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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:

$\displaystyle \underline{\hat{x}}(n) \isdefs \underline{\hat{x}}(n-1) + T\, \dot{\underline{\hat{x}}}(n-1) \eqsp \underline{\hat{x}}(n-1) + T\, f[n-1,\underline{\hat{x}}(n-1),\underline{u}(n-1)] \protect$ (8.11)

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.$ \,$ (7.11) 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.$ \,$ (7.9).

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 [452]. 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University