# File ms4body.rb, line 88
  def yo8(dt)
    d = [0.104242620869991e1, 0.182020630970714e1, 0.157739928123617e0, 
         0.244002732616735e1, -0.716989419708120e-2, -0.244699182370524e1, 
         -0.161582374150097e1, -0.17808286265894516e1]
    for i in 0..6 do leapfrog(dt*d[i]) end
    leapfrog(dt*d[7])
    for i in 0..6 do leapfrog(dt*d[6-i]) end
  end

  def ms2(dt)
    if @nsteps == 0
      @prev_acc = acc
      rk2(dt)
    else
      old_acc = acc
      jdt = old_acc - @prev_acc
      @pos += vel*dt + old_acc*0.5*dt*dt
      @vel += old_acc*dt + jdt*0.5*dt
      @prev_acc = old_acc
    end
  end

  def ms4(dt)
    if @nsteps == 0
      @ap3 = acc
      rk4(dt)
    elsif @nsteps == 1
      @ap2 = acc
      rk4(dt)
    elsif @nsteps == 2
      @ap1 = acc
      rk4(dt)
    else
      ap0 = acc
      jdt = ap0*(11.0/6.0) - @ap1*3 + @ap2*1.5 - @ap3/3.0
      sdt2 = ap0*2 - @ap1*5 + @ap2*4 - @ap3
      cdt3 = ap0 - @ap1*3 + @ap2*3 - @ap3
      @pos += vel*dt + (ap0/2.0 + jdt/6.0 + sdt2/24.0)*dt*dt
      @vel += ap0*dt + (jdt/2.0 + sdt2/6.0 + cdt3/24.0)*dt
      @ap3 = @ap2
      @ap2 = @ap1
      @ap1 = ap0
    end
  end

  def ms4pc(dt)
    if @nsteps == 0
      @ap3 = acc
      rk4(dt)
    elsif @nsteps == 1
      @ap2 = acc
      rk4(dt)
    elsif @nsteps == 2
      @ap1 = acc
      rk4(dt)
      @ap0 = acc
    else
      old_pos = pos
      old_vel = vel
      jdt = @ap0*(11.0/6.0) - @ap1*3 + @ap2*1.5 - @ap3/3.0
      sdt2 = @ap0*2 - @ap1*5 + @ap2*4 - @ap3
      cdt3 = @ap0 - @ap1*3 + @ap2*3 - @ap3
      @pos += vel*dt + (@ap0/2.0 + jdt/6.0 + sdt2/24.0)*dt*dt
      @ap3 = @ap2
      @ap2 = @ap1
      @ap1 = @ap0
      @ap0 = acc
      jdt = @ap0*(11.0/6.0) - @ap1*3 + @ap2*1.5 - @ap3/3.0
      sdt2 = @ap0*2 - @ap1*5 + @ap2*4 - @ap3
      cdt3 = @ap0 - @ap1*3 + @ap2*3 - @ap3
      @vel = old_vel + @ap0*dt + (-jdt/2.0 + sdt2/6.0 - cdt3/24.0)*dt
      @pos = old_pos + vel*dt + (-@ap0/2.0 + jdt/6.0 - sdt2/24.0)*dt*dt
    end
  end

  def ekin                        # kinetic energy
    0.5*(@vel*@vel)               # per unit of reduced mass
  end

  def epot                        # potential energy
    -@mass/sqrt(@pos*@pos)        # per unit of reduced mass
  end

  def e_init                      # initial total energy
    @e0 = ekin + epot             # per unit of reduced mass
  end

  def write_diagnostics(time)
    etot = ekin + epot
    STDERR.print "at time t = \#{sprintf(\"%g\", time)}, after \#{@nsteps} steps :\n  E_kin = \#{sprintf(\"%.3g\", ekin)} ,\\\n E_pot =  \#{sprintf(\"%.3g\", epot)} ,\\\n E_tot = \#{sprintf(\"%.3g\", etot)}\n             E_tot - E_init = \#{sprintf(\"%.3g\", etot-@e0)}\n  (E_tot - E_init) / E_init =\#{sprintf(\"%.3g\", (etot - @e0) / @e0 )}\nEND\n  end\n\n  def to_s\n    \"  mass = \" + @mass.to_s + \"\\n\" +\n    \"   pos = \" + @pos.join(\", \") + \"\\n\" +\n    \"   vel = \" + @vel.join(\", \") + \"\\n\"\n  end\n\n  def pp               \# pretty print\n    print to_s\n  end\n\n  def simple_print\n    printf(\"%24.16e\\n\", @mass)\n    @pos.each{|x| printf(\"%24.16e\", x)}; print \"\\n\"\n    @vel.each{|x| printf(\"%24.16e\", x)}; print \"\\n\"\n  end\n\n  def simple_read\n    @mass = gets.to_f\n    @pos = gets.split.map{|x| x.to_f}.to_v\n    @vel = gets.split.map{|x| x.to_f}.to_v\n  end\n\nend\n"