RombergMethod Class 
Namespace: Accord.Math.Integration
The RombergMethod type exposes the following members.
Name  Description  

RombergMethod 
Constructs a new Romberg's integration method.
 
RombergMethod(FuncDouble, Double) 
Constructs a new Romberg's integration method.
 
RombergMethod(Int32) 
Constructs a new Romberg's integration method.
 
RombergMethod(Int32, FuncDouble, Double) 
Constructs a new Romberg's integration method.
 
RombergMethod(FuncDouble, Double, Double, Double) 
Constructs a new Romberg's integration method.
 
RombergMethod(Int32, FuncDouble, Double, Double, Double) 
Constructs a new Romberg's integration method.

Name  Description  

Area 
Gets the numerically computed result of the
definite integral for the specified function.
 
Function 
Gets or sets the unidimensional function
whose integral should be computed.
 
Range 
Gets or sets the input range under
which the integral must be computed.
 
Steps 
Gets or sets the number of steps used
by Romberg's method. Default is 6.

Name  Description  

Clone 
Creates a new object that is a copy of the current instance.
 
Compute  
Equals  Determines whether the specified object is equal to the current object. (Inherited from Object.)  
Finalize  Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)  
GetHashCode  Serves as the default hash function. (Inherited from Object.)  
GetType  Gets the Type of the current instance. (Inherited from Object.)  
Integrate(FuncDouble, Double, Double, Double) 
Computes the area under the integral for the given function,
in the given integration interval, using Romberg's method.
 
Integrate(FuncDouble, Double, Double, Double, Int32) 
Computes the area under the integral for the given function,
in the given integration interval, using Romberg's method.
 
MemberwiseClone  Creates a shallow copy of the current Object. (Inherited from Object.)  
ToString  Returns a string that represents the current object. (Inherited from Object.) 
Name  Description  

HasMethod 
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.)  
IsEqual 
Compares two objects for equality, performing an elementwise
comparison if the elements are vectors or matrices.
(Defined by Matrix.)  
To(Type)  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)  
ToT  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.) 
In numerical analysis, Romberg's method (Romberg 1955) is used to estimate the definite integral ∫_a^b(x) dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate.
References:
Let's say we would like to compute the definite integral of the function f(x) = cos(x) in the interval 1 to +1 using a variety of integration methods, including the TrapezoidalRule, RombergMethod and NonAdaptiveGaussKronrod. Those methods can compute definite integrals where the integration interval is finite:
// Declare the function we want to integrate Func<double, double> f = (x) => Math.Cos(x); // We would like to know its integral from 1 to +1 double a = 1, b = +1; // Integrate! double trapez = TrapezoidalRule.Integrate(f, a, b, steps: 1000); // 1.6829414 double romberg = RombergMethod.Integrate(f, a, b); // 1.6829419 double nagk = NonAdaptiveGaussKronrod.Integrate(f, a, b); // 1.6829419
Moreover, it is also possible to calculate the value of improper integrals (it is, integrals with infinite bounds) using InfiniteAdaptiveGaussKronrod, as shown below. Let's say we would like to compute the area under the Gaussian curve from infinite to +infinite. While this function has infinite bounds, this function is known to integrate to 1.
// Declare the Normal distribution's density function (which is the Gaussian's bell curve) Func<double, double> g = (x) => (1 / Math.Sqrt(2 * Math.PI)) * Math.Exp((x * x) / 2); // Integrate! double iagk = InfiniteAdaptiveGaussKronrod.Integrate(g, Double.NegativeInfinity, Double.PositiveInfinity); // Result should be 0.99999...