...

static double	Point (size_t N, int j)
	Specific Chebyshev point, within [-1,1] interval. More...

static double	Point (size_t N, int j, double a, double b)
	Specific Chebyshev point, within [a,b] interval. More...

static Vector	Points (size_t N)
	All Chebyshev points. More...

static Vector	Points (size_t N, double a, double b)
	All Chebyshev points, within [a,b] interval. More...

static Vector	vector (std::function< double(double)> f, size_t N, double a=-1, double b=1)
	Create matrix of values at Chebyshev points given vector-valued function. More...

Static Public Member Functions inherited from gtsam::Basis< Chebyshev2 >
static Matrix	WeightMatrix (size_t N, const Vector &X)

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...

Detailed Description

Chebyshev Interpolation on Chebyshev points of the second kind Note that N here, the number of points, is one less than N from 'Approximation Theory and Approximation Practice by L. N. Trefethen (pg.42)'.

Definition at line 46 of file Chebyshev2.h.

Member Typedef Documentation

◆ Base

using gtsam::Chebyshev2::Base = Basis<Chebyshev2>

Definition at line 50 of file Chebyshev2.h.

◆ DiffMatrix

using gtsam::Chebyshev2::DiffMatrix = Eigen::Matrix<double, -1, -1>

Definition at line 52 of file Chebyshev2.h.

◆ Parameters

using gtsam::Chebyshev2::Parameters = Eigen::Matrix<double, -1, 1>

Definition at line 51 of file Chebyshev2.h.

Member Function Documentation

◆ CalculateWeights()

Weights gtsam::Chebyshev2::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) These weights implement barycentric interpolation at a specific x. More precisely, f(x) ~ [w0;...;wN] * [f0;...;fN], where the fj are the values of the function f at the Chebyshev points. As such, for a given x we obtain a linear map from parameter vectors f to interpolated values f(x). Optional [a,b] interval can be specified as well.

Definition at line 121 of file Chebyshev2.cpp.

◆ DerivativeWeights()

Weights gtsam::Chebyshev2::DerivativeWeights	(	size_t	N,
		double	x,
		double	a = `-1`,
		double	b = `1`
	)

static

Evaluate derivative of barycentric weights. This is easy and efficient via the DifferentiationMatrix.

Definition at line 151 of file Chebyshev2.cpp.

◆ DifferentiationMatrix() [1/2]

Chebyshev2::DiffMatrix gtsam::Chebyshev2::DifferentiationMatrix ( size_t N )

static

Compute D = differentiation matrix, Trefethen00book p.53 When given a parameter vector f of function values at the Chebyshev points, D*f are the values of f'. https://people.maths.ox.ac.uk/trefethen/8all.pdf Theorem 8.4

Definition at line 202 of file Chebyshev2.cpp.

◆ DifferentiationMatrix() [2/2]

Chebyshev2::DiffMatrix gtsam::Chebyshev2::DifferentiationMatrix	(	size_t	N,
		double	a,
		double	b
	)

static

Compute D = differentiation matrix, for interval [a,b].

Definition at line 216 of file Chebyshev2.cpp.

◆ DoubleIntegrationWeights() [1/2]

Weights gtsam::Chebyshev2::DoubleIntegrationWeights ( size_t N )

static

Calculate Double Clenshaw-Curtis integration weights We compute them as W * P, where W are the Clenshaw-Curtis weights and P is the integration matrix.

Definition at line 293 of file Chebyshev2.cpp.

◆ DoubleIntegrationWeights() [2/2]

Weights gtsam::Chebyshev2::DoubleIntegrationWeights	(	size_t	N,
		double	a,
		double	b
	)

static

Calculate Double Clenshaw-Curtis integration weights, for interval [a,b].

Definition at line 299 of file Chebyshev2.cpp.

◆ IntegrationMatrix() [1/2]

Matrix gtsam::Chebyshev2::IntegrationMatrix ( size_t N )

static

IntegrationMatrix returns the (N+1)×(N+1) matrix P such that for any f, F = P * f recovers F (the antiderivative) satisfying f = D * F and F(0)=0.

Definition at line 227 of file Chebyshev2.cpp.

◆ IntegrationMatrix() [2/2]

Matrix gtsam::Chebyshev2::IntegrationMatrix	(	size_t	N,
		double	a,
		double	b
	)

static

IntegrationMatrix returns the (N+1)×(N+1) matrix P for interval [a,b].

Definition at line 244 of file Chebyshev2.cpp.

◆ IntegrationWeights() [1/2]

Weights gtsam::Chebyshev2::IntegrationWeights ( size_t N )

static

Calculate Clenshaw-Curtis integration weights. Trefethen00book, pg 128, clencurt.m Note that N in clencurt.m is 1 less than our N

Definition at line 264 of file Chebyshev2.cpp.

◆ IntegrationWeights() [2/2]

Weights gtsam::Chebyshev2::IntegrationWeights	(	size_t	N,
		double	a,
		double	b
	)

static

Calculate Clenshaw-Curtis integration weights, for interval [a,b].

Definition at line 289 of file Chebyshev2.cpp.

◆ matrix()

template<size_t M>

static Matrix gtsam::Chebyshev2::matrix	(	std::function< Eigen::Matrix< double, M, 1 >(double)>	f,
		size_t	N,
		double	a = `-1`,
		double	b = `1`
	)

inlinestatic

Create matrix of values at Chebyshev points given vector-valued function.

Definition at line 138 of file Chebyshev2.h.

◆ Point() [1/2]

double gtsam::Chebyshev2::Point	(	size_t	N,
		int	j
	)

static

Specific Chebyshev point, within [-1,1] interval.

Parameters

N	The degree of the polynomial
j	The index of the Chebyshev point

Returns: double

Definition at line 27 of file Chebyshev2.cpp.

◆ Point() [2/2]

double gtsam::Chebyshev2::Point	(	size_t	N,
		int	j,
		double	a,
		double	b
	)

static

Specific Chebyshev point, within [a,b] interval.

Parameters

N	The degree of the polynomial
j	The index of the Chebyshev point
a	Lower bound of interval (default: -1)
b	Upper bound of interval (default: 1)

Returns: double

Definition at line 34 of file Chebyshev2.cpp.

◆ Points() [1/2]

Vector gtsam::Chebyshev2::Points ( size_t N )

static

All Chebyshev points.

Definition at line 39 of file Chebyshev2.cpp.

◆ Points() [2/2]

Vector gtsam::Chebyshev2::Points	(	size_t	N,
		double	a,
		double	b
	)

static

All Chebyshev points, within [a,b] interval.

Definition at line 58 of file Chebyshev2.cpp.

◆ vector()

Vector gtsam::Chebyshev2::vector	(	std::function< double(double)>	f,
		size_t	N,
		double	a = `-1`,
		double	b = `1`
	)

static

Create matrix of values at Chebyshev points given vector-valued function.

Create vector of values at Chebyshev points given scalar-valued function.

Definition at line 306 of file Chebyshev2.cpp.

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

Public Types

Static Public Member Functions

Detailed Description

Member Typedef Documentation

◆ Base

◆ DiffMatrix

◆ Parameters

Member Function Documentation

◆ CalculateWeights()

◆ DerivativeWeights()

◆ DifferentiationMatrix() [1/2]

◆ DifferentiationMatrix() [2/2]

◆ DoubleIntegrationWeights() [1/2]

◆ DoubleIntegrationWeights() [2/2]

◆ IntegrationMatrix() [1/2]

◆ IntegrationMatrix() [2/2]

◆ IntegrationWeights() [1/2]

◆ IntegrationWeights() [2/2]

◆ matrix()

◆ Point() [1/2]

◆ Point() [2/2]

◆ Points() [1/2]

◆ Points() [2/2]

◆ vector()