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 . 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  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
It is possible to convert this nonlinear eigenvalue problem into a linear
one of order (see e.g. ),