gradient_descent1 ( f(), fp(), x, r, dxtol, fptol, itmax  )

# gradient descent for a function of 1 parameter

  it = 0

  Loop1:

    if |fp(x)| < fptol ) return (success)

    xold = x
    beta = 1.0
    Loop2:
      it = it + 1
      if ( itmax < it ) return (failure)
      dx = - beta * r * fp(xold)
      if ( |dx| < dxtol ) return (success)
      x = xold + dx
      if f(x) < f(xold) break Loop2
      beta = beta / 2
      if ( beta < 1 / 1024 ) return (failure)
    Loop2 end

  Loop1 end

gradient_descent1 end