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













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