Jacobi eigenvalues

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

A comparison of the performance of the three algorithms for eigenvalue decomposition has been made on a PC (IBM AT) equipped with a mathematical coprocessor [38]. The results which are displayed in Fig. 31.14 show that the Householder-QR algorithm outperforms Jacobi s by a factor of about 4 and is superior to the power method by a factor of about 20. The time for diagonalization of a square symmetric value required by Householder-QR increases with the power 2.6 of the dimension of the matrix. 

The eigenvalues of A can be find by solving the characteristic equation of (1.61). It is much more efficient to look for similarity transformations that will translate A into the diagonal form with the eigenvalues in the diagonal. The Jacobi method involves a sequence of orthonormal similarity transformations, 12,... such that A(<+1 = TTkAkTk. The matrix Tk differs from the identity... 

In the Jacobi method, a series of similarity transformations is carried out. It is easily proven that similar matrices have the same eigenvalues. Let A = P, BP. The eigenvalues of A satisfy the secular equation (2.38) ... 

The Jacobi method is generally slower than these other methods unless the matrix is nearly diagonal. In SCF calculations one is faced with the non-orthogonal eigenvalue equation... 

The eigenanalysis of the MIL tensor is run via Jacobi method to calculate the main characteristics values, that is, eigenvalues (eo-g), and characteristic directions, that is, eigenvectors (co-s)-... 

This is an ordinary eigenvalue problem in which the tridiagonal Jacobi matrix Jxj is given in Eq. (60) with M = oo. The residues dk) are defined by Eq. (15), where IT ) is the exact complete state vector normalized to Co 0. The same type of definition for dk is valid for an approximation such as Eq. (67), provided that normalization is properly included according to Eq. (69) ... 

This recurrence relation and the available triple set an, fin uk] are sufficient to completely determine the state vector Q without any diagonalization of the associated Jacobi matrix U = c0J, which is given in Eq. (60). Of course, diagonalization of J might be used to obtain the eigenvalues uk, but this is not the only approach at our disposal. Alternatively, the same set uk is also obtainable by rooting the characteristic polynomial or eigenpolynomial from... 

G. L. G. Sleijpen and H. A. van der Vorst, A generalized Jacobi-Davidson iteration method for linear eigenvalue problems, Technical Report Preprint 856, Dept. Math., Utrecht University, 1994. 

Equation (3.12) clearly illustrates that the roots of Pn(0, namely the nodes of the quadrature approximation a, are the eigenvalues of the tridiagonal matrix appearing in the equation. This matrix can be made symmetric (preserving the eigenvalues) by a diagonal similarity transformation to give a Jacobi matrix ... 

This procedure transforms the ill-conditioned problem of finding the roots of a polynomial into the well-conditioned problem of finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix. As shown by Wilf (1962), the N weights can then be calculated as Wa = OToV ai where tpai is the first component of the ath eigenvector (pa of the Jacobi matrix. 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

Secondly, the canonical orthonormalization procedure to diagonalize the overlap matrix and then the application of the Jacobi transformation to diagonalize the Fock matrix in the eigenfunctions of the overlap matrix, returns two eigenvalues, the values —0.50000 and —0.12352 Hartrees, in canonical B 18 and B 19. This is the important elementary point that we can make two linear combinations of two functions and so there are two possible eigenvalues to be calculated. These eigenvalues, of course, are present in the calculation set out in the other worksheet, based on the Schmidt procedure. The Is... 

In eq. (2.39) a general equation has been given which allows the calculation of eigenvalues of the Jacobi matrix (that means reaction constants or their combinations) taking the degree of advancement. In the case of consecutive reactions, special solutions have been given. However, this system of differential equations has a general solution [17a],... 

The elements of the Jacobi-matrix are named. The trace T of the Jacobi matrix K is equal to the sum of all eigenvalues. It amounts to... 

K and K are assumed to be regular Jacobi matrices of two different linear systems of equal rank s. Furthermore, all eigenvalues are assumed to be different. Under these conditions, similarity transformations S and S are possible ... 

The Jacobi matrix remains a triangular matrix. The only difference from eq. (2.88) is given by a change in the rate constant which amounts to k22 =-( 2 + 3)- concentration-time curve is given in Fig. 2.8. The eigenvalues are the elements of the diagonal. By use of the final values calculated in Example 2.20 in Section 2.2.1.3 one finds the following time functions ... 

If the absorption coefficients in a quasi-linear photoreaction are not known, the rank of the Jacobi matrix, the eigenvalues of the reactions and the final absorbances can be calculated. Furthermore the initial values of the absorbances and their derivatives with respect to time can be determined. Any additional statements require additional information since for linear systems the following theorem is valid [15] ... 

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