QUESO::UniformLegendre1DQuadrature Class Reference

Class for Legendre-Gauss quadrature rule for one-dimensional functions. More...

#include <1DQuadrature.h>

Inheritance diagram for QUESO::UniformLegendre1DQuadrature:

Public Member Functions
Constructor/Destructor methods
	UniformLegendre1DQuadrature (double minDomainValue, double maxDomainValue, unsigned int order, bool densityIsNormalized)
	Default constructor. More...
	~UniformLegendre1DQuadrature ()
	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 Legendre-Gauss quadrature rule for one-dimensional functions.

In a general Gaussian quadrature rule, an definite integral of \( f(x)\) is first approximated over the interval [-1,1] by a polynomial approximable function \( g(x)\) and a known weighting function \( W(x)\):

\[\int_{-1}^1 f(x) \, dx = \int_{-1}^1 W(x) g(x) \, dx\]

Those are then approximated by a sum of function values at specified points \( x_i \) multiplied by some weights \( w_i \):

\[ \int_{-1}^1 W(x) g(x) \, dx \approx \sum_{i=1}^n w_i g(x_i) \]

In the case of Gauss-Legendre quadrature, the weighting function \( W(x) = 1 \), so we can approximate an integral of \( f(x) \) with:

\[ \int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) \]

The abscissas for quadrature order \( n \) are given by the roots of the Legendre polynomials \( P_n(x)\), which occur symmetrically about 0. The weights are

\[ w_i = \frac{2}{(1-x_i^2)[P'_n(x_i)]^2}=\frac{2(1-x_i^2)}{(n+1)^2[P_{n+1}(x_i)]^2} \]

Several authors give a table of abscissas and weights:

\( n \)	\( x_i \)	\( w_i \)
2	\( \pm \frac{1}{3}\sqrt{3} \)	\( 1 \)
3	\( 0 \)	\( \frac{8}{9} \)
	\( \pm \frac{1}{5} \sqrt{15} \)	\( \frac{5}{9} \)
4	\( \pm \frac{1}{35}\sqrt{525-70\sqrt{30}} \)	\( \frac{1}{36}(18+\sqrt{30})\)
	\( \pm \frac{1}{35}\sqrt{525+70\sqrt{30}} \)	\( \frac{1}{36}(18-\sqrt{30})\)
5	\( 0 \)	\( \frac{128}{225}\)
	\( \pm \frac{1}{21}\sqrt{245-14\sqrt{70}} \)	\( \frac{1}{900}(322+13\sqrt{70})\)
	\( \pm \frac{1}{21}\sqrt{245+14\sqrt{70}} \)	\( \frac{1}{900}(322-13\sqrt{70})\)

See Also

Weisstein, Eric W. "Legendre-Gauss Quadrature." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Legendre-GaussQuadrature.html.

Definition at line 148 of file 1DQuadrature.h.

Constructor & Destructor Documentation

QUESO::UniformLegendre1DQuadrature::UniformLegendre1DQuadrature	(	double	minDomainValue,
		double	maxDomainValue,
		unsigned int	order,
		bool	densityIsNormalized
	)

Default constructor.

Constructs a Gaussian-Legendre quadrature of order order, in the interval [minDomainValue,maxDomainValue]. Valid values for the order of the quadrature rule are: 1-7, 10-12, 16. This method scales the abscissas (positions) of the quadrature from the interval [-1,1] to [minDomainValue,maxDomainValue], and the parameter densityIsNormalized determines whether the weights should be scaled as well.

Definition at line 98 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://www.holoborodko.com/pavel/?page_id=679
  switch (m_order) { // eep2011
    case 1:
      m_weights  [0] =  1.;
      m_weights  [1] =  1.;

      m_positions[0] = -1./sqrt(3.);
      m_positions[1] =  1./sqrt(3.);
    break;

    case 2:
      m_weights  [0] =  5./9.;
      m_weights  [1] =  8./9.;
      m_weights  [2] =  5./9.;

      m_positions[0] = -sqrt(.6);
      m_positions[1] =  0.;
      m_positions[2] =  sqrt(.6);
    break;

    case 3:
      m_weights  [0] =  0.5 - sqrt(30.)/36.;
      m_weights  [1] =  0.5 + sqrt(30.)/36.;
      m_weights  [2] =  0.5 + sqrt(30.)/36.;
      m_weights  [3] =  0.5 - sqrt(30.)/36.;

      m_positions[0] = -sqrt(3.+2.*sqrt(1.2))/sqrt(7.);
      m_positions[1] = -sqrt(3.-2.*sqrt(1.2))/sqrt(7.);
      m_positions[2] =  sqrt(3.-2.*sqrt(1.2))/sqrt(7.);
      m_positions[3] =  sqrt(3.+2.*sqrt(1.2))/sqrt(7.);
    break;

    case 4:
      m_weights  [0] =  (322.-13.*sqrt(70.))/900.; // 0.236926885
      m_weights  [1] =  (322.+13.*sqrt(70.))/900.; // 0.478628670
      m_weights  [2] =  128./225.;                 // 0.568888889
      m_weights  [3] =  (322.+13.*sqrt(70.))/900.;
      m_weights  [4] =  (322.-13.*sqrt(70.))/900.;

      m_positions[0] = -sqrt(5.+2.*sqrt(10./7.))/3.;
      m_positions[1] = -sqrt(5.-2.*sqrt(10./7.))/3.;
      m_positions[2] =  0.;
      m_positions[3] =  sqrt(5.-2.*sqrt(10./7.))/3.;
      m_positions[4] =  sqrt(5.+2.*sqrt(10./7.))/3.;
    break;

    case 5:
      m_weights  [0] =  0.1713244923791703450402961;
      m_weights  [1] =  0.3607615730481386075698335;
      m_weights  [2] =  0.4679139345726910473898703;
      m_weights  [3] =  0.4679139345726910473898703;
      m_weights  [4] =  0.3607615730481386075698335;
      m_weights  [5] =  0.1713244923791703450402961;

      m_positions[0] = -0.9324695142031520278123016;
      m_positions[1] = -0.6612093864662645136613996;
      m_positions[2] = -0.2386191860831969086305017;
      m_positions[3] =  0.2386191860831969086305017;
      m_positions[4] =  0.6612093864662645136613996;
      m_positions[5] =  0.9324695142031520278123016;
    break;

    case 6:
      m_weights  [0] =  0.1294849661688696932706114;
      m_weights  [1] =  0.2797053914892766679014678;
      m_weights  [2] =  0.3818300505051189449503698;
      m_weights  [3] =  0.4179591836734693877551020;
      m_weights  [4] =  0.3818300505051189449503698;
      m_weights  [5] =  0.2797053914892766679014678;
      m_weights  [6] =  0.1294849661688696932706114;

      m_positions[0] = -0.9491079123427585245261897;
      m_positions[1] = -0.7415311855993944398638648;
      m_positions[2] = -0.4058451513773971669066064;
      m_positions[3] =  0.;
      m_positions[4] =  0.4058451513773971669066064;
      m_positions[5] =  0.7415311855993944398638648;
      m_positions[6] =  0.9491079123427585245261897;
    break;

    case 7:
      m_weights  [0] =  0.10122854;
      m_weights  [1] =  0.22238103;
      m_weights  [2] =  0.31370665;
      m_weights  [3] =  0.36268378;
      m_weights  [4] =  0.36268378;
      m_weights  [5] =  0.31370665;
      m_weights  [6] =  0.22238103;
      m_weights  [7] =  0.10122854;

      m_positions[0] = -0.96028986;
      m_positions[1] = -0.79666648;
      m_positions[2] = -0.52553241;
      m_positions[3] = -0.18343464;
      m_positions[4] =  0.18343464;
      m_positions[5] =  0.52553241;
      m_positions[6] =  0.79666648;
      m_positions[7] =  0.96028986;
    break;

    case 10:
      m_weights  [ 0] =  0.0556685671161736664827537;
      m_weights  [ 1] =  0.1255803694649046246346943;
      m_weights  [ 2] =  0.1862902109277342514260976;
      m_weights  [ 3] =  0.2331937645919904799185237;
      m_weights  [ 4] =  0.2628045445102466621806889;
      m_weights  [ 5] =  0.2729250867779006307144835;
      m_weights  [ 6] =  0.2628045445102466621806889;
      m_weights  [ 7] =  0.2331937645919904799185237;
      m_weights  [ 8] =  0.1862902109277342514260976;
      m_weights  [ 9] =  0.1255803694649046246346943;
      m_weights  [10] =  0.0556685671161736664827537;

      m_positions[ 0] = -0.9782286581460569928039380;
      m_positions[ 1] = -0.8870625997680952990751578;
      m_positions[ 2] = -0.7301520055740493240934163;
      m_positions[ 3] = -0.5190961292068118159257257;
      m_positions[ 4] = -0.2695431559523449723315320;
      m_positions[ 5] =  0.;
      m_positions[ 6] =  0.2695431559523449723315320;
      m_positions[ 7] =  0.5190961292068118159257257;
      m_positions[ 8] =  0.7301520055740493240934163;
      m_positions[ 9] =  0.8870625997680952990751578;
      m_positions[10] =  0.9782286581460569928039380;
    break;

    case 11:
      m_weights  [ 0] =  0.0471753363865118271946160;
      m_weights  [ 1] =  0.1069393259953184309602547;
      m_weights  [ 2] =  0.1600783285433462263346525;
      m_weights  [ 3] =  0.2031674267230659217490645;
      m_weights  [ 4] =  0.2334925365383548087608499;
      m_weights  [ 5] =  0.2491470458134027850005624;
      m_weights  [ 6] =  0.2491470458134027850005624;
      m_weights  [ 7] =  0.2334925365383548087608499;
      m_weights  [ 8] =  0.2031674267230659217490645;
      m_weights  [ 9] =  0.1600783285433462263346525;
      m_weights  [10] =  0.1069393259953184309602547;
      m_weights  [11] =  0.0471753363865118271946160;

      m_positions[ 0] = -0.9815606342467192506905491;
      m_positions[ 1] = -0.9041172563704748566784659;
      m_positions[ 2] = -0.7699026741943046870368938;
      m_positions[ 3] = -0.5873179542866174472967024;
      m_positions[ 4] = -0.3678314989981801937526915;
      m_positions[ 5] = -0.1252334085114689154724414;
      m_positions[ 6] =  0.1252334085114689154724414;
      m_positions[ 7] =  0.3678314989981801937526915;
      m_positions[ 8] =  0.5873179542866174472967024;
      m_positions[ 9] =  0.7699026741943046870368938;
      m_positions[10] =  0.9041172563704748566784659;
      m_positions[11] =  0.9815606342467192506905491;
    break;

    case 12:
      m_weights  [ 0] =  0.0404840047653158795200216;
      m_weights  [ 1] =  0.0921214998377284479144218;
      m_weights  [ 2] =  0.1388735102197872384636018;
      m_weights  [ 3] =  0.1781459807619457382800467;
      m_weights  [ 4] =  0.2078160475368885023125232;
      m_weights  [ 5] =  0.2262831802628972384120902;
      m_weights  [ 6] =  0.2325515532308739101945895;
      m_weights  [ 7] =  0.2262831802628972384120902;
      m_weights  [ 8] =  0.2078160475368885023125232;
      m_weights  [ 9] =  0.1781459807619457382800467;
      m_weights  [10] =  0.1388735102197872384636018;
      m_weights  [11] =  0.0921214998377284479144218;
      m_weights  [12] =  0.0404840047653158795200216;

      m_positions[ 0] = -0.9841830547185881494728294;
      m_positions[ 1] = -0.9175983992229779652065478;
      m_positions[ 2] = -0.8015780907333099127942065;
      m_positions[ 3] = -0.6423493394403402206439846;
      m_positions[ 4] = -0.4484927510364468528779129;
      m_positions[ 5] = -0.2304583159551347940655281;
      m_positions[ 6] =  0.;
      m_positions[ 7] =  0.2304583159551347940655281;
      m_positions[ 8] =  0.4484927510364468528779129;
      m_positions[ 9] =  0.6423493394403402206439846;
      m_positions[10] =  0.8015780907333099127942065;
      m_positions[11] =  0.9175983992229779652065478;
      m_positions[12] =  0.9841830547185881494728294;
    break;
#if 0
    case 13:
      m_weights  [ 0] =  ;
      m_weights  [ 1] =  ;
      m_weights  [ 2] =  ;
      m_weights  [ 3] =  ;
      m_weights  [ 4] =  ;
      m_weights  [ 5] =  ;
      m_weights  [ 6] =  ;
      m_weights  [ 7] =  ;
      m_weights  [ 8] =  ;
      m_weights  [ 9] =  ;
      m_weights  [10] =  ;
      m_weights  [11] =  ;
      m_weights  [12] =  ;
      m_weights  [13] =  ;

      m_positions[ 0] = -;
      m_positions[ 1] = -;
      m_positions[ 2] = -;
      m_positions[ 3] = -;
      m_positions[ 4] = -;
      m_positions[ 5] = -;
      m_positions[ 6] = -;
      m_positions[ 7] =  ;
      m_positions[ 8] =  ;
      m_positions[ 9] =  ;
      m_positions[10] =  ;
      m_positions[11] =  ;
      m_positions[12] =  ;
      m_positions[13] =  ;
    break;
#endif
#if 0
    14 ±0.1080549487073436620662447 0.2152638534631577901958764
    ±0.3191123689278897604356718 0.2051984637212956039659241
    ±0.5152486363581540919652907 0.1855383974779378137417166
    ±0.6872929048116854701480198 0.1572031671581935345696019
    ±0.8272013150697649931897947 0.1215185706879031846894148
    ±0.9284348836635735173363911 0.0801580871597602098056333
    ±0.9862838086968123388415973 0.0351194603317518630318329
#endif

    case 16:
      m_weights  [ 0] =  0.0241483028685479319601100;
      m_weights  [ 1] =  0.0554595293739872011294402;
      m_weights  [ 2] =  0.0850361483171791808835354;
      m_weights  [ 3] =  0.1118838471934039710947884;
      m_weights  [ 4] =  0.1351363684685254732863200;
      m_weights  [ 5] =  0.1540457610768102880814316;
      m_weights  [ 6] =  0.1680041021564500445099707;
      m_weights  [ 7] =  0.1765627053669926463252710;
      m_weights  [ 8] =  0.1794464703562065254582656;
      m_weights  [ 9] =  0.1765627053669926463252710;
      m_weights  [10] =  0.1680041021564500445099707;
      m_weights  [11] =  0.1540457610768102880814316;
      m_weights  [12] =  0.1351363684685254732863200;
      m_weights  [13] =  0.1118838471934039710947884;
      m_weights  [14] =  0.0850361483171791808835354;
      m_weights  [15] =  0.0554595293739872011294402;
      m_weights  [16] =  0.0241483028685479319601100;

      m_positions[ 0] = -0.9905754753144173356754340;
      m_positions[ 1] = -0.9506755217687677612227170;
      m_positions[ 2] = -0.8802391537269859021229557;
      m_positions[ 3] = -0.7815140038968014069252301;
      m_positions[ 4] = -0.6576711592166907658503022;
      m_positions[ 5] = -0.5126905370864769678862466;
      m_positions[ 6] = -0.3512317634538763152971855;
      m_positions[ 7] = -0.1784841814958478558506775;
      m_positions[ 8] =  0.;
      m_positions[ 9] =  0.1784841814958478558506775;
      m_positions[10] =  0.3512317634538763152971855;
      m_positions[11] =  0.5126905370864769678862466;
      m_positions[12] =  0.6576711592166907658503022;
      m_positions[13] =  0.7815140038968014069252301;
      m_positions[14] =  0.8802391537269859021229557;
      m_positions[15] =  0.9506755217687677612227170;
      m_positions[16] =  0.9905754753144173356754340;
    break;

    default:
      std::cerr << "In UniformLegendre1DQuadrature::constructor()"
                << ": m_order = " << m_order
                << std::endl;
      queso_error_msg("order not supported");
    break;
  }

  // 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);
    if (densityIsNormalized) {
      // Since \rho is "1/\Delta", we just multiply by ".5"
      m_weights[j] *= .5;
    }
    else {
      // Since \rho is "1", we multiply by ".5 * \Delta"
      m_weights[j] *= .5*(m_maxDomainValue - m_minDomainValue);
    }
  }
}

QUESO::UniformLegendre1DQuadrature::~UniformLegendre1DQuadrature

(

)

Destructor.

Definition at line 395 of file 1DQuadrature.C.

{
}

---

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

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