Namespace for quadrature functions. More...

Functions
Real	Quadrature (RealFunction &f, const Real c, const Real x, int k)
	General purpose summation routine, returns sum[i=0..k-1]ci*f(xi)

void	Quadrature (VectorFunction &f, const Real c, const Real x, int k, Vector &res)

Real	trapezoidal (RealFunction &f, Real a, Real b)

Real	trapezoidal (Real f0, Real f1)

Real	simpsons (RealFunction &f, Real a, Real b)

Real	simpsons (Real f0, Real f1, Real f2)

Real	simpsons_3_8 (RealFunction &f, Real a, Real b)

Real	simpsons_3_8 (Real f0, Real f1, Real f2, Real f3)

Real	NC4 (RealFunction &f, Real a, Real b)

Real	NC4 (Real f0, Real f1, Real f2, Real f3, Real f4)

Real	Gaussian (RealFunction &f, int k)
	Gaussian quadrature rule with k points on [-1,1], [a,b] respectively.

Real	Gaussian (RealFunction &f, Real a, Real b, int k)

Real	compositeTrapezoidal (RealFunction &f, Real a, Real b, int n)

Real	compositeSimpsons (RealFunction &f, Real a, Real b, int n)

Real	composite (QuadratureFunction q, RealFunction &f, Real a, Real b, int n)
	composite integration for an arbitrary quadrature function

Real	trapezoidal2D (RealFunction2 &f, Real a, Real b, Real c, Real d)
	integral[a,b]x[c,d] f dA = 1/4(b-a)(c-d) (f(a,c) + f(b,c) + f(a,d) + f(b,d)) + O(h^2)

Real	simpsons2D (RealFunction2 &f, Real a, Real b, Real c, Real d)

Real	simpsons2D (RealFunction2 &f, Real a, Real b, RealFunction &c, RealFunction &d)

Variables
const Real	xGaussian2 [1] = {0.57735026918962576450914878050196}

const Real	cGaussian2 [1] = {1.0}

const Real	xGaussian3 [2] = {0,0.77459667}

const Real	cGaussian3 [2] = {0.88888889,0.55555555}

const Real	xGaussian4 [2] = {0.33998104,0.65214515}

const Real	cGaussian4 [2] = {0.65214515,0.34785485}

const Real	xGaussian5 [3] = {0.0,0.53846931,0.90617985}

const Real	cGaussian5 [3] = {0.56888889,0.47862867,0.23692689}

const Real	xGaussian6 [3] = {0.23861918,0.66120939,0.93246951}

const Real	cGaussian6 [3] = {0.46791393,0.36076157,0.17132449}

const Real	xGaussian7 [4] = {0.0,0.40584515,0.74153119,0.94910791}

const Real	cGaussian7 [4] = {0.41795918,0.38183005,0.27970539,0.12948497}

const Real	xGaussian8 [4] = {0.18343464,0.52553241,0.79666648,0.96028986}

const Real	cGaussian8 [4] = {0.36268378,0.31370665,0.22238103,0.10122854}

const Real	xGaussian10 [5] = {0.14887434,0.43339539,0.67940957,0.86506337,0.97390653}

const Real	cGaussian10 [5] = {0.29552422,0.26926672,0.21908636,0.14945135,0.06667134}

const Real	cGaussian2_x_unit [2] ={0.1819586182,0.3180413818}

const Real	xGaussian2_x_unit [2] ={0.3550510257,0.8449489743}

const Real	cGaussian3_x_unit [3] ={0.2009319140,0.06982697994,0.2292411061}

const Real	xGaussian3_x_unit [3] ={0.9114120405,0.2123405382,0.5905331354}

const Real	cGaussian2_x2_unit [2] ={0.1007858821,0.2325474512}
	f*x^2 on [0,1]

const Real	xGaussian2_x2_unit [2] ={0.4558481560,0.8774851770}

const Real	cGaussian3_x2_unit [3] ={0.1571363612,0.02995070308,0.1462462695}

const Real	xGaussian3_x2_unit [3] ={0.9270059759,0.2949977900,0.6529962340}

const Real	cGaussian2_x1x_unit [2] ={0.08333333333,0.08333333333}
	f*x(x-1) on [0,1]

const Real	xGaussian2_x1x_unit [2] ={0.2763932023,0.7236067977}

const Real	cGaussian3_x1x_unit [3] ={0.08888888899,0.03888888889,0.03888888889}

const Real	xGaussian3_x1x_unit [3] ={0.5,0.8273268354,0.1726731646}

const int	maxGaussianDegree = 10

const Real *	xGaussian [11]

const Real *	cGaussian [11]

Detailed Description

Namespace for quadrature functions.

Function Documentation

Real Math::Quadrature::compositeSimpsons	(	RealFunction &	f,
		Real	a,
		Real	b,
		int	n
	)

composite simpsons rule on n segments (n assumed to be even) integral[a,b] f dx ~= 1/3h(f(a) + 4*sum[i=1:n-1, odd]f(a+ih) + 2*sum[i=1:n-1, even]f(a+ih) + f(b)) + (b-a)O(h^4)

Real Math::Quadrature::compositeTrapezoidal	(	RealFunction &	f,
		Real	a,
		Real	b,
		int	n
	)

composite trapezoidal rule on n segments integral[a,b] f dx ~= 1/2h(f(a) + 2*sum[i=1:n-1]f(a+ih) + f(b)) + (b-a)O(h^2)

Real Math::Quadrature::NC4	(	RealFunction &	f,
		Real	a,
		Real	b
	)

Newton-Cotes order 4 integral[a,b] f dx ~= 2/45h(7f(a) + 32f(a+h) + 12f(a+2h) + 32f(a+3h) + 7f(b)) + O(h^7)

Real Math::Quadrature::simpsons	(	RealFunction &	f,
		Real	a,
		Real	b
	)

simpsons rule integral[a,b] f dx ~= 1/3h(f(a) + 4f(a+h) + f(b)) + O(h^5)

Real Math::Quadrature::simpsons2D	(	RealFunction2 &	f,
		Real	a,
		Real	b,
		Real	c,
		Real	d
	)

integral[a,b]x[c,d] f dA = 1/9(b-a)(c-d) (f(a,c) + f(a,d) + f(b,c) + f(b,d)

4[f(a+h,c)+f(a+h,d)+f(a,c+h)+f(b,c+h)] + 16*f(a+h,c+h) ) + O(h^4)

Real Math::Quadrature::simpsons2D	(	RealFunction2 &	f,
		Real	a,
		Real	b,
		RealFunction &	c,
		RealFunction &	d
	)

integral[a,b]x[c(x),d(x)] f dA = 1/9 (f(a,c(a)) + f(a,d(a)) + f(b,c(b)) + f(b,d(b))

4[f(a+h,c(a+h))+f(a+h,d(a+h))+f(a,c(a)+h(a))+f(b,c(b)+h(b))] + 16*f(a+h,c(a+h)+h(a+h)) ) + O(h^4)

Real Math::Quadrature::simpsons_3_8	(	RealFunction &	f,
		Real	a,
		Real	b
	)

simpsons 3/8 rule integral[a,b] f dx ~= 3/8h(f(a) + 3f(a+h) + 3f(a+2h) + f(b)) + O(h^5)

Real Math::Quadrature::trapezoidal	(	RealFunction &	f,
		Real	a,
		Real	b
	)

trapezoidal rule integral[a,b] f dx ~= 1/2h(f(a) + f(b)) + O(h^3)

Variable Documentation

const Real* Math::Quadrature::cGaussian[11]

Initial value:

= {NULL,NULL,cGaussian2,cGaussian3,cGaussian4,

cGaussian5,cGaussian6,cGaussian7,cGaussian8,NULL,cGaussian10 }

Math::Quadrature::cGaussian2

const Real cGaussian2[1]

Definition: quadrature.cpp:77

const Real Math::Quadrature::cGaussian2 = {1.0}

Gaussian quadrature: The coefficient versions Gaussiank give coefficients for the k-point gaussian quadrature rule in the arrays c and x The resulting rule is int[-1,1]f(x) = sum ci*f(xi) note: only the positive half is provided

const Real Math::Quadrature::cGaussian2_x_unit ={0.1819586182,0.3180413818}

Warped gaussian quadrature: Same as above, but integral of a warped function f*x on [0,1]

const Real* Math::Quadrature::xGaussian[11]

Initial value:

= {NULL,NULL,xGaussian2,xGaussian3,xGaussian4,

xGaussian5,xGaussian6,xGaussian7,xGaussian8,NULL,xGaussian10 }

Functions

Variables

Detailed Description

Function Documentation

Variable Documentation