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The backwardEuler method also truncates the Taylor series after two
terms. The difference is that the derivative is evaluated at point
instead of at point .

(1.3) 
Assuming that the value at point is correct, the backwardEuler
method computes the value at point with a local truncation error
that scales with . The backwardEuler method always gives
undershoots on the original curve.
Normally we do not know the derivative at point , although we
need it to compute the function value at point . In practice
this requires a rearrangement of the equation. We call such a
numerical scheme an implicit numerical scheme. For most
equations implicit schemes are more stable than explicit schemes
because of the undershoots.
Figure:
Graphical illustration of
the backwardEuler method. To obtain point 2 from point 1, we
take the derivative at point 2 and extrapolate it at point 1. To
obtain point 3 starting at point 2, we do the same: take the
derivative at point 3 and extrapolate it at point 2.

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20021115