romberg Namespace Reference

Univariate and multivariate numerical integration. More...

Functions
template<class Type , class F >
Type	integrate (F f, Type a, Type b, int n=7, int p=2)
	1D numerical integration using Romberg's method. More...
template<class Type , class F >
Type	integrate (F f, vector< Type > a, vector< Type > b, int n=7, int p=2)
	Multi-dimensional numerical integration using Romberg's method. Dimension need not be known at compile time. More...

Detailed Description

Univariate and multivariate numerical integration.

• Based on CppAD's Romberg integration routines. Adapted so easy to use with TMB, e.g. allow multivariate dimension to be unknown at compile time.

Function Documentation

integrate() [1/2]

template<class Type , class F >

Type romberg::integrate	(	F	f,
		Type	a,
		Type	b,
		int	n = 7,
		int	p = 2
	)

1D numerical integration using Romberg's method.

Parameters

f	Univariate functor
a	Lower scalar integration limit
b	Upper scalar integration limit
n	Subdivisions ( \(2^{n-1}+1\) function evaluations).

Note

The method is not adaptive and the user may have to specify a larger n.

Example:

#include <TMB.hpp>

template<class Type>
struct univariate {
  Type theta;               // Parameter in integrand
  univariate(Type theta_)   // Constructor of integrand
  : theta (theta_) {}       // Initializer list
  Type operator()(Type x){  // Evaluate integrand
    return exp( -theta * (x * x) );
  }
};

template<class Type>
Type objective_function<Type>::operator() () {
  DATA_SCALAR(a);
  DATA_SCALAR(b);
  PARAMETER(theta);
  univariate<Type> f(theta);
  Type res = romberg::integrate(f, a, b);
  return res;
}

Examples:

adaptive_integration.cpp, register_atomic.cpp, and register_atomic_parallel.cpp.

Definition at line 52 of file romberg.hpp.

integrate() [2/2]

template<class Type , class F >

Type romberg::integrate	(	F	f,
		vector< Type >	a,
		vector< Type >	b,
		int	n = 7,
		int	p = 2
	)

Multi-dimensional numerical integration using Romberg's method. Dimension need not be known at compile time.

Parameters

f	Multivariate functor
a	Lower vector integration limit
b	Upper vector integration limit
n	Subdivisions per dimension ( \((2^{n-1}+1)^d\) function evaluations).

Note

The method is not adaptive and the user may have to specify a larger n.

Example:

#include <TMB.hpp>

template<class Type>
struct multivariate {
  Type theta;               // Parameter in integrand
  multivariate(Type theta_) // Constructor of integrand
  : theta (theta_) {}       // Initializer list
  Type operator()           // Evaluate integrand
  (vector<Type> x){
    return exp( -theta * (x * x).sum() );
  }
};

template<class Type>
Type objective_function<Type>::operator() () {
  DATA_VECTOR(a);
  DATA_VECTOR(b);
  PARAMETER(theta);
  multivariate<Type> f(theta);
  Type res = romberg::integrate(f, a, b);
  return res;
}

Definition at line 154 of file romberg.hpp.