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where
Continuing this process, one can show that
The cost for evaluating (4.66) can be greatly reduced if
is chosen
to be
where
is an integer. For instance, if
can
be obtain in only ten matrix multiplications.
Similarly,
is obtained from
where
The preceding analysis shows that both and can be
computed efficiently over a large number of fine steps to achieve high
accuracy.
Stability
To investigate the stability of numerical Laplace inversion, we follow
the approach for Euler formulae and the same benchmark equation
(4.5) is used. The response of (4.5) at
is obtained by using
numerical Laplace inversion
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