JacobiOrthogPolynomial Class Reference

Derived orthogonal polynomial class for Jacobi polynomials. More...

Inheritance diagram for JacobiOrthogPolynomial:

List of all members.

Public Member Functions
	JacobiOrthogPolynomial ()
	default constructor
	JacobiOrthogPolynomial (const Real &alpha_stat, const Real &beta_stat)
	standard constructor
	~JacobiOrthogPolynomial ()
	destructor
Protected Member Functions
const Real &	get_value (const Real &x, unsigned short order)
	retrieve the Jacobi polynomial value for a given parameter x
const Real &	get_gradient (const Real &x, unsigned short order)
	retrieve the Jacobi polynomial gradient for a given parameter x
const Real &	norm_squared (unsigned short order)
	||P^(alpha,beta)_n||^2
const RealVector &	gauss_points (unsigned short order)
	polynomial order n
const RealVector &	gauss_weights (unsigned short order)
	polynomial order n
void	alpha_stat (const Real &alpha)
	set betaPoly using the conversion betaPoly = alpha_stat - 1.
void	beta_stat (const Real &beta)
	set alphaPoly using the conversion alphaPoly = beta_stat - 1.
Private Attributes
Real	alphaPoly
	Abramowitz and Stegun (differs from statistical PDF notation).
Real	betaPoly
	Abramowitz and Stegun (differs from statistical PDF notation).

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Detailed Description

Derived orthogonal polynomial class for Jacobi polynomials.

The JacobiOrthogPolynomial class evaluates a univariate Jacobi polynomial P^(alpha,beta)_n of a particular order. These polynomials are orthogonal with respect to the weight function (1-x)^alpha (1+x)^beta when integrated over the support range of [-1,+1]. This corresponds to the probability density function f(x) = (1-x)^alpha (1+x)^beta / (2^(alpha+beta+1) B(alpha+1,beta+1)) for the beta distribution for [L,U]=[-1,1], where common statistical PDF notation conventions (see, e.g., the uncertain variables section in the DAKOTA Reference Manual) and the Abramowiz and Stegun orthogonal polynomial conventions are inverted and require conversion in this case (alpha_poly = beta_stat - 1; beta_poly = alpha_stat - 1 with the poly definitions used in both cases above). It enables (mixed) multidimensional orthogonal polynomial basis functions within OrthogPolyApproximation. A special case is the LegendreOrthogPolynomial (implemented separately), for which alpha_poly = beta_poly = 0.

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The documentation for this class was generated from the following files:

• JacobiOrthogPolynomial.H
• JacobiOrthogPolynomial.C

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