INCONSISTENT INITIAL CONDITIONS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 84

It is seen that the response contains both a Dirac impulse function and
an exponentially decaying function. Also Its time-domain

response at using { N , M } = {2, 4} is obtained from

If we evaluate (5.3) by opening the brackets

then with 15 digits, the first term of (5.4) vanishes and the second term
is identical to (4.38) except the negative sign. The relative error between
the analytical results and those from numerical Laplace inversion is the
curve-b in Fig.5.2, which is identical to Fig.4.4.
However, if (5.3) is evaluated without separating the terms associated
with the Dirac impulse function and those associated with exponentially
decaying function, the relative error is the curve-a in Fig.5.2. It is ob-
served that the relative error is much higher in this case, as compared
with curve-b.
Because for arbitrary networks, it is general not trivial to separate
the terms associated with the Dirac impulse function and those with-
out, the above results reveal that the existence of a Dirac impulse will
significantly increase the error of numerical Laplace inversion when the
time step is small. Because a Dirac impulse may exist at switching
instants in mixed-mode switching circuits, special algorithms are needed
to minimize the error in computing from Also observed is
that the relative error is large when the step size is small and decreases
monotonically with the increase in the step size until a minimum point
is reached. The error then increases rapidly if the step size is further
increased. This observation suggests that to achieve high accuracy in
the presence of impulses, the step size should not be too large, nor too