Lanczos filtered

McCormack D A, Kroes G J and Neuhauser D 1998 Resonance affected scattering Comparison of two hybrid methods involving filter diagonalization and the Lanczos method J. Chem. Phys. 109 5177... 

However, there is a price to pay in a spectral transform Lanczos algorithm At each recursion step, the action of the filter operator onto the Lanczos vectors has to be evaluated. In the original version, Ericsson and Ruhe update the Lanczos vectors by solving the following linear equation ... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

PIST distinguishes itself from other spectral transform Lanczos methods by using two important innovations. First, the linear equation Eq. [38] is solved by QMR but not to a high degree of accuracy. In practice, the QMR recursion is terminated once a prespecified (and relatively large) tolerance is reached. Consequently, the resulting Lanczos vectors are only approximately filtered. This inexact spectral transform is efficient because many less matrix-vector multiplications are needed, and its deficiencies can subsequently... 

The filtering, namely the construction of energy local bases, can also be carried out using the Lanczos recursion or similar recursive methods. However, filtered vectors at E/ can only be obtained using the Green filter ... 

Smith and Yu have used MINRES to construct filtered vectors in Lanczos-based FD calculations for Hermitian/real-symmetric matrices.76,77,181,182 To this end, these authors demonstrated that the filtered vectors can be written as a linear combination of the Lanczos vectors ... 

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... 

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. 

Distributions from Resonance Decay via Lanczos Subspace Filter Diagonalization Application to H02. 

Lanczos Based Forward Substitution and Filter-Diagonalization. 

H. Zhang and S. C. Smith, Phys. Chem. Chem. Phys., 3, 2282 (2001). Lanczos Subspace Filter... 

Resonance Affected Scattering Comparison of Two Hybrid Methods Involving Filter Diagonalization and the Lanczos Method. 

Filter Diagonalization Methods with the Lanczos Method for Calculating Vibrational Energy Levels. 

Strategies for Spectral Analysis in Dissipative Systems Filter Diagonalization in the Lanczos Representation and Harmonic Inversion of the Chebychev Order Domain Autocorrelation Function. 

In recent years the solution of problems of large amplitude motions (LAM s) has usually been based on grid representations, such as DVR,[11, 12] of the Hamiltonians coupled with solution by sequential diagonalization and truncation (SDT[13, 9]) of the basis or by Lanczos[2] or other iterative nicthods[14]. More recently, filter diagonalization (ED) [5, 4] and spectral transforms of the iterative operator[15] have also been used. There has usually been a trade-off between the use of a compact basis with a dense Hamiltonian matrix, or a simple but very large D R with a sparse H and a fast matrix-vector product. 

Several filters have been used with the Lanczos algorithm these are described below. 

Table 1 Energies Associated with the Eigenvectors Obtained from the Initial State nv,)( Either by Performing Four Iterations of the Wave Operator Contraction Algorithm or by Performing an Exact Calculation (based upon filtered Lanczos calculations).

Table 3 Number of Filtered Lanczos Recursion Steps Required to Obtain an Energy with an Error Smaller than 0.1 cm and a Residue Known with Three Significant Digits...

Eigenvalues Number of filtered Lanczos recursions required ... 

There is considerable interest in the development and testing of spectral filters for use in Lanczos recursion procedures. Although progress has been made in the testing of a Green function filter for incorporation in Lanczos codes (74), it remains to be seen how well this will perform in difficult molecular problems having many coupled states (such as the benzene overtones). 

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