The philosophy of the deflation method is to start above the first
eigenvalue and use the secant method to find the first eigenvalue.
Once that eigenvalue is located it is deflated, that is, divided out
of the characteristic equation, and the process is repeated for each
eigenvalue in turn. Interestingly, one may formally prove that the
secant method will * always* converge to the first eigenvalue if
started at a point above that first eigenvalue. (See Wilkinson,
[62].) Naturally, there are some footnotes to this sweeping
statement. The key one is that the characteristic function should be
a polynomial. Some care is required to avoid violating this
requirement. Secondly, the eigenvalues should all be real. In
practice, the deflation procedure works very well for most realistic
ocean acoustic problems--- even for complex eigenvalues. However, if
branch cuts are present, problems are not unlikely.

The deflation of previous eigenvalues is a trivial process. Instead of computing the characteristic function, , one computes:

where are the previously computed eigenvalues which are to be removed or deflated.

Tue Oct 28 13:27:38 PST 1997