Numerical Integration Type  Integration Formulas  Methods Available  Description 

Trapezoidal Rule 
*
*  2point NewtonCotes trapezoidal rule:
* @f[
* \int_{a}^b f(x)dx \approx \frac{h}{2}[f(a) + f(b)]
* @f]
* where @e h = @a b  @a a.
*
* 
*
* 
* The 2point closed rule, known as the trapezoidal
* rule, uses straight line segments between known data
* points to estimate the area under the curve (integral).
* This NewtonCotes formula uses a uniform stepsize of
* @e h = @a b  @a a and is derived by integrating the
* Lagrange polynomials over the closed interval [@a a,
* @a a+h] for @a x_{i} =
* @a x_{0} + @e ih.
* 
Simpson's Rule, or Simpson's 3Point Rule 
*
*  3point NewtonCotes, Simpson's Rule
* @f[
* \int_{a}^b f(x)dx \approx \frac{h}{3}[f(a) +
* 4f(a+h) + f(a+2h)]
* @f]
* where @e h = (@a b  @a a)/2.
*
* 
*
* 
* The 3point closed rule, known as Simpson's Rule or
* Simpson's 3Point Rule, uses parabolic arcs between
* known data points to estimate the area under the curve
* (integral). This NewtonCotes formula uses a uniform
* stepsize of @e h = (@a b  @a a)/2 and is derived by
* integrating the Lagrange polynomials over the closed
* interval [@a a, @a a+2h] for @a x_{i} =
* @a x_{0} + @e ih.
* 
Simpson's 3/8 Rule, or Simpson's 4Point Rule 
*
*  4point NewtonCotes, Simpson's 3/8 Rule
* @f[
* \int_{a}^b f(x)dx \approx \frac{3h}{8}[f(a) +
* 3f(a+h) + 3f(a+2h) + f(a+3h)]
* @f]
* where @e h = (@a b  @a a)/3.
*
* 
*
* 
* The 4point closed rule, known as Simpson's 3/8 Rule or
* Simpson's 4Point Rule, uses cubic curves between
* known data points to estimate the area under the curve
* (integral). This NewtonCotes formula uses a uniform
* stepsize of @e h = (@a b  @a a)/3 and is derived by
* integrating the Lagrange polynomials over the closed
* interval [@a a, @a a+3h] for @a x_{i} =
* @a x_{0} + @e ih.
* 
Boole's Rule 
*
*  5point NewtonCotes, Boole's Rule
* @f[
* \int_{a}^b f(x)dx \approx \frac{2h}{45}[7f(a) +
* 32f(a+h) + 12f(a+2h) + 32f(a+3h) + 7f(a+4h)]
* @f]
* where @e h = (@a b  @a a)/4.
*
* 
*
* 
* The 5point closed rule, known as Boole's Rule, uses
* quartic curves between known data points to estimate
* the area under the curve (integral). This NewtonCotes
* formula uses a uniform stepsize of @e h = (@a b 
* @a a)/4 and is derived by integrating the Lagrange
* polynomials over the closed interval [@a a, @a a+4h]
* for @a x_{i} = @a x_{0} +
* @a ih.
* 
Refinements of Extended Trapezoidal Rule 
*
*  Extended closed trapezoidal rule
* @f[
* \int_{a}^b f(x)dx \approx h[\frac12 f(x_0) +
* f(x_1) + ... + f(x_{n2} + \frac12
* f(x_{n1})]
* @f]
* where @e h = (@a b  @a a)/4, @a x_{0} =
* @a a, and @a x_{n1} = @a b.
*
* 
*
* 
* The extended (or composite) trapezoidal rule can be
* used to with a series of refinements to approximate the
* integral. The first stage of refinement returns the
* ordinary trapezoidal estimate of the integral.
* Subsequent stages will improve the accuracy, where, for
* the @a n^{th} stage 2^{@a n2 } interior
* points are added.
* 
Romberg's Method 
*
*  Romberg's Method for Integration

*
* 
* Romberg's Method is a potent numerical integration
* tool. It uses the extended trapezoidal rule and
* Neville's algorithm for polynomial interpolation.
* 
GSL Integration  Unknown 
*
* 
* No documentation was found about the algorithm used for
* this methods. They may only be called when
* interpolation type is one of the following: @a Linear,
* @a Polynomial, @a CubicNatural, @a CubicNatPeriodic, @a Akima, or
* @a AkimaPeriodic.
* 