Civil Engineering Reference
In-Depth Information
More sophisticated methods do exist‚ such as the conjugate gradient method
or the GMRES (generalized minimum residual) method‚ and for a detailed
explanation‚ the reader is referred to [Gv96‚ SS86].
A technique that is often used to hasten the convergence of an iterative
method involves the use of
preconditioners.
Given a system of equations in
variables‚
A
x
=
b‚
and an matrix
P
that acts as a preconditioner‚ one
may premultiply both sides of the matrix equation by
P
to obtain
If
P
were to be exactly then the method would conclude in a single
iteration; this however‚ involves the substantial computational effort of finding
Instead‚ if
P
were to be a “good” approximation to then
PA
would
be “close” to the identity matrix‚ and the iterations would converge relatively
soon. The notion of preconditioning essentially involves the choice of a suitable
P
that is both easy to compute and satisfies the requirement of being a “good”
approximation to
For a detailed discussion on preconditioning‚ the reader is referred to [Gv96].
2.8
SUMMARY
This chapter has presented an overview of techniques used for circuit simulation.
For elements represented by linear device equations‚ the MNA formulation is
employed‚ and we have seen how it may be constructed by inspection of the
circuit. Nonlinear elements are iteratively linearized at a guess solution and
folded into this formulation‚ and elements described by differential equations
are numerically integrated‚ so that the system is solved at successive time
points. This represents the slowest and practically most exact way of solving a
circuit. In the succeeding chapters‚ we will next see methods that can be used
for faster simulation.
