midpoint_explicit <- function ( f, x0, y0, h, n )

#*****************************************************************************80
#
## midpoint_explicit(): explicit midpoint method for to solve an ODE.
#
#  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:
#
#    26 April 2021
#
#  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.
#
{
  x <- x0
  y <- y0

  for ( i in 1 : n )
  {
    s1 <- h * f ( x0, y0 )
    s2 <- h * f ( x0+h/2, y0+s1/2 )
    y0 <- y0 + s2
    x0 <- x0 + h
    x <- c ( x, x0 )
    y <- c ( y, y0 )
  }

  return ( data.frame ( x = x, y = y ) )
}