Methods iterative

When dealing with large sets of equations, especially if the coefficient matrix is sparse, the iterative methods provide an attractive option in getting the solution. In the iterative methods, an initial solution vector is assumed, and the process is iterated to reduce the error between the iterated solution and the exact solution x, where k is the iteration number. Since the exact solution is not known, the iteration process is stopped by using the difference Ax, = - 

The disadvantage of the iterative methods is that they may not provide a convergent solution. Diagonal dominance (Eqs. B.8 and B.9) is the sufficient condition for convergence. The stronger the diagonal dominance the fewer number of iterations required for the convergence. 

There are three commonly used iterative methods which we will briefly present here. They are Jacobi, Gauss-Seidel and the successive overrelaxation methods. 

Matrix M is chosen such that its inverse is easy to compute. One successively computes 

There are many iterative methods (Jacobi, Gauss—Seidel, successive overrelaxations, conjugate gradients, conjugate directions, etc.) characterized by various choices of the matrix M. However, very often the most successful iterative processes result from physico-chemical considerations and, hence, corresponding subroutines cannot be found in normal computer libraries. 

If this algorithm is convergent, it converges to the solution. Generally, computations are stopped when x and are sufficiently close 

In the case of intermolecular interactions between polar molecules Otto and Ladik proposed the so-called mutually consistent field (MCF) method [135-136]. They discussed various aspects of the MCF approach in a series of papers [137-139], and compared it with the conventional SCF supermolecule and perturbational calculations. 

The main advantage of the MCF method as compared to the supermolecule calculations is that the dimensionality of the variational calculation can be considerably reduced. Nevertheless a large portion of the intermolecular integrals are still 

Although this latter approximation was found to be unsatisfactory from a numerical aspect and it was later abandoned by the authors, it is interesting to discuss it briefly. It can be shown, that one can arrive to this Mulliken point charge approximation by introducing the following asymmetrical Mulliken integral approximation [140]  

The MCF model has been refined to take into account the exchange potential by a simplified local representation  

The MCF method was successfully applied, for example, to the study of hydration of the glycine z witter ion with 6 and 12 water molecules [138]. It has been concluded that mutual polarization effects were quite important in the first hydration shell, but presumably the water molecules in more distant hydration shells are not polarized appreciably by the solute therefore their effect can be represented by the electrostatic potential of the corresponding unperturbed charge distributions. 

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W.C. Chew and Y.M. Wang. Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method. IEEE Transaetions on Medical Imaging, 9, 1990. 

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... 

For large Cl calculations, the frill matrix is not fonned and stored in the computer s memory or on disk rather, direct CF methods [ ] identify and compute non-zero and inunediately add up contributions to the sum jCj. Iterative methods [, in which approximate values for the Cj coefficients are refined tlirough sequential application of to the preceding estimate of the vector, are employed to solve... 

An alternative is to use iterative methods. The simplest iterative teclniique for calculating bound state or resonances is to pick a random initial wavefimction vi/q(a ) and propagate it forward in time, producing a wavepacket ... 

When considering the construction of exactly symmetric schemes, we are obstructed by the requirement to find exactly symmetric approximations to exp(—ir/f/(2fi,)). But it is known [10], that the usual stepsize control mechanism destroys the reversibility of the discrete solution. Since we are applying this mechanism, we now may use approximations to exp —iTH/ 2h)) which are not precisely symmetric, i.e., we are free to take advantage of the superior efficiency of iterative methods for evaluating the matrix exponential. In the following, we will compare three different approaches. 

Zienkiewicz, O. C. et al, 1985. Iterative method for constrained and mixed approximation, an inexpensive improvement to f.e.m. performance. Comput. Methods Appl. Meek Eng. 51, 3-29. 

One of the most important methods of modem computation is solution by iteration. The method has been known for a very long time but has come into widespread use only with the modem computer. Normally, one uses iterative methods when ordinary analytical mathematical methods fail or are too time-consuming to be... 

The Gauss-Seidel Iterative Method. The Gauss-Seidel iterative method uses substitution in a way that is well suited to maehine eomputation and is quite easy to eode. One guesses a solution for xi in Eqs. (2-44)... 

The purpose of this projeet is to gain familiarity with the strengths and limitations of the Gauss-Seidel iterative method (program QGSEID) of solving simultaneous equations. 

As part of the same HMO method but with various approximations, appeared the calculations of Zahradnik and Koutecki (117). Vincent and Metzger (118), Vitry-Raymond and Metzger (119), Bonnier and Gelus (120), and Bonnier et al. (121). In 1966, Vincent et al. applied to thiazole the iterative methods restricted to the tt system in the following approximations w" (122), w (123). and P.P.P. (124, 123), This last method was later employed with different approximations and for various purposes by Chowdhury and Basu (125), J. Devanneaux and Labarre (126), Yoshida and Kobayashi (127). Witanowskiet et al. (128). and E. Corradi et al. (129). 

We consider penalized operator equations approximating variational inequalities. For equations with strongly monotonous operators we construct an iterative method, prove convergence of solutions, and obtain error estimates. 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

Kovtunenko V.A. (1993) An iterative methods for solving variational inequalities of the contact elastoplastic problem by the penalty method. Comp. Maths. Math. Phys. 33 (9), 1245-1249. 

Computer solutions entail setting up component equiUbrium and component mass and enthalpy balances around each theoretical stage and specifying the required design variables as well as solving the large number of simultaneous equations required. The expHcit solution to these equations remains too complex for present methods. Studies to solve the mathematical problem by algorithm or iterational methods have been successflil and, with a few exceptions, the most complex distillation problems can be solved. 

A theoretical, comparative study of the tautomerism of 56 five-membered heterocyclic rings announced in (76AHC(Sl)l) has appeared (81MI40402). The stabilities of the three forms for 5-pyrazolones, 5-pyrazolethiones and 5-aminopyrazoles have been calculated by a simple Hiickel o) iterative method. The relative energies and the substituent and solvent effects are in agreement with the experimental results. 

Kincaid, D. R., and D. M. Young. Survey of Iterative Methods, in Encyclopedia of Computer Science and Technology, Marcel Dekker, New York (1979). 

Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. 

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