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Forward Euler Method
The finitedifference approximation (Eq.
(7.2)) with the
derivative evaluated at time
yields the forward Euler
method of numerical integration:

(8.10) 
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, timevarying 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
[452]. Methods that depend on current and/or future
solution samples (i.e.,
for
) are
called implicit methods. When a nonlinear
numericalintegration 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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