romberg <- function ( f, a, b, m, tab = FALSE )

#*****************************************************************************80
#
## romberg computes the intercept and slope of a line.
#
#  Licensing:
#
#    Copyright 2016 James P. Howard, II
#
#    The computer code and data files on this web page are distributed under
#    https://opensource.org/licenses/BSD-2-Clause, the BSD-2-Clause license.
#
#  Modified:
#
#    08 March 2020
#
#  Author:
#
#    Original R code by James Howard;
#    Modifications by John Burkardt.
#
#  Reference:
#
#    James Howard,
#    Computational Methods for Numerical Analysis with R,
#    CRC Press, 2017,
#    ISBN13: 978-1-4987-2363-3.
#
{
  source ( "/home/burkardt/public_html/r_src/trap/trap.R" )

  R <- matrix ( NA, nrow = m, ncol = m )

  R[1,1] <- trap ( f, a, b, m = 1 )

  for ( j in 2 : m )
  {
    R[j,1] <- trap ( f, a, b, m = 2^(j-1) )
    for ( k in 2 : j )
    {
      k4 = 4 ^ ( k - 1 )
      R[j,k] <- k4 * R[j,k-1] - R[j-1,k-1]
      R[j,k] <- R[j,k] / ( k4 - 1 )
    }
  }

  if ( tab == TRUE )
  {
    return ( R )
  }
  else
  {
    return ( R[m,m] )
  }
}