proportional to . This intuitive reasoning can be made precise. If the solution has a bounded second derivative and is Lipschitz continuous in its second argument, then the global truncation error (denoted as ) is bounded by where is an upper bound on the second derivative of on the given interval and is the Lipschitz constant of . Or more simply, when , the value (such that is treated as a constant). In contrast, where function is the exact solution which only contains the variable. The precise form of this bound is of little practical importance, as in most cases the bound vastly overestimates the actual error committed by the Euler method. What is important is that it shows that the global truncation error is (approximately) proportional to . For this reason, the Euler method is said to be first order. Numerical stability The Euler method can also be numerically unstable, especially for stiff equations, meaning that the numerical solution grows very large for equations where the exact solution does not. This can be illustrated using the linear equation The exact solution is , which decays to zero as . However, if the Euler method is applied to this equation with step size , then the numerical solution is qualitatively wrong: It oscillates and grows (see the figure). This is what it means to be unstable. If a smaller step size is used, for instance , then the numerical solution does decay to zero. If the Euler method is applied to the linear equation , then the numerical solution is unstable if the product is outside the region illustrated on the right. This region is called the (linear) stability region. In the example, , so if then which is outside the stability region, and thus the numerical solution is unstable. This limitation — along with its slow convergence of error with — means that the Euler method is not often used, except as a simple example of numerical integration. Rounding errors In step of the Euler method, the rounding error is roughly of the magnitude where is the machine epsilon. Assuming that the rounding errors are independent random variables, the expected total rounding error is proportional to . Thus, for extremely small values of the step size the truncation error will be small but the effect of rounding error may be big. Most of the effect of rounding error can