QUESO::WignerChebyshev2nd1DQuadrature Class Reference

Class for second type Chebyshev-Gauss quadrature rule for one-dimensional functions. More...

#include <1DQuadrature.h>

Inheritance diagram for QUESO::WignerChebyshev2nd1DQuadrature:

Public Member Functions
Constructor/Destructor methods
	WignerChebyshev2nd1DQuadrature (double minDomainValue, double maxDomainValue, unsigned int order)
	Default constructor. More...
	~WignerChebyshev2nd1DQuadrature ()
	Destructor. More...
Public Member Functions inherited from QUESO::Base1DQuadrature
	Base1DQuadrature (double minDomainValue, double maxDomainValue, unsigned int order)
	Default constructor. More...
virtual	~Base1DQuadrature ()=0
	Pure virtual destructor, forcing this to be an abstract object. More...
double	minDomainValue () const
	Returns the minimum value of the domain of the (one-dimensional) function. More...
double	maxDomainValue () const
	Returns the maximum value of the domain of the (one-dimensional) function. More...
unsigned int	order () const
	Returns the order of the quadrature rule. More...
const std::vector< double > &	positions () const
	Array of the positions for the numerical integration. More...
Public Member Functions inherited from QUESO::BaseQuadrature
	BaseQuadrature ()
virtual	~BaseQuadrature ()=0
	Pure virtual destructor, forcing this to be an abstract object. More...
const std::vector< double > &	weights () const
	Array of the weights used in the numerical integration. More...

Additional Inherited Members
Protected Attributes inherited from QUESO::Base1DQuadrature
double	m_minDomainValue
double	m_maxDomainValue
unsigned int	m_order
std::vector< double >	m_positions
Protected Attributes inherited from QUESO::BaseQuadrature
std::vector< double >	m_weights

Detailed Description

Class for second type Chebyshev-Gauss quadrature rule for one-dimensional functions.

Chebyshev-Gauss quadrature, also called Chebyshev Type 2 quadrature, is a Gaussian quadrature over the interval [-1,1] with weighting function \( W(x)=\sqrt{1-x^2}\).
The abscissas for quadrature order \( n \) are given by the roots of the Chebyshev polynomial of the second kind \( U_n(x) \), which occur symmetrically about 0.
The abscissas are given explicitly by \( x_i=\cos[\frac{i\pi}{n+1}].\) and all the weights are \( w_i=\frac{\pi}{n+1}\sin^2[\frac{i\pi}{n+1}]. \)

See Also

Weisstein, Eric W. "Gaussian Quadrature." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GaussianQuadrature.html.

http://en.wikipedia.org/wiki/Chebyshev-Gauss_quadrature.

Definition at line 267 of file 1DQuadrature.h.

Constructor & Destructor Documentation

QUESO::WignerChebyshev2nd1DQuadrature::WignerChebyshev2nd1DQuadrature	(	double	minDomainValue,
		double	maxDomainValue,
		unsigned int	order
	)

Default constructor.

Constructs a Gaussian-Chebyshev quadrature (of second type) of order order, in the interval [minDomainValue,maxDomainValue]. This method scales the abscissas (positions) of the quadrature from the interval [-1,1] to [minDomainValue,maxDomainValue].

Definition at line 653 of file 1DQuadrature.C.

References QUESO::Base1DQuadrature::m_maxDomainValue, QUESO::Base1DQuadrature::m_minDomainValue, QUESO::Base1DQuadrature::m_order, QUESO::Base1DQuadrature::m_positions, and QUESO::BaseQuadrature::m_weights.

  :
  Base1DQuadrature(minDomainValue,maxDomainValue,order)
{
  m_positions.resize(m_order+1,0.); // Yes, '+1'
  m_weights.resize  (m_order+1,0.); // Yes, '+1'

  // http://en.wikipedia.org/wiki/Chebyshev-Gauss_quadrature
  unsigned int n = m_order+1;
  for (unsigned int i = 0; i < n; ++i) {
    double angle = M_PI*((double)(i+1))/((double)(n+1));
    double cosValue = cos(angle);
    double sinValue = sin(angle);
    m_positions[i] = cosValue;
    m_weights[i] = ( M_PI/((double)(n+1)) )*sinValue*sinValue;
  }

  // Scale positions from the interval [-1, 1] to the interval [min,max]
  for (unsigned int j = 0; j < m_positions.size(); ++j) {
    m_positions[j] = .5*(m_maxDomainValue - m_minDomainValue)*m_positions[j] + .5*(m_maxDomainValue + m_minDomainValue);
    m_weights[j] *= .5*(m_maxDomainValue - m_minDomainValue);
  }
}

QUESO::WignerChebyshev2nd1DQuadrature::~WignerChebyshev2nd1DQuadrature

(

)

Destructor.

Definition at line 680 of file 1DQuadrature.C.

{
}

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

• src/misc/inc/1DQuadrature.h
• src/misc/src/1DQuadrature.C