gtsam::Chebyshev2 Class Reference

#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, double a=-1, double b=1)
static Weights	IntegrationWeights (size_t N, double a=-1, double b=1)
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)
static double	Point (size_t N, int j, double a=-1, double b=1)
	Specific Chebyshev point, within [a,b] interval. Default interval is [-1, 1]. 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 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 23 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 62 of file Chebyshev2.cpp.

DifferentiationMatrix()

Chebyshev2::DiffMatrix gtsam::Chebyshev2::DifferentiationMatrix ( size_t  N, double  a = -1, double  b = 1  )

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 147 of file Chebyshev2.cpp.

IntegrationWeights()

Weights gtsam::Chebyshev2::IntegrationWeights ( size_t  N, double  a = -1, double  b = 1  )

static

Evaluate Clenshaw-Curtis integration weights. Trefethen00book, pg 128, clencurt.m Note that N in clencurt.m is 1 less than our N K = N-1; theta = pi*(0:K)'/K; w = zeros(1,N); ii = 2:K; v = ones(K-1, 1); if mod(K,2) == 0 w(1) = 1/(K^2-1); w(N) = w(1); for k=1:K/2-1, v = v-2*cos(2*k*theta(ii))/(4*k^2-1); end v = v - cos(K*theta(ii))/(K^2-1); else w(1) = 1/K^2; w(N) = w(1); for k=1:K/2, v = v-2*cos(2*k*theta(ii))/(4*k^2-1); end end w(ii) = 2*v/K;

Definition at line 183 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()

static double gtsam::Chebyshev2::Point ( size_t  N, int  j, double  a = -1, double  b = 1  )

inlinestatic

Specific Chebyshev point, within [a,b] interval. Default interval is [-1, 1].

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 64 of file Chebyshev2.h.

Points() [1/2]

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

inlinestatic

All Chebyshev points.

Definition at line 73 of file Chebyshev2.h.

Points() [2/2]

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

inlinestatic

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

Definition at line 82 of file Chebyshev2.h.

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

• Chebyshev2.h
• Chebyshev2.cpp