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Forward Euler Method

The finite-difference approximation (Eq.$ \,$(7.2)) with the derivative evaluated at time $ n-1$ yields the forward Euler method of numerical integration:

$\displaystyle \underline{\hat{x}}(n) \isdefs \underline{\hat{x}}(n-1) + T\, \do...
...\hat{x}}(n-1) + T\, f[n-1,\underline{\hat{x}}(n-1),\underline{u}(n-1)] \protect$ (8.10)

where $ \underline{\hat{x}}(n)$ denotes the approximation to $ \underline{x}(nT)$ computed by the forward Euler method. Note that the ``driving function'' $ f$ is evaluated at time $ n-1$, not $ n$. As a result, given, $ \underline{\hat{x}}(0)=\underline{x}(0)$ and the input vector $ \underline{u}(n)$ for all $ n\ge0$, Eq.$ \,$(7.10) can be iterated forward in time to compute $ \underline{\hat{x}}(n)$ for all $ n>0$. Since $ f$ 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., $ \underline{\hat{x}}(n)$ for $ n\ge0$) 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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General Nonlinear ODE