cmp::HermiteBasis Struct Reference

Hermite 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 mean, double stdDev)
static double	mapToCanonical (double x, double mean, double stdDev)

Detailed Description

Hermite orthogonal polynomial basis.

Mathematical Formulation Probabilist's Hermite polynomials \(H_n(\xi)\) are orthogonal with respect to the standard Gaussian density function \(w(\xi) = \frac{1}{\sqrt{2\pi}} e^{-\xi^2/2}\). The recurrence relation is:

\[ H_0(\xi) = 1, \quad H_1(\xi) = \xi, \quad H_{n+1}(\xi) = \xi H_n(\xi) - n H_{n-1}(\xi) \]

with \(\mathbb{E}[H_m(\xi) H_n(\xi)] = n! \delta_{mn}\).

Implementation Algorithm

1. evaluate() computes polynomial values using the 3-term recurrence relation.
2. getJacobiMatrix() constructs the symmetric tridiagonal Jacobi matrix \(\mathbf{J}\) with diagonal \(\alpha_i = 0\) and off-diagonal \(\beta_i = \sqrt{i}\) to evaluate quadrature nodes via spectral decomposition.

Member Function Documentation

evaluate()

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

inlinestatic

getJacobiMatrix()

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

inlinestatic

getPDFWeight()

static double cmp::HermiteBasis::getPDFWeight ( )

inlinestatic

mapToCanonical()

static double cmp::HermiteBasis::mapToCanonical ( double  x, double  mean, double  stdDev  )

inlinestatic

mapToPhysical()

static double cmp::HermiteBasis::mapToPhysical ( double  xi, double  mean, double  stdDev  )

inlinestatic

normSquared()

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

inlinestatic

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

• include/poly.h