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)'.
Inheritance diagram for gtsam::Chebyshev2:
Static Public Member Functions | |
| static double | Point (size_t N, int j) |
| Specific Chebyshev point. | |
| static double | Point (size_t N, int j, double a, double b) |
| Specific Chebyshev point, within [a,b] interval. | |
| static Vector | Points (size_t N) |
| All Chebyshev points. | |
| static Vector | Points (size_t N, double a, double b) |
| All Chebyshev points, within [a,b] interval. | |
| static Weights | CalculateWeights (size_t N, double x, double a=-1, double b=1) |
| 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... | |
| static Weights | DerivativeWeights (size_t N, double x, double a=-1, double b=1) |
| Evaluate derivative of barycentric weights. More... | |
| static DiffMatrix | DifferentiationMatrix (size_t N, double a=-1, double b=1) |
| 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'. More... | |
| static Weights | IntegrationWeights (size_t N, double a=-1, double b=1) |
| Evaluate Clenshaw-Curtis integration weights. More... | |
| template<size_t M> | |
| static Matrix | matrix (boost::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. | |
| Static Public Member Functions inherited from gtsam::Basis< Chebyshev2 > | |
| 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 Types | |
| using | Base = Basis< Chebyshev2 > |
| using | Parameters = Eigen::Matrix< double, -1, 1 > |
| using | DiffMatrix = Eigen::Matrix< double, -1, -1 > |
Member Function Documentation
◆ CalculateWeights()
|
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.
◆ DerivativeWeights()
|
static |
Evaluate derivative of barycentric weights.
This is easy and efficient via the DifferentiationMatrix.
◆ DifferentiationMatrix()
|
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
◆ IntegrationWeights()
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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;
The documentation for this class was generated from the following files:
- /Users/dellaert/git/github/gtsam/basis/Chebyshev2.h
- /Users/dellaert/git/github/gtsam/basis/Chebyshev2.cpp
