subroutine bisection_rc ( a, b, x, fx, job )

!*****************************************************************************80
!
!! bisection_rc() seeks a zero of f(x) in a change of sign interval.
!
!  Discussion:
!
!    The bisection method is used.
!
!    This routine uses reverse communication, so that the function is always
!    evaluated in the calling program.
!
!    On the first call, the user sets JOB = 0, and the values of A and B.
!    Thereafter, the user checks the returned value of JOB and follows 
!    directions.
!
!  Licensing:
!
!    This code is distributed under the MIT license.
!
!  Modified:
!
!    03 September 2021
!
!  Author:
!
!    John Burkardt
!
!  Input:
!
!    real ( kind = rk ) A, B: the change of sign interval.
!
!    real ( kind = rk ) FX:
!    on first call, with JOB = 0, a value of FX is not needed.
!    Thereafter, FX should be set to f(x), where x is output of previous call.
!
!    integer JOB: set to 0 on first call, to request initialization.
!
!  Output:
!
!    real ( kind = rk ) A, B: the updated change of sign interval.
!
!    real ( kind = rk ) X: the next point at which a function value is needed.
!
!    integer JOB: reset to 1, indicating that initialization was completed.
!  
  implicit none

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

  real ( kind = rk ) a
  real ( kind = rk ) b
  real ( kind = rk ), save :: fa
  real ( kind = rk ), save :: fb
  real ( kind = rk ) fx
  integer job
  integer, save :: state
  real ( kind = rk ) x
!
!  Initialize.  
!    Accept value of a.
!    Request value of f(a).
!
  if ( job == 0 ) then

    fa = 0.0D+00
    fb = 0.0D+00
    state = 1
    x = a
    job = 1
!
!  Accept value of f(a).
!  Request value of f(b).
!
  else if ( state == 1 ) then

    if ( fx == 0.0 ) then
      b = a
      state = 3
      return
    end if

    fa = fx
    x = b
    state = 2
!
!  Accept value of f(b).
!  Request value of f at midpoint.
!
  else if ( state == 2 ) then

    if ( fx == 0.0 ) then
      a = b
      state = 3
      return
    end if

    fb = fx

    if ( 0.0 < fa * fb ) then
      write ( *, '(a)' ) ' '
      write ( *, '(a)' ) 'bisection_rc(): Fatal error!'
      write ( *, '(a)' ) '  f(a) and f(b) have the same sign.'
      stop ( 1 )
    end if

    x = ( a + b ) / 2.0D+00
    state = 3
!
!  Use sign of function value to bisect current interval.
!  Determine new midpoint and request function value.
!
  else

    if ( fx == 0.0 ) then
      a = x
      b = x
      return
    end if

    if ( 0.0 < fx * fa ) then
      a = x
      fa = fx
    else
      b = x
      fb = fx
    end if
    x = ( a + b ) / 2.0D+00
    state = 3

  end if

  return
end