Lanczos

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... 

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. 

C. Lanczos, The Variational Principles of Mechanics , University of Toronto Press, Toronto, 1970... 

Krylov Approximation of the Matrix Exponential The iterative approximation of the matrix exponential based on Krylov subspaces (via the Lanczos method) has been studied in different contexts [12, 19, 7]. After the iterative construction of the Krylov basis ui,..., Vn j the matrix exponential is approximated by using the representation A oi H(g) in this basis ... 

In most cases, this Lanczos-based technique proves to be superior to the Chebyshev method introduced above. It is the method of choice for the application problems of class 2b of Sec. 2. The Chebyshev method is superior only in the case that nearly all eigenstates of the Hamiltonian are substantially occupied. 

Fig. 3. Stepsize r used in the simulation of the collinear photo dissociation of ArHCl the adaptive Verlet-baaed exponential integrator using the Lanczos iteration (dash-dotted line) for the quantum propagation, and a stepsize controlling scheme based on PICKABACK (solid line). For a better understanding we have added horizontal lines marking the collisions (same tolerance TOL). We observe that the quantal H-Cl collision does not lead to any significant stepsize restrictions.

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

T. J. Park and J.C. Light Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85 (1986) 5870-5876... 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

We here describe the alternative of approximating <,c(S)b via Lanczos method. The Lanczos process [18, 22] recursively generates an orthonormal basis Qm = [qi,.., qm] of the mth Krylov subspace... 

This construction requires one matrix-vector multiplication with S and two inner products in each recursive step. Therefore, it is not necessary to store S explicitly as a matrix. The Lanczos process yields the approximation [21, 7, 12]... 

C. Lanczos. Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bureau Standards, 49 33-53, 1952. 

Lanczos, C. Applied Analysis, Prentice Hah, Englewood Cliffs, NJ... 

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. 

Although the Lanczos is a fast efficient algorithm, it does not necessarily give savings in memory. To save memory a number of techniques divide the molecule into smaller parts that correspond to subspaces within which the Hessian can be expressed as a matrix of much lower order. These smaller matrices are then diagonalized. The methods described below show how one then proceeds to achieve good approximations to the true low frequency modes by combining results from subspaces of lower dimension. 

Lagrange Multiplier Method for programming problems, 289 for weapon allocation, 291 Lamb and Rutherford, 641 Lamb shift, 486,641 Lanczos form, 73 Landau, L. D., 726,759, 768 Landau-Lifshitz theory applied to magnetic structure, 762 Large numbers, weak law of, 199 Law of large numbers, weak, 199 Lawson, J. L., 170,176 Le Cone, Y., 726... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

Yang-Mills fields, 249-250, 255-257 Lagrangian multiplier, conical intersection location, 488-489, 565 Laguerre polynomials, Renner-Teller effect, triatomic molecules, 589—598 Lanczos reduction ... 

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