euler <- function ( f, t0, y0, tstop, n )

#*****************************************************************************80
#
## euler() uses the Euler method to solve an ordinary differential equation (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:
#
#    23 March 2020
#
#  Author:
#
#    Original R code by James Howard;
#    This version by John Burkardt.
#
#  Reference:
#
#    James Howard,
#    Computational Methods for Numerical Analysis with R,
#    CRC Press, 2017
#    ISBN13: 978-1-4987-2363-3.
#
#  Input:
#
#    function f ( t, y ): evaluates the right hand side of the ODE at (x,y).
#
#    real T0: the initial time.
#
#    real Y0: the initial solution value.
#
#    real TSTOP: the final time.
#
#    integer N: the number of steps to take.
#
#  Output:
#
#    list ( T(1:n+1), Y(1:n+1) ): the solution estimates.
#
{
  t <- t0
  y <- y0

  dt <- ( tstop - t0 ) / ( n - 1 )

  for ( i in 2 : n )
  {
    y0 <- y0 + dt * f ( t0, y0 )
    t0 <- t0 + dt
    t <- c ( t, t0 )
    y <- c ( y, y0 )
  }

  return ( list ( t = t, y = y ) )
}