Eigenvalue filtering

Some of the linear combinations will be well defined and others poorly defined. The latter may be eliminated in a filtering procedure, referred to in the literature under the names characteristic value filtering, eigenvalue filtering, and principal component analysis. If the parameter set is not homogeneous, but includes different types, relative scaling is important. Watkin (1994) recommends that the unit be scaled such that similar shifts in all parameters lead to similar changes in the error function S. 

To simplify FECO evaluation, it is conmion practice to experimentally filter out one of the components by the use of a linear polarizer after the interferometer. Mica bireftingence can, however, be useftil to study thin films of birefringent molecules [49] between the surfaces. Rabinowitz [53] has presented an eigenvalue analysis of birefringence in the multiple beam interferometer. 

This direct filter teclnrique is very powerfiil [, 59] in extracting highly excited states, since only the propagation of a wavepacket is required. However, it is inefficient when there are closely-lying eigenvalues T... 

Wall M R and Neuhauser D 1995 Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. 

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

The major shortcoming of the spectral method is the rate of convergence. Its ability to resolve eigenvalues is restricted by the width of the filter, which in turn is inversely proportional to the length of the Fourier series (the uncertainty principle). Thus, to accurately characterize an eigenpair in a dense spectrum, one might have to use a very long Chebyshev recursion. 

It should be pointed out that a low-storage version of the Lanczos-FD can also be formulated without explicit recourse to the filtered vectors.77,164 Such a low-storage version is preferred if only eigenvalues are needed. For example, Yu and Smith have shown that the overlap and Hamiltonian matrices in a prespecified energy range can be directly obtained as follows 77... 

Residual Filter Diagonalization Method for Interior Eigenvalues of Physical Systems. 

With Eqs. (32)-(35), one can construct the basis functions (Ej), filtered out of the initial state near the energies Ei. Before applying the modified Chebyshev expansion, one has to rescale the original Hamiltonian H, because Chebyshev polynomials converge only for la < 1 [236]. In fact, H in the equations above has to be substituted with the operator. ffnorm H H))/AH, whose eigenvalues lie between —1 and 1. Then, a basis function can be written as [208]... 

In quantum scattering calculations we are typically interested in the eigenstates in the low energy end of the spectrum. For this purpose tm exponential filter f Hs) = cxp[—— Hmin)] is useful as it will dilate the eigenvalues at low energies. The action of the filter on the vectors can be performed via the Chebyshev polynoinicds[63, 64],... 

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