cmp::LaguerreBasis Struct Reference

Laguerre 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 location, double scale)
static double	mapToCanonical (double x, double location, double scale)

Detailed Description

Laguerre orthogonal polynomial basis.

Mathematical Formulation Laguerre polynomials \(L_n(\xi)\) are orthogonal with respect to the exponential weight function \(w(\xi) = e^{-\xi}\) on \([0, \infty)\). The recurrence relation is:

\[ (n+1) L_{n+1}(\xi) = (2n + 1 - \xi) L_n(\xi) - n L_{n-1}(\xi) \]

Implementation Algorithm

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

Member Function Documentation

evaluate()

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

inlinestatic

getJacobiMatrix()

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

inlinestatic

mapToCanonical()

static double cmp::LaguerreBasis::mapToCanonical ( double  x, double  location, double  scale  )

inlinestatic

mapToPhysical()

static double cmp::LaguerreBasis::mapToPhysical ( double  xi, double  location, double  scale  )

inlinestatic

normSquared()

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

inlinestatic

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

• include/poly.h