Inverse iteration

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

Tan, T.B. and J.P. Letkeman, "Application of D4 Ordering and Minimization in an Effective Partial Matrix Inverse Iterative Method", paper SPE 10493 presented at the 1982 SPE Symposium on Reservoir Siumulation, San Antonio, TX (1982). 

The diagonal elements of the matrix A are af1 and the off-diagonal elements of Aij are Ty. Equation (9-21) determines how the dipoles are coupled to the static electric field. There are three major methods to determine the dipoles matrix inversion, iterative methods and predictive methods. 

The determination of eigenvalues and eigenvectors of the matrix A is based on a routine by Grad and Brebner (1968). The matrix is first scaled by a sequence of similarity transformations and then normalized to have the Euclidian norm equal to one. The matrix is reduced to an upper Hessenberg form by Householder s method. Then the QR double-step iterative process is performed on the Hessenberg matrix to compute the eigenvalues. The eigenvectors are obtained by inverse iteration. 

Einding the inducible dipoles requires a self-consistent method, because the field that each dipole feels depends on all of the other induced dipoles. There exist three methods for determining the dipoles matrix inversion, iterative methods, and predictive methods. We describe each of these in turn. 

A defining feature of the models discussed in the previous section, regardless of whether they are implemented via matrix inversion, iterative techniques, or predictive methods, is that they all treat the polarization response in each polarizable center using point dipoles. An alternative approach is to model the polarizable centers using dipoles of finite length, represented by a pair of point charges. A variety of different models of polarizability have used this approach, but especially noteworthy are the shell models frequently used in simulations of solid-state ionic materials. 

In most electronegativity equalization models, if the energy is quadratic in the charges (as in Eq. [36]), the minimization condition (Eq. [41]) leads to a coupled set of linear equations for the charges. As with the polarizable point dipole and shell models, solving for the charges can be done by matrix inversion, iteration, or extended Lagrangian methods. 

A powerful tool for EM modeling and inversion is the integral equation (IE) method and the corresponding linear and nonlinear approximations, introduced in the previous chapter. One important advantage which the IE method has over the finite difference (FD) and finite element (FE) methods is its greater suitability for inversion. Integral equation formulation readily contains a sensitivity matrix, which can be recomputed at each inversion iteration at little expense. With finite differences, however, this matrix has to be established anew on each iteration at a cost at least equal to the cost of the full forward simulation. 

We denote by the element of the vector corresponding to the r, -th receiver position. It can be treated as an electric field, generated by an electric source Sae, located in the cell Vq. The Frechet derivative matrix, F, is formed by the components 6ej /6a. Therefore, the direct, brute force method of computing the Frechet matrix would require Nm forward modeling solutions for each inversion iteration. 

Huthnance, J. M., 1978. On coastal trapped waves analysis and numerical calculation by inverse iteration. Journal of Physical Oceanography, 8, 74-92. 

This correction can be inserted into the Boltzmann-inversion iterations to adjust the pressure to the target value. 

Select ji = 300 and implement the inverse iteration equations to obtain Table P10.31a. 

This is efficiently performed by block inverse iteration [4]. The continuum states so obtained are then normalized, fitting the asymptotic part to the regular and irregular Coulomb wavefunctions. 

In connection with NFC method in its matrix-block form, it is noteworthy that the inverse iteration technique enables one to com-... 

In Section 4.2.2, dealing with the simple example of disordered finite hydrogen rings, the wave functions were calculated directly at the Hartree-Fock level and so their localization properties could be determined by simple inspection. The same is true for states determined by much larger chains with the aid of the inverse iteration technique (see Section 4.4.2). 

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