### Forward Euler Method

The finite-difference approximation (Eq.(7.2)) with the derivative evaluated at time yields the*forward Euler method*of numerical integration:

where denotes the

*approximation*to computed by the forward Euler method. Note that the ``driving function'' is evaluated at time , not . As a result, given, and the input vector for all , Eq.(7.10) can be iterated forward in time to compute for all . Since is an arbitrary function, we have a solver that is applicable to nonlinear, time-varying ODEs Eq.(7.8). Because each iteration of the forward Euler method depends only on past quantities, it is termed an

*explicit method*. In the LTI case, an explicit method corresponds to a

*causal*digital filter [449]. Methods that depend on current and/or future solution samples (

*i.e.*, for ) are called

*implicit methods*. When a nonlinear numerical-integration method is implicit, each step forward in time typically uses some number of iterations of Newton's Method (see §7.4.5 below).

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Backward Euler Method

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General Nonlinear ODE