Eigensystem

J.J. Dongarra, C.B. Moler, J.R. Bunch and G.W. Stewart, LINPACK. SIAM, Philadelphia, 1979. B.T. Smith, Matrix Eigensystem Routines — EISPACK Guide. 2nd edn.. Lecture Notes in Computer Science, Vol. 6. Springer, New York, 1976. 

It is often convenient to select the coordinate system for which the only nonzero elements of the property tensor lie on its diagonal. This is the eigensystem. To find the eigensystem, the general rules for transformation of a tensor must be identified. The transformation of Ohm s law (Eq. 1.24) illustrates the way in which the material properties tensor xold transforms to xnew and serves to demonstrate the general rule for transforming rank-two tensors ... 

The diagonal elements of D are the eigenvalues of D, and the coordinate system of D defines the principal axes Xi, 2, 3 (the eigensystem). In the principal axes coordinate system, the diffusion equation then has the relatively simple form... 

We will start out by assuming that S = 1 and leave the more general case until later. When the entire eigensystem is required, the eigenvectors are collected as columns into a matrix U, and the eigenvalues A as diagonal elements into a diagonal matrix D. The equation then takes the form... 

Finally, we shall describe an idea which at first seems odd Instead of solving the linear equation problem, we solve a related eigensystem problem. It is easily verified that the equation to be solved can be written on the form... 

In this formula, p. stands for gTd This is certainly an eigensystem equation. However, we must add a small correction to H, and moreover, this correction is not known in advance. It turns out that this correction can be left out, unless it is important that the linear equation is exactly solved. This is not necessary if the object is to find a good step for a macroiteration. Moreover, it turns out that, in such a context, the discrepancy introduced between this method and the exact NR-steps has the same asymptotic dependence as the error. Therefore, the method is still a second-order method with this modification, and there is no way to say a priori that this method is better or worse than the exact NR-iterations. This method is called the augmented Hessian (AH-)method. It is seen to be equivalent to a Newton-Raphson using a shifted hessian. This can be very advantageous, since this shift tends to keep the step down, and to keep the shifted hessian positive definite, when one is far from a solution. The size of... 

Naturally, the success of this method, which would have seemed a bit implausible some years ago, reflects the advances made in eigensystem solution methods. 

B.T. Smith, J.M. Boyle, J.J. Dongarra, B. S. Garbow, Y. Ikebe, V.C. Klema and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide, Springer-Verlag, 1976. 

The plan of this chapter is as follows. The next section briefly reviews the CC formalism for the ground state. This is necessary since the LR-CC and EOM-CC approaches start from the CC ground state description. It also introduces some notation that will be used in later sections. Next, the basics of the exact EOM-CC approach are derived, showing how an eigensystem is arrived at. After some aspects of characterizing an electronic transition, EOM-/LR-CC methods that have been developed and implemented are surveyed. The next section presents a numerical assessment of some of the main methods. Finally, a few illustrative applications are summarized. Some aspects of EOM-CC methods are discussed in Chapter 2. The symmetry-adapted cluster configuration interaction (SAC-CI) method can be related to EOM-CC methods. The SAC-CI method and several impressive applications thereof are described in Chapter 4. 

In the linear response TDDFT formalism the excitation energies of a molecular system are determined as poles of the linear response of the ground state electron density to a time dependent perturbation [2], After Fourier transformation from the time to frequency domain, and some algebra, the excitation energies can be obtained as eigenvalues of the non-Hermitian eigensystem [23]... 

ARM 98] ARMSTRONG N., KALCEFF W., Eigensystem analysis of x-ray diffraction profile deconvolution methods explains ill conditioning , J. Appl. Cryst, vol. 31, p. 453-460, 1998. 

And, after having explicitly described the eigenstates i/i) and of fhe reaction and bath DoFs, respectively, one obtains a complete (approximate) eigensystem for the original reactive scattering problem. 

For the simulation of ESR spectra one has to solve the spin Hamiltonian of Eq. (10). The easiest way to do this is to regard all the different terms in the spin Hamiltonian as small compared with the electron Zeeman interaction and to use perturbation theory of the first order. The Zeeman term can easily be solved within the eigensystem of the Sz operator (in the main axis system of the g-tensor or S 2=5 for isotropic cases), for instance in the isotropic case ... 

Unfortunately, in most cases this simplification is not applicable. Therefore, the use of perturbation theory of higher order is recommended, or in more complicated situations, the diagonalization of the spin Hamiltonian within the eigensystem of its spin operators. 

Pappa, R. S. and Juang, J. N. Galileo spacecraft modal identification using an eigensystem realization algorithm. Journal of Astronautical Sciences 33(1) (1985), 95-118. 

R. Carbo-Dorca, ]. Math. Chem., 27, 357 (2000). Quantum QSAR and the Eigensystems of... 

S is a symmetric matrix [since correlation Xi.Xj) - correlation Xj,Xi)] of real values. Therefore, from the theory of eigenanalysis, it follows that P is the matrix whose columns are the orthogonal eigenvectors of S and that, ..., are the corresponding eigenvalues. For a more detailed discussion of eigensystem theory see, for example, Morris (1982). 

The problem of finding the components of X is therefore that of deriving the eigensystem of S, the correlation matrix of X. 

Extracting principal components. The mathematical literature details a number of methods for determining the eigensystem of a matrix. In the case of principal component extraction, the matrix X X is square and symmetric, as... 

Unfortunately, Jacobi s method is computationally intensive, requiring of the order of 6n arithmetic operations to calculate the eigensystem. 

Subsequent eigenvalues can be found by transforming the matrix, A, such that the remaining eigensystem is retained, but the influence of z is removed. This process is known as deflation and may be achieved in a number of ways. Again, see Gourlay and Watson (1973) for further details. 

G.H.Golub, J.H.Wilkinson El-conditioned Eigensystems and Computation of the Jordan Canonical Form. SIAM Review 18, pp. 578-619 (1976)... 

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