Lanczos method applications

Vibrational Eigenfunctions Using a Modified Single Lanczos Method Application to Acetylene (HCCH). 

Note that the most expensive part of the numerical calculations is the determination of the matrix Q using the Lanczos method. This matrix depends only on the coefficients of the matrix and the vector fic. Therefore, due to the fact that matrix does not depend on frequency, we should apply this decomposition only once for all frequency ranges (if also vector c does not depend on frequency, which is typical for many practical problems). The calculation of the inverse of the matrix (T+iujfj,(T) is computationally a much simpler problem, because T is a tri-diagonal matrix, and jj, and diagonal matrices. As a result, one application of SLDM allows us to solve forward problems for the entire frequency range. That is why SLDM increases the speed of solution of the forward problem by an order for multifrequency data. This is the main advantage of this method over any other approach. 

In this paper, we have presented an application of the Lanczos method to the IR spectrum calculation of a polyatomic molecule The power of this approach comes from the use of the Lanczos algorithm, in conjunction with tailored dipole functions, which allows to tune selectively the calculation to some specific components of the spectrum it is equivalent to split the whole spectrum calculation into smaller, nearly independent parts This has been shown explicitly in the case of the C-H stretch overtones, which have been converged up to c a 16000 cm, in a region where the density of states is about 50 per cm Contrary to a previous conclusion drawn by Slbert 19, the Lanczos method is able to provide highly excited overtone line positions, by choosing an ad hoc dipole function in eq.(l), as discussed in sections 2 and 4 3 ... 

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. 

Wu, W. and Manne, R., Fast regression methods in a Lanczos (PLS-1) basis theory and applications, Chemom. Intell. Lab. Syst., 51, 145, 2000. 

Application of the spectral Lanczos decomposition method (SLDM) for solving the linear system of equations for discrete electromagnetic fields... 

The spectral Lanczos decomposition method is designed for effective calculation of the matrix functions. This method has found a useful application in the solution of linear equations in electromagnetic modeling (Druskin and Knizhnerman, 1994 Druskin et ah, 1999)). Let us consider, for example, the matrix equation (12.35). The formal solution of this equation has the form... 

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

In the historical survey of the spectral methods given by Canute et al [22], it was assumed that Lanczos [101] was the first to reveal that a proper choice of trial functions and distribution of collocation points is crucial to the accuracy of the solution of ordinary differential equations. Villadsen and Stewart [203] developed this method for boundary value problems. The earliest applications of the spectral collocation method to partial differential equations were made for spatially periodic problems by Kreiss and Oliger [94] and Orszag [139]. However, at that time Kreiss and Oliger [94] termed the novel spectral method for the Fourier method while Orszag [139] termed it a pseudospectral method. 

The numerical treatment is again based on the matrix representation of the operator on a coupled basis set of functions, followed by the application of the Lanczos algorithm. Following the same method used in the previous section, we define the two tittle angular momentum operators (one for each body) and the overall little angular momentum operator... 

Noether s theorem is derived in E. Noether, Goett. Nachr. 235 (1918). It is explained further in C. Lanczos, The Variational Principles of Mechanics, Toronto, 1966, Appendix II and in R. Courant and D. Hilbert, Mathematical Methods of Physics, Interscience, 1953, Vo. I, p. 262. Application to fields is clearly demonstrated in N. N. Bogolubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed., Interscience, 1980, Sec. E2. 

In cases where equations of motion are desired for deformable bodies, methods such as the extended Hamilton s principle may be employed. The energy is written for the system and, in addition to the terms used in Lagrange s equation, strain energy would be included. Application of Hamilton s principle will yield a set of equations of motion in the form of partial differential equations as well as the corresponding boundary conditions. Derivations and examples can be found in other sources (Baruh, 1999 Benaroya, 1998). Hamilton s principle employs the calculus of variations, and there are many texts that will be of benefit (Lanczos, 1970). 

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