Chebyshev recursion

One can extend Eq. [63] to compute the entire eigenspectrum of H. This can be achieved by calculating and storing, along the Chebyshev recursion, the... 

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. 

One can also compute any selected eigenvector and its overlap with the initial vector used in the Chebyshev recursion ... 

We note that the Chebyshev recursion-based LSFD can be used to extract frequencies from a time signal in the following form ... 

Unfortunately, the symmetry adaptation scheme described above for the Chebyshev recursion cannot be applied directly to the Lanczos recursion. 

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

HOC1,309,310 HArF,311 and C1HC1.71 Most of these calculations were carried out using either the complex-symmetric Lanczos algorithm or filter-diagonali-zation based on the damped Chebyshev recursion. The convergence behavior of these two algorithms is typically much less favorable than in Hermitian cases because the matrix is complex symmetric. 

In this chapter, we also discussed several schemes that allow for the computation of scalar observables without explicit construction and storage of the eigenvectors. This is important not only numerically for minimizing the core memory requirement but also conceptually because such a strategy is reminiscent of the experimental measurement, which almost never measures the wave function explicitly. Both the Lanczos and the Chebyshev recursion-based methods for this purpose have been developed and applied to both bound-state and scattering problems by various groups. 

Techniques similar to the those described in this section and in Ref. 133, but used within a time-independent framework, have been developed by Kouri and coworkers [188,189] and by Mandelshtam and Taylor [62,63]. Kroes and Neuhauser [65-68] have used the methods developed in these papers to perform time-independent wavepacket calculations using only real arithmetic. The iterative equation that lies at the heart of the real wavepacket method, Eq. (4.68), is in fact simply the Chebyshev recursion relationship [187]. This was realized by Guo, who developed similar techniques based on Chebyshev iterations [50,51]. 

B. Iterative Calculation of (E - H) 1 Using Modified Chebyshev Recursion Relations... 

To numerically absorb the wavepacket components at the boundaries, which are set beforehand to prevent the packet from flowing out to the asymptotic ranges, a damping function e is introduced in the Chebyshev recursion scheme such that... 

Even for fairly short spectral ranges, the computing time involved for this operation can be quite high, especially if each cosine term required is calculated from a series. Cosine tables can be stored in the computer memory, but this does not provide an adequate solution. It has been found that a successfiil method of circumventing this problem is by the use of recursion relationships, whereby a value for cos 2i6i kh can be calculated from the value of cos 2n ih. In this way, only one cosine value has to be calculated for each spectral wavenumber. The Chebyshev recursion formula. 

---