Jacobi eigenvalue problem

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) ... 

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. 

While the polynomial method can be used for solving small eigenvalue problems by hand, all computational implementations rely on iterative similarity transform methods for bringing the matrix to a diagonal form. The simplest of these is the Jacobi method, where a sequence of 2 x 2 rotations analogous to eqs (16.28)-(16.30) can be used to bring all the off-diagonal elements below a suitable threshold value. 

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. 

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. 

The optimum over-relaxation parameter is known only for a small class of linear problems and for select boundary conditions. The iteration matrix has eigenvalues each one of which reflects the factor by which the amplitude of an eigenmode of undesired residual is suppressed for each iterative step. Obviously the modulus of all these modes must be less than 1. The modulus of the factor with the largest amplitude is called the spectral radius and determines the overall long term convergence of the procedure for many iterative steps. If Pj is the spectral radius of the Jacobi iteration then the optimum value of X is known to be ... 

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