Spectral transform Lanczos

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 formal properties of operator L eq 2.18 (known as the symplectic structure ) allow the introduction of a variational principle eq D3, " a scalar product (eq Bl), and ultimately to reduce the original non-Hermitian eigenvalue problem (eq 2.18) to the equivalent Hermitian problem which may be solved using standard numerical algorithms (Appendices B—E). For example, F is a Hermitian operator. Lowdin s symmetric orthogonalization procedure " " leads to the Hermitian eigenvalue problem as well (eq E5), which may be subsequently solved by Davidson s algorithm (Appendix E). The spectral transform Lanczos method developed by Ruhe and Ericsson is another example of such transformation. 

STLM, A Software Package for the Spectral Transform Lanczos Method (Ume University, Ume4, Sweden, 1984). 

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. 

Dual Lanczos transformation theory is a projection operator approach to nonequilibrium processes that was developed by the author to handle very general spectral and temporal problems. Unlike Mori s memory function formalism, dual Lanczos transformation theory does not impose symmetry restrictions on the Liouville operator and thus applies to both reversible and irreversible systems. Moreover, it can be used to determine the time evolution of equilibrium autocorrelation functions and crosscorrelation functions (time correlation functions not describing self-correlations) and their spectral transforms for both classical and quantum systems. In addition, dual Lanczos transformation theory provides a number of tools for determining the temporal evolution of the averages of dynamical variables. Several years ago, it was demonstrated that the projection operator theories of Mori and Zwanzig represent special limiting cases of dual Lanczos transformation theory. 

In other words, the Chebyshev polynomials are essentially a cosine function in disguise. This duality underscores the effectiveness of the Chebyshev polynomials in numerical analysis, which has been recognized long ago by many,[7] including the great Hungarian applied mathematician C. Lanczos.[9] In particular, Fourier transform (and FFT) can be readily implemented in the spectral method involving the Chebyshev polynomials. 

Apart from being used to construct contracted equations of motion, projection techniques have been used to develop powerful analytic and numerical tools that erable one to solve spectral and temporal problems without resorting to the solution of global equations of motion. In this respect, we mention Mori s memory function formalism and dual Lanczos transformation theory. 

The character of a dynamical system is usually described in terms of only a few spectral and temporal properties. One would like to determine these properties without resorting to the difficult, if not impossible, task of solving the equations of motion for the system. This goal may be achieved, or approximately achieved, by utilizing the formal apparatus of the author s dual Lanczos transformation theory, whichprovides anumberof universal formulas and other results for handling such problems. Unlike more traditional approaches to spectral and temporal problems, dual Lanczos transformation theory is written in such a way that the same formal apparatus applies to classical and quantum systems, regardless of whether the imderly-ing dynamics is reversible or irreversible. 

Within the context of dual Lanczos transformation theory, the basic steps involved in the determination of the spectral and temporal properties of interest are as follows (i) Build a dual Lanczos vector space embedded with the appropriate dynamical information, (ii) Extract the information embedded in the dual Lanczos vector space, (iii) Utilize the extracted information to determine the spectral and temporal properties of interest. 

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