\name{romberg}
\alias{romberg}
\title{
  Romberg Integration
}
\description{
  Romberg Integration
}
\usage{
romberg(f, a, b, maxit = 50, tol = 1e-6, ...)
}
\arguments{
  \item{f}{function to be integrated.}
  \item{a, b}{end points of the interval.}
  \item{maxit}{maximum number of iterations.}
  \item{tol}{requested tolerance.}
  \item{...}{variables to be passed to the function.}
}
\details{
  Simple Romberg integration with an explicit Richardson method applied
  to a series of trapezoidal integrals. This scheme works best with very
  smmoth functions and needs the least number of function calls among all
  integration routines.

  The function does \emph{not} need to be vectorized.
}
\value{
  List of value, number or iterations, and relative error.
}
\references{
  Chapra, S. C., and R. P. Canale (2006). Numerical Methods for Engineers.
  Fifth Edition, McGraw-Hill, New York.
}
\note{
  Using a trapezoid formula Romberg integration will use
  \code{2*(2^iter-1)+iter} function calls. By remembering function values
  this could be reduced to \code{2^iter+1} calls.
}
\seealso{
  \code{\link{integrate}}, \code{\link{quadgr}}
}
\examples{
romberg(sin, 0, pi, tol = 1e-12)    #  2.000000000000001 , rel.error 0
romberg(exp, 0, 1,  tol = 1e-12)    #  1.718281828458432 , rel error 3.6e-13
                                    #  1.718281828459045 , i.e. exp(1) - 1

f <- function(x, p) sin(x) * cos(p*x)
romberg(f, 0, pi, tol = 1e-10, p = 2)   # 2/3
# value: -0.6666667, iter: 6, rel.error: 6.57967e-11
}
\keyword{ math }