The discrete eigenvalue problems are each of the form (4).
The method employed for solving (4)
requires that in an interval [kA, kB] of values of k
the matrix Ak is approximated by a matrix polynomial in k
Ak » A[0] + k A[1] + ... + kmA[m]forkreal.
(9)
Similar methods of approximation are considered in references
[24],
[25] and [26], but with regard to reducing the computational
cost in the standard boundary element solution of Helmholtz
problems over a range of wavenumbers. These references do not consider the
eigenvalue problem.
The non-linear eigenvalue problem (4) can be replaced
with the following eigenvalue problem
[A[0] + k A[1] + ... + kmA[m]]m = 0.
(10)
The solutions of (10) are the same as those of the following
generalised linear eigenvalue problem
é ê ê ê
ê ê ê ë
A[0]
A[1]
A[2]
.
.
A[m-2]
A[m-1]
0
I
0
.
.
0
0
:
:
:
:
:
:
:
0
0
0
.
.
I
0
0
0
0
.
.
0
I
ù ú ú ú
ú ú ú û
é ê ê ê
ê ê ê ë
m
k m
:
km-2 m
km-1 m
ù ú ú ú
ú ú ú û
(11)
= k
é ê ê ê
ê ê ê ë
0
0
0
.
.
0
-A[m]
I
0
0
.
.
0
0
:
:
:
:
:
:
:
0
0
0
.
.
0
0
0
0
0
.
.
I
0
ù ú ú ú
ú ú ú û
é ê ê ê
ê ê ê ë
m
k m
:
km-2 m
km-1 m
ù ú ú ú
ú ú ú û
.
Equation (11) is amenable to solution by the QZ algorithm
[27], which may be invoked, for example, via NAG routine
F02GJF [28].
Methods for solving problems of the form (10)
are considered in references
[29], [11], [30] and [13].
The authors are not aware of any special methods of solving (11)
which takes advantage of its special structure or the sparsity of the
matrices.
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 (9) for the positive real
numbers k. However, due to the application of the boundary element method
and the approximation (9), the solutions of (11) that
correspond to the solution of the underlying Helmholtz eigenvalue
problem will tend to have small imaginary parts.
The generalised eigenvalue problem (11)
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 (9). 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 [kA, kB] may also be produced. These approximations
will generally be poor and they can be excluded from the results.
Let [^k], [^(m)] be a typical non-spurious solution to (11).
The eigenvalue [^k] is an approximations
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 (5) or [^(m)]
for j (Dirichlet) or
[(¶j)/(¶n)] (Neumann) in equations
(7)-(8).
In this paper the method employed for deriving the polynomial approximation
(9)
involves computing Ak at the m+1 Chebyshev (¥ norm)
interpolation points for any selected range [kA, kB]. The
coefficient matrices
A[0], A[1], ..., A[m] in (9) are obtained through
Newton's divided differences using the value of Ak at the selected values of
k in [kA, kB]. The generalised eigenvalue problem
(11) is then solved through invoking NAG routine F02GJF.