cmp::ChebyshevBasis Struct Reference

Chebyshev orthogonal polynomial basis. More...

#include <poly.h>

Static Public Member Functions
static double	evaluate (size_t deg, double xi)
static double	normSquared (size_t deg)
static Eigen::MatrixXd	getJacobiMatrix (int numPoints)
static double	mapToPhysical (double xi, double lowerBound, double upperBound)
static double	mapToCanonical (double x, double lowerBound, double upperBound)

Detailed Description

Chebyshev orthogonal polynomial basis.

Mathematical Formulation Chebyshev polynomials of the first kind \(T_n(\xi)\) are orthogonal with respect to the weight function \(w(\xi) = \frac{1}{\sqrt{1-\xi^2}}\) on \([-1, 1]\). The recurrence relation is:

\[ T_0(\xi) = 1, \quad T_1(\xi) = \xi, \quad T_{n+1}(\xi) = 2\xi T_n(\xi) - T_{n-1}(\xi) \]

Implementation Algorithm

1. evaluate() computes polynomial values using the 3-term recurrence relation.
2. getJacobiMatrix() constructs the symmetric tridiagonal Jacobi matrix \(\mathbf{J}\) with \(\alpha_i = 0\) and off-diagonal \(\beta_1 = \frac{1}{\sqrt{2}}\), \(\beta_i = 0.5\) for \(i \ge 2\).

Member Function Documentation

evaluate()

static double cmp::ChebyshevBasis::evaluate ( size_t  deg, double  xi  )

inlinestatic

getJacobiMatrix()

static Eigen::MatrixXd cmp::ChebyshevBasis::getJacobiMatrix ( int  numPoints )

inlinestatic

mapToCanonical()

static double cmp::ChebyshevBasis::mapToCanonical ( double  x, double  lowerBound, double  upperBound  )

inlinestatic

mapToPhysical()

static double cmp::ChebyshevBasis::mapToPhysical ( double  xi, double  lowerBound, double  upperBound  )

inlinestatic

normSquared()

static double cmp::ChebyshevBasis::normSquared ( size_t  deg )

inlinestatic

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The documentation for this struct was generated from the following file:

• include/poly.h