108 lines
3.7 KiB
C++
108 lines
3.7 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file Chebyshev.h
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* @brief Chebyshev basis decompositions
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* @author Varun Agrawal, Jing Dong, Frank Dellaert
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* @date July 4, 2020
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*/
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#pragma once
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#include <gtsam/base/Manifold.h>
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#include <gtsam/basis/Basis.h>
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namespace gtsam {
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/**
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* Basis of Chebyshev polynomials of the first kind
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* https://en.wikipedia.org/wiki/Chebyshev_polynomials#First_kind
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* These are typically denoted with the symbol T_n, where n is the degree.
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* The parameter N is the number of coefficients, i.e., N = n+1.
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*/
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struct GTSAM_EXPORT Chebyshev1Basis : Basis<Chebyshev1Basis> {
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using Parameters = Eigen::Matrix<double, -1, 1 /*Nx1*/>;
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Parameters parameters_;
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/**
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* @brief Evaluate Chebyshev Weights on [-1,1] at x up to order N-1 (N values)
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*
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* @param N Degree of the polynomial.
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* @param x Point to evaluate polynomial at.
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* @param a Lower limit of polynomial (default=-1).
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* @param b Upper limit of polynomial (default=1).
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*/
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static Weights CalculateWeights(size_t N, double x, double a = -1,
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double b = 1);
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/**
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* @brief Evaluate Chebyshev derivative at x.
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* The derivative weights are pre-multiplied to the polynomial Parameters.
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*
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* From Wikipedia we have D[T_n(x),x] = n*U_{n-1}(x)
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* I.e. the derivative fo a first kind cheb is just a second kind cheb
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* So, we define a second kind basis here of order N-1
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* Note that it has one less weight.
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*
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* The Parameters pertain to 1st kind chebs up to order N-1
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* But of course the first one (order 0) is constant, so omit that weight.
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*
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* @param N Degree of the polynomial.
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* @param x Point to evaluate polynomial at.
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* @param a Lower limit of polynomial (default=-1).
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* @param b Upper limit of polynomial (default=1).
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* @return Weights
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*/
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static Weights DerivativeWeights(size_t N, double x, double a = -1,
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double b = 1);
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}; // Chebyshev1Basis
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/**
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* Basis of Chebyshev polynomials of the second kind.
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* https://en.wikipedia.org/wiki/Chebyshev_polynomials#Second_kind
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* These are typically denoted with the symbol U_n, where n is the degree.
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* The parameter N is the number of coefficients, i.e., N = n+1.
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* In contrast to the templates in Chebyshev2, the classes below specify
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* basis functions, weighted combinations of which are used to approximate
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* functions. In this sense, they are like the sines and cosines of the Fourier
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* basis.
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*/
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struct GTSAM_EXPORT Chebyshev2Basis : Basis<Chebyshev2Basis> {
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using Parameters = Eigen::Matrix<double, -1, 1 /*Nx1*/>;
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/**
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* Evaluate Chebyshev Weights on [-1,1] at any x up to order N-1 (N values).
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*
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* @param N Degree of the polynomial.
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* @param x Point to evaluate polynomial at.
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* @param a Lower limit of polynomial (default=-1).
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* @param b Upper limit of polynomial (default=1).
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*/
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static Weights CalculateWeights(size_t N, double x, double a = -1,
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double b = 1);
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/**
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* @brief Evaluate Chebyshev derivative at x.
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*
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* @param N Degree of the polynomial.
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* @param x Point to evaluate polynomial at.
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* @param a Lower limit of polynomial (default=-1).
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* @param b Upper limit of polynomial (default=1).
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* @return Weights
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*/
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static Weights DerivativeWeights(size_t N, double x, double a = -1,
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double b = 1);
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}; // Chebyshev2Basis
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} // namespace gtsam
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