Lanczos Algorithms

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

A block Lanczos algorithm (where one starts with more than one vector) has been used to calculate the first 120 normal modes of citrate synthase [4]. In this calculation no apparent use was made of symmetry, but it appears that to save memory a short cutoff of 7.5 A was used to create a sparse matrix. The results suggested some overlap between the low frequency normal modes and functional modes detennined from the two X-ray conformers. 

Sometimes, the eigenvectors are needed in addition to the eigenvalues, and they can also be obtained by the Lanczos algorithm. For a K-step recursion, an eigenvector of FI can be expressed as a linear combination of the Lanczos vectors ... 

It has been long recognized that the convergence rate of the Lanczos algorithm depends on the spectral range of the Hamiltonian matrix (AH). Recently, it was shown from several numerical examples that the convergence rate is actually inversely proportional to the square root of AH.41,42 This... 

The implementation of various forms of the Lanczos algorithm is straightforward, and a library of routines has been collected in the second volume of Cullum and Willoughby s book.27 Applications of the Lanczos algorithm to solve molecular vibration problems, pioneered by Wyatt,43 7 Carrington,12,13,15 48 49 and others,50-55 have been reviewed by several authors56-60 and will be discussed in more detail below. A list of other applications of the Lanczos algorithm in different fields of science can be found in the review by Wyatt.56... 

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. 

Such a method was first proposed by Wyatt and co-workers.43 7,56 In their so-called recursive residue generation method (RRGM), both eigenvalues and overlaps are obtained using the Lanczos algorithm, without explicit calculation and storage of eigenvectors. In particular, the residue in Eq. [41] can be expressed as a linear combination of two residues ... 

For real-symmetric systems, the above relation holds up to machine precision despite the fact that individual copies are not normalized.38 It also works reasonably well for complex-symmetric Hamiltonians.102 Unfortunately, there has not yet been a formal proof of this striking observation of the Lanczos algorithm. 

It was demonstrated in the above subsection that the Lanczos algorithm can be used to compute scalar quantities such as transition amplitudes without explicit calculation and storage of the eigenvectors. We discuss here another... 

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