cmp::LegendreBasis Struct Reference

Legendre 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	getPDFWeight ()
static double	mapToPhysical (double xi, double lowerBound, double upperBound)
static double	mapToCanonical (double x, double lowerBound, double upperBound)

Detailed Description

Legendre orthogonal polynomial basis.

Mathematical Formulation Legendre polynomials \(P_n(\xi)\) are orthogonal with respect to the uniform density function \(w(\xi) = 1/2\) on \([-1, 1]\). The recurrence relation is:

\[ (n+1) P_{n+1}(\xi) = (2n+1) \xi P_n(\xi) - n P_{n-1}(\xi) \]

with \(\int_{-1}^1 P_m(\xi) P_n(\xi) d\xi = \frac{2}{2n+1} \delta_{mn}\).

Implementation Algorithm

1. evaluate() computes polynomial values using Bonnet's recurrence relation.
2. getJacobiMatrix() constructs the symmetric tridiagonal Jacobi matrix \(\mathbf{J}\) with \(\alpha_i = 0\) and off-diagonal \(\beta_i = \frac{i}{\sqrt{(2i-1)(2i+1)}}\).

Member Function Documentation

evaluate()

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

inlinestatic

getJacobiMatrix()

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

inlinestatic

getPDFWeight()

static double cmp::LegendreBasis::getPDFWeight ( )

inlinestatic

mapToCanonical()

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

inlinestatic

mapToPhysical()

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

inlinestatic

normSquared()

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

inlinestatic

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

• include/poly.h