The solution of the above matrix eigenvalue problem proceeds in two steps. First, the eigenvalues are found by applying a root finder to the characteristic equation,

where denotes the determinant of the matrix. Finally, once the eigenvalues are found, the eigenvectors are computed using inverse iteration. The inverse iteration is given by,

The matrix **A** is (nearly) singular because is (nearly) an
eigenvalue of **A**. One may easily show that the effect of this
iteration is to continually amplify the eigenvector component in the
starting vector that corresponds to the eigenvalue .
Thus, the starting vector may be picked fairly arbitrarily.
After each iteration, the new iterate is renormalized to
avoid overflow. In addition, the growth of the iterates is used as a
check for convergence. (Typically 2 iterations are sufficient.)

This inverse iteration technique is described in numerous texts on
algebraic eigenvalue problems, see for instance, Wilkinson [62].
The usual development applies for this nonlinear eigenvalue problem
with some minor restrictions. In particular, the inverse iteration will
* not* be reliable in cases where one-sided shooting would be
unstable for constructing the impedance conditions. There are two cases
where this could be an issue: 1) when the `internal reflection
coefficient' option is used recklessly and 2) for certain elastic
problems with internal ducts.

One of the most difficult aspects of normal mode computations is that
of finding the roots of the characteristic equation. The fundamental
difficulty is that many familiar root-finding algorithms (such as the
secant method or Newton's method) will only converge to a particular
root if an initial guess is provided which is sufficiently close.
Unfortunately, even though the eigenvalues of a purely acoustic
problem are guaranteed to be distinct, they can be very nearly
degenerate. As a result, many existing mode codes provide accurate
but * incomplete* mode sets.

Two root-finding techniques are used in the * KRAKEN*
program which we
shall describe next. The first is efficient and robust, in fact
fool-proof, but is applicable only to purely acoustic problems, or
acoustic problems with an elastic half-space. The second is less
reliable but valid for problems with elastic layers.

Tue Oct 28 13:27:38 PST 1997