#include <Chebyshev2.h>
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... | |
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.
using gtsam::Chebyshev2::Base = Basis<Chebyshev2> |
Definition at line 50 of file Chebyshev2.h.
using gtsam::Chebyshev2::DiffMatrix = Eigen::Matrix<double, -1, -1> |
Definition at line 52 of file Chebyshev2.h.
using gtsam::Chebyshev2::Parameters = Eigen::Matrix<double, -1, 1> |
Definition at line 51 of file Chebyshev2.h.
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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.
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Evaluate derivative of barycentric weights. This is easy and efficient via the DifferentiationMatrix.
Definition at line 62 of file Chebyshev2.cpp.
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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.
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.
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inlinestatic |
Create matrix of values at Chebyshev points given vector-valued function.
Definition at line 138 of file Chebyshev2.h.
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inlinestatic |
Specific Chebyshev point, within [a,b] interval. Default interval is [-1, 1].
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) |
Definition at line 64 of file Chebyshev2.h.
All Chebyshev points.
Definition at line 73 of file Chebyshev2.h.
All Chebyshev points, within [a,b] interval.
Definition at line 82 of file Chebyshev2.h.