6.3 Solution of the Nonlinear Eigenvalue Problem
The discrete eigenvalue problems are each of the form (6.4).
The method employed for solving the general problem (6.4)
requires that in an interval [k_{A}, k_{B}] of values of k
the matrix A_{k} is approximated by a matrix polynomial in k
A_{k} » A_{[0]} + k A_{[1]} + ... + k^{m} A_{[m]} for k real. 
 (6.12) 
The nonlinear eigenvalue problem (6.4) can be replaced
with the following eigenvalue problem:
[A_{[0]} + k A_{[1]} + ... + k^{m} A_{[m]}]m = 0. 
 (6.13) 
The solutions of (6.13) are the same as those of the following
generalised linear eigenvalue problem:

é ê ê ê ê
ê ê ê ë


 
ù ú ú ú ú
ú ú ú û


é ê ê ê ê
ê ê ê ë


 
ù ú ú ú ú
ú ú ú û


 (6.14) 
= k 
é ê ê ê ê
ê ê ê ë


 
ù ú ú ú ú
ú ú ú û


é ê ê ê ê
ê ê ê ë


 
ù ú ú ú ú
ú ú ú û

. 

Equation (6.14) is amenable to solution by the QZ algorithm
[61], which is available on a number of numerical libraries.
Methods for solving problems of the form (6.13)
are considered in references
[67], [55], [28] and [84].
Since the eigenvalues k^{*} of the underlying Helmholtz problem are all real
and we are interested only in positive values, it
is sufficient to compute the interpolant (6.12) for the positive real
numbers k. However, as a result of numerical error,
the computed wavenumber and mode shape will have small imaginary parts.
The generalised eigenvalue problem (6.14)
will generally have m ×n solutions. Half of these can be immediately
discounted since the eigenvalues occur in pairs [^k], [^k]. The full set
of solutions will contain approximations to the true eigenvalues of the
underlying Helmholtz problem. However, many spurious solutions are generally
produced as a result of the collocation method and approximation (6.12). These spurious
eigenvalues do not have small imaginary parts and hence they can be
sorted from the true eigenvalues. Approximations to the true eigenvalues
lying outside the range [k_{A}, k_{B}] may also be produced. These approximations
will generally be poor and they can be excluded from the results.
Let [^k], [^(m)] be a typical nonspurious
solution to (6.14).
The eigenvalue [^k] is an approximation
to the eigenfrequencies k^{*} of the Helmholtz problem.
The approximation
to the eigenfunctions in D ÈS can be recovered through the
substitution of the approximation [^(m)] for s
in equation (6.5) or [^(m)]
for f (Dirichlet) or
[(¶f)/( ¶n)] (Neumann) in equations
(6.8)(6.9).
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