program main

c*********************************************************************72
c
cc rkf45_test() tests rkf45().
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    17 June 2006
c
c  Author:
c
c    John Burkardt
c
      implicit none

      call timestamp ( )
      write ( *, '(a)' ) ' '
      write ( *, '(a)' ) 'rkf45_test():'
      write ( *, '(a)' ) '  Fortran77 version'
      write ( *, '(a)' ) '  Test rkf45().'

      call test04 ( )
      call test05 ( )
      call test06 ( )
c
c  Terminate.
c
      write ( *, '(a)' ) ' '
      write ( *, '(a)' ) 'rkf45_test():'
      write ( *, '(a)' ) '  Normal end of execution.'
      write ( *, '(a)' ) ' '
      call timestamp ( )

      stop
      end
      subroutine test04 ( )

c*********************************************************************72
c
cc TEST04 solves a scalar ODE.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    17 June 2006
c
c  Author:
c
c    John Burkardt
c
      implicit none

      integer neqn
      parameter ( neqn = 1 )

      double precision abserr
      integer flag
      integer i_step
      integer iwork(5)
      integer n_step
      double precision r8_epsilon
      external r8_f1
      double precision r8_y1x
      double precision relerr
      double precision t
      double precision t_out
      double precision t_start
      double precision t_stop
      double precision work(3+6*neqn)
      double precision y(neqn)

      write ( *, '(a)') ' '
      write ( *, '(a)' ) 'TEST04'
      write ( *, '(a)' ) '  Solve a scalar equation using RKF_D:'
      write ( *, '(a)') ' '
      write ( *, '(a)' ) '  Y'' = 0.25 * Y * ( 1 - Y / 20 )'
      write ( *, '(a)') ' '

      abserr = sqrt ( r8_epsilon ( ) )
      relerr = sqrt ( r8_epsilon ( ) )

      flag = 1

      t_start = 0.0D+00
      t_stop = 20.0D+00

      n_step = 5

      t_out = 0.0D+00
      t = t_out
      y(1) = 1.0D+00

      write ( *, '(a)' ) ' '
      write ( *, '(a,a)' ) '  FLAG     T             Y            ',
     &  'Y_Exact         Error'
      write ( *, '(a)' ) ' '
      write ( *, '(i4,2x,5g14.6)' ) flag, t, y(1), r8_y1x ( t ), 
     &  y(1) - r8_y1x ( t )

      do i_step = 1, n_step

        t = ( dble ( n_step - i_step + 1 ) * t_start 
     &      + dble (          i_step - 1 ) * t_stop ) 
     &      / dble ( n_step              )

        t_out = ( real ( n_step - i_step ) * t_start 
     &          + real (          i_step ) * t_stop ) 
     &          / real ( n_step          )

        call r8_rkf45 ( r8_f1, neqn, y, t, t_out, relerr, abserr, 
     &    flag, work, iwork )

        write ( *, '(i4,2x,5g14.6)' ) flag, t, y(1),  
     &    r8_y1x ( t ), y(1) - r8_y1x ( t )

      end do

      return
      end
      subroutine test05 ( )

c*********************************************************************72
c
cc TEST05 solves a vector ODE.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    17 June 2006
c
c  Author:
c
c    John Burkardt
c
      implicit none

      integer neqn
      parameter ( neqn = 2 )

      double precision abserr
      integer flag
      integer i_step
      integer iwork(5)
      integer n_step
      double precision r8_epsilon
      external r8_f2
      double precision relerr
      double precision t
      double precision t_out
      double precision t_start
      double precision t_stop
      double precision work(3+6*neqn)
      double precision y(neqn)

      write ( *, '(a)') ' '
      write ( *, '(a)' ) 'TEST05'
      write ( *, '(a)' ) '  Solve a vector equation using RKF_D:'
      write ( *, '(a)') ' '
      write ( *, '(a)' ) '  Y''(1) =  Y(2)'
      write ( *, '(a)' ) '  Y''(2) = -Y(1)'
      write ( *, '(a)') ' '

      abserr = sqrt ( r8_epsilon ( ) )
      relerr = sqrt ( r8_epsilon ( ) )

      flag = 1

      t_start = 0.0D+00
      t_stop = 2.0D+00 * 3.14159265D+00

      n_step = 12

      t = 0.0D+00
      t_out = 0.0D+00

      y(1) = 1.0D+00
      y(2) = 0.0D+00

      write ( *, '(a)' ) ' '
      write ( *, '(a)' ) '  FLAG       T          Y(1)          Y(2)'
      write ( *, '(a)' ) ' '
      write ( *, '(i4,2x,4g14.6)' ) flag, t, y(1), y(2)

      do i_step = 1, n_step

        t = ( dble ( n_step - i_step + 1 ) * t_start 
     &      + dble (          i_step - 1 ) * t_stop ) 
     &      / dble ( n_step              )

        t_out = ( real ( n_step - i_step ) * t_start 
     &          + real (          i_step ) * t_stop ) 
     &          / real ( n_step          )

        call r8_rkf45 ( r8_f2, neqn, y, t, t_out, relerr, abserr, 
     &    flag, work, iwork )

        write ( *, '(i4,2x,4g14.6)' ) flag, t, y(1), y(2)

      end do

      return
      end
      subroutine test06 ( )

c*********************************************************************72
c
cc TEST06 solves a scalar ODE and uses one-step integration.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    17 June 2006
c
c  Author:
c
c    John Burkardt
c
      implicit none

      integer neqn
      parameter ( neqn = 1 )

      double precision abserr
      integer flag
      integer i_step
      integer iwork(5)
      integer n_step
      double precision r8_epsilon
      external r8_f1
      double precision r8_y1x
      double precision relerr
      double precision t
      double precision t_out
      double precision t_start
      double precision t_stop
      double precision work(3+6*neqn)
      double precision y(neqn)

      write ( *, '(a)') ' '
      write ( *, '(a)' ) 'TEST06'
      write ( *, '(a)' ) '  Solve a scalar equation using RKF_D:'
      write ( *, '(a)') ' '
      write ( *, '(a)' ) '  Y'' = 0.25 * Y * ( 1 - Y / 20 )'
      write ( *, '(a)') ' '
      write ( *, '(a)' ) '  Use the special SINGLE_STEP mode'
      write ( *, '(a)' ) '  which returns after every step.'

      abserr = sqrt ( r8_epsilon ( ) )
      relerr = sqrt ( r8_epsilon ( ) )

      flag = -1

      t_start = 0.0D+00
      t_stop = 20.0D+00

      n_step = 5

      t = 0.0D+00
      t_out = 0.0D+00
      y(1) = 1.0D+00

      write ( *, '(a)' ) ' '
      write ( *, '(a,a)' ) '  FLAG     T             Y           ',
     &  'Y_Exact        Error'
      write ( *, '(a)' ) ' '
      write ( *, '(i4,2x,5g14.6)' ) flag, t, y(1), r8_y1x ( t ), 
     &  y(1) - r8_y1x ( t )

      do i_step = 1, n_step

        t = ( dble ( n_step - i_step + 1 ) * t_start 
     &      + dble (          i_step - 1 ) * t_stop ) 
     &      / dble ( n_step              )

        t_out = ( real ( n_step - i_step ) * t_start 
     &          + real (          i_step ) * t_stop ) 
     &          / real ( n_step          )
c
c  As long as FLAG is negative, we are heading towards T_OUT, but
c  have not reached itc
c
10      continue

        if ( flag < 0 ) then

          call r8_rkf45 ( r8_f1, neqn, y, t, t_out, relerr, abserr, 
     &    flag, work, iwork )

          write ( *, '(i4,2x,4g14.6)' ) flag, t, y(1), 
     &      r8_y1x ( t ), y(1) - r8_y1x ( t )

          go to 10

        end if
c
c  FLAG is returned as +2 when we reach T_OUT.  Reset it to -2
c  to continue to the next T_OUT in one step mode.
c
        flag = -2

      end do

      return
      end
      subroutine r8_f1 ( t, y, yp )

c*********************************************************************72
c
cc R8_F1 evaluates the derivative for the ODE.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    26 March 2004
c
c  Author:
c
c    John Burkardt
c
c  Parameters:
c
c    Input, double precision T, the value of the independent variable.
c
c    Input, double precision Y(NEQN), the value of the dependent variable.
c
c    Output, double precision YP(NEQN), the value of the derivative
c    dY(1:NEQN)/dT.
c
      implicit none

      double precision t
      double precision y(1)
      double precision yp(1)

      yp(1) = 0.25D+00 * y(1) * ( 1.0D+00 - y(1) / 20.0D+00 )

      return
      end
      function r8_y1x ( t )

c*********************************************************************72
c
cc R8_Y1X evaluates the exact solution of the ODE.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    26 March 2004
c
c  Author:
c
c    John Burkardt
c
c  Parameters:
c
c    Input, double precision T, the value of the independent variable.
c
c    Output, double precision Y1X_D, the exact solution.
c
      implicit none

      double precision t
      double precision r8_y1x

      r8_y1x = 20.0D+00 / ( 1.0D+00 + 19.0D+00 
     &  * exp ( - 0.25D+00 * t ) )

      return
      end
      subroutine r8_f2 ( t, y, yp )

c*********************************************************************72
c
cc R8_F2 evaluates the derivative for the ODE.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    26 March 2004
c
c  Author:
c
c    John Burkardt
c
c  Parameters:
c
c    Input, double precision T, the value of the independent variable.
c
c    Input, double precision Y(NEQN), the value of the dependent variable.
c
c    Output, double precision YP(NEQN), the value of the derivative
c    dY(1:NEQN)/dT.
c
      implicit none

      double precision t
      double precision y(2)
      double precision yp(2)

      yp(1) =  y(2)
      yp(2) = -y(1)

      return
      end
      subroutine timestamp ( )

c*********************************************************************72
c
cc timestamp() prints out the current YMDHMS date as a timestamp.
c
c  Licensing:
c
c    This code is distributed under the MIT license.
c
c  Modified:
c
c    12 January 2007
c
c  Author:
c
c    John Burkardt
c
c  Parameters:
c
c    None
c
      implicit none

      character * ( 8 ) ampm
      integer d
      character * ( 8 ) date
      integer h
      integer m
      integer mm
      character * ( 9 ) month(12)
      integer n
      integer s
      character * ( 10 ) time
      integer y

      save month

      data month /
     &  'January  ', 'February ', 'March    ', 'April    ', 
     &  'May      ', 'June     ', 'July     ', 'August   ', 
     &  'September', 'October  ', 'November ', 'December ' /

      call date_and_time ( date, time )

      read ( date, '(i4,i2,i2)' ) y, m, d
      read ( time, '(i2,i2,i2,1x,i3)' ) h, n, s, mm

      if ( h .lt. 12 ) then
        ampm = 'AM'
      else if ( h .eq. 12 ) then
        if ( n .eq. 0 .and. s .eq. 0 ) then
          ampm = 'Noon'
        else
          ampm = 'PM'
        end if
      else
        h = h - 12
        if ( h .lt. 12 ) then
          ampm = 'PM'
        else if ( h .eq. 12 ) then
          if ( n .eq. 0 .and. s .eq. 0 ) then
            ampm = 'Midnight'
          else
            ampm = 'AM'
          end if
        end if
      end if

      write ( *, 
     &  '(i2,1x,a,1x,i4,2x,i2,a1,i2.2,a1,i2.2,a1,i3.3,1x,a)' ) 
     &  d, month(m), y, h, ':', n, ':', s, '.', mm, ampm

      return
      end