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Backward Euler Method

An example of an implicit method is the backward Euler method:

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

Because the derivative is now evaluated at time $ n$ instead of $ n-1$ , the backward Euler method is implicit. Notice, however, that if time were reversed, it would become explicit; in other words, backward Euler is implicit in forward time and explicit in reverse time.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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