subroutine compass_search ( function_handle, m, x0, delta_tol, delta_init, &
  k_max, x, fx, k )

!*****************************************************************************80
!
!! COMPASS_SEARCH carries out a direct search minimization algorithm.
!
!  Licensing:
!
!    This code is distributed under the GNU LGPL license.
!
!  Modified:
!
!    05 January 2012
!
!  Author:
!
!    John Burkardt
!
!  Reference:
!
!    Tamara Kolda, Robert Michael Lewis, Virginia Torczon,
!    Optimization by Direct Search: New Perspectives on Some Classical 
!    and Modern Methods,
!    SIAM Review,
!    Volume 45, Number 3, 2003, pages 385-482. 
!
!  Parameters:
!
!    Input, external real ( kind = rk ) FUNCTION_HANDLE, the name of
!    a FORTRAN90 function which evaluates the function to be minimized, of the
!    form FUNCTION FUNCTION_HANDLE ( M, X ).
!
!    Input, integer M, the number of variables.
!
!    Input, real ( kind = rk ) X0(M), a starting estimate for the minimizer.
!
!    Input, real ( kind = rk ) DELTA_TOL, the smallest step size that is allowed.
!
!    Input, real ( kind = rk ) DELTA_INIT, the starting stepsize.  
!
!    Input, integer K_MAX, the maximum number of steps allowed.
!
!    Output, real ( kind = rk ) X(M), the estimated minimizer.
!
!    Output, real ( kind = rk ) FX, the function value at X.
!
!    Output, integer K, the number of steps taken.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m

  logical decrease
  real ( kind = rk ) delta
  real ( kind = rk ) delta_init
  real ( kind = rk ) delta_tol
  real ( kind = rk ), external :: function_handle
  real ( kind = rk ) fx
  real ( kind = rk ) fxd
  integer i
  integer ii
  integer k
  integer k_max
  real ( kind = rk ) s
  real ( kind = rk ) x(m)
  real ( kind = rk ) x0(m)
  real ( kind = rk ) xd(m)

  k = 0
  x(1:m) = x0(1:m)
  fx = function_handle ( m, x )

  if ( delta_tol <= 0 ) then
    write ( *, '(a)' ) ' '
    write ( *, '(a)' ) 'COMPASS_SEARCH - Fatal error!'
    write ( *, '(a)' ) '  DELTA_TOL <= 0.0.'
    write ( *, '(a,g14.6)' ) '  DELTA_TOL = ', delta_tol
    stop
  end if

  if ( delta_init <= delta_tol ) then
    write ( *, '(a)' ) ' '
    write ( *, '(a)' ) 'COMPASS_SEARCH - Fatal error!'
    write ( *, '(a)' ) '  DELTA_INIT < DELTA_TOL.'
    write ( *, '(a,g14.6)' ) '  DELTA_INIT = ', delta_init
    write ( *, '(a,g14.6)' ) '  DELTA_TOL = ', delta_tol
    stop
  end if

  delta = delta_init

  do while ( k < k_max )

    k = k + 1
!
!  For each coordinate direction I, seek a lower function value
!  by increasing or decreasing X(I) by DELTA.
!
    decrease = .false.
    s = + 1.0D+00
    i = 1

    do ii = 1, 2 * m

      xd = x
      xd(i) = xd(i) + s * delta
      fxd = function_handle ( m, xd )
!
!  As soon as a decrease is noticed, accept the new point.
!
      if ( fxd < fx ) then
        x = xd
        fx = fxd
        decrease = .true.
        exit
      end if

      s = - s
      if ( s == + 1.0D+00 ) then
        i = i + 1
      end if

    end do
!
!  If no decrease occurred, reduce DELTA.
!
    if ( .not. decrease ) then
      delta = delta / 2.0D+00
      if ( delta < delta_tol ) then
        exit
      end if
    end if

  end do

  return
end
subroutine r8vec_print ( n, a, title )

!*****************************************************************************80
!
!! R8VEC_PRINT prints an R8VEC.
!
!  Discussion:
!
!    An R8VEC is a vector of R8's.
!
!  Licensing:
!
!    This code is distributed under the GNU LGPL license.
!
!  Modified:
!
!    22 August 2000
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the number of components of the vector.
!
!    Input, real ( kind = rk ) A(N), the vector to be printed.
!
!    Input, character ( len = * ) TITLE, a title.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n)
  integer i
  character ( len = * ) title

  write ( *, '(a)' ) ' '
  write ( *, '(a)' ) trim ( title )
  write ( *, '(a)' ) ' '

  do i = 1, n
    write ( *, '(2x,i8,a,1x,g16.8)' ) i, ':', a(i)
  end do

  return
end
subroutine timestamp ( )

!*****************************************************************************80
!
!! TIMESTAMP prints the current YMDHMS date as a time stamp.
!
!  Example:
!
!    31 May 2001   9:45:54.872 AM
!
!  Licensing:
!
!    This code is distributed under the GNU LGPL license.
!
!  Modified:
!
!    06 August 2005
!
!  Author:
!
!    John Burkardt
!
  implicit none

  character ( len = 8 ) ampm
  integer d
  integer h
  integer m
  integer mm
  character ( len = 9 ), parameter, dimension(12) :: month = (/ &
    'January  ', 'February ', 'March    ', 'April    ', &
    'May      ', 'June     ', 'July     ', 'August   ', &
    'September', 'October  ', 'November ', 'December ' /)
  integer n
  integer s
  integer values(8)
  integer y

  call date_and_time ( values = values )

  y = values(1)
  m = values(2)
  d = values(3)
  h = values(5)
  n = values(6)
  s = values(7)
  mm = values(8)

  if ( h < 12 ) then
    ampm = 'AM'
  else if ( h == 12 ) then
    if ( n == 0 .and. s == 0 ) then
      ampm = 'Noon'
    else
      ampm = 'PM'
    end if
  else
    h = h - 12
    if ( h < 12 ) then
      ampm = 'PM'
    else if ( h == 12 ) then
      if ( n == 0 .and. s == 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, trim ( month(m) ), y, h, ':', n, ':', s, '.', mm, trim ( ampm )

  return
end