What are the prospects for a more intelligent `hunt' procedure?
It is interesting that the individual matrix elements of and
are linear up to a
-scale of
, the inverse system length
(this is because the basis functions generally oscillate at a wavenumber
).
This scale is
times larger than the average level-spacing
in
.
This implies that information about all the tension minima in
a
range
might be contained in
and
and their
-derivatives
at a single
value.
To take a `toy problem' example,
imagine the values of are desired such that a parameter-dependent
order-
symmetric matrix
has a zero eigenvalue.
We assume linear dependence
.
The zero eigenvalue condition is written
.
This is simply the generalized eigenvalue equation between
and
,
whose once-off diagonalization predicts all
solutions of
.
This sounds promising, however the types of zero-crossings produced in
the eigenvalues of
are linear (passing through zero with finite
slope, changing sign in the process). Unfortunately the methods of this
chapter require detection of eigenvalues of positive definite matrices
which reach (close to) zero in a quadratic fashion (they cannot change sign).
Therefore the above trick is no help.
In fact, the above linearization of and
is deceptive, since
their positive-definiteness cannot be maintained without considering
higher-order powers of
.
The positive-definite nature of
and
arises because they are the square
of other matrices (e.g.
, see Appendix G).
It is these other matrices whose entries can unproblematically be linearized
in
.
If we imagine again the toy problem now with
and
.
Eigenvalues of
are given by squares of singular values of
[81].
So now the problem is to predict the singular value zero-intersections
of
(of course, this is the problem common to all Class b) methods, and
is therefore of huge interest).
Unfortunately the generalized singular value decomposition
[81] of
and
does not predict these
values.
Even though all the information about the
values is contained in
and
, I am unaware of a suitable decomposition which returns these values.
It is an area for future research, and would have a huge impact on the large
physics and engineering community currently using Class b) methods.
One untested idea on this front is the presentation of the toy problem
as
, where
and
.
It is possible to convert this nonlinear eigenvalue problem into a linear
one of order
(see e.g. [161]),
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