Gauss Seidel

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

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. 

Gaussian mode Gaussian plume model Gauss-Seidel Gauzes Gaylusite... 

If P = I, this is the Gauss-Seidel method. If > I, it is overrelaxation if P < I it is underrelaxation. The value of may be chosen empirically, 0 < P < 2, but it can be selected theoretically tor simple problems hke this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. 

In another important class of cases, the matrix A is positive definite. When this is so, both the Gauss-Seidel iteration and block relaxation converge, but the Jacobi iteration may or may not. 

The choice ya = ra is the method of steepest descent. If the ya are taken to be the vectors et in rotation the method turns out to be the Gauss-Seidel iteration. If each ya is taken to be that e, for which e ra is greatest, the method is the method of relaxation (often attributed to Southwell but actually known to Gauss). An alternative choice is the et for which the reduction Eq. (2-10) in norm is greatest. 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

The detailed 3D model of porous catalyst is solved in pseudo-steady state. A large set of non-linear algebraic equations is obtained after equidistant discretization of spatial derivatives. This set can be solved by the Gauss-Seidel iteration method (cf. Koci et al., 2007a). 

First, we have an abundancy of relaxation methods. Recall from chapter 3 that such a method is characterized by the subsequent variation of only a few parameters at a time, and that they demand an efficient bookkeeping of which entries of r to update, when the current subset of elements of d are varied. Of this class, the simplest methods in use are the Gauss-Seidel family of methods. Essentially only one element at a time gets updated. Let us simplify by using an algorithmic notation, where the iteration counter is dropped, and we use the replacement operator = instead of equalities ... 

There are iterative methods (e.g., Jacobi, Gauss-Seidel, Newton) whose purpose is simply to provide solutions for the steady-state equations, others (e.g., Euler and its improved versions) aim to give trajectories. Cycling will be felt as a disagreeable iteration artifact in the first case, as an indication of a probably cyclic trajectory in the second case. The relation between the behavior in a simple iteration method (e.g., Jacobi) and the real trajectory is interesting, if not simple. Consider, for instance, a simple negative loop comprising three inhibitory elements ... 

A Jacobi or Gauss-Seidel iteration on (6) will provide us with the coordinates of the steady state, or it will cycle indefinitely, depending on the slope of the functions. On the other hand, determining the trajectory by numerical integration of equation (5) will lead to a stable steady state or to a limit cycle depending on the slope of the functions. There is thus an obvious formal similarity between the two situations. However, the steepness corresponding to the transition from a punctual to a cyclic attractor is much smaller in the first case (in which the cyclic attractor is an iteration artifact) as in the second case (in which the cyclic attractor is close to the real trajectory). 

The situation is rather similar if one applies iterative methods to the Boolean description. As noticed by, for example, Robert39 and by Goles,40 Boolean iterations in parallel and in series correspond, respectively, to the Jacobi and Gauss-Seidel iterations used in the quantitative description. In the first case (Jacobi), from an initial Boolean state (ot0 Jo yo.. . . ) one computest the values of the functions a, b, c,. . . which are reintroduced, respectively, as ot,Pi-y,.. . , and so on in the second case (Gauss-Seidel), the new value of each variable is reintroduced in a defined (but arbitrary) order. 

In the Jacobi iteration, one introduces arbitrary initial values (say jcj, y, zi) of jc, y, z in the right-hand sides of equations (6) this provides (left side) a new set of values (jc2, v , z2) which are reintroduced in the right sides and so on. In the Gauss-Seidel iteration the new value of a variable is reintroduced in the next equation as soon as it has been computed from xi, y i, Zi one calculates x2, fronts, y i, Zi, one calculates y2, and so on. 

Note that the synchronous treatment of a Boolean system is in fact a Jacobi (parallel) iteration. Our treatment may also be considered a kind of iteration, but it is neither a Jacobi iteration (in which all commutations are synchronous) nor a Gauss-Seidel iteration (in which the commutations take place one at a time but in a predetermined, arbitrary, order). We consider all the successions of states implicitly contained in the state table which one is followed depends on the values of the delays. 

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

Note that the first two terms on the right-hand side of the preceding equation are from iteration n and the last two terms are from the current iteration. The Gauss-Seidel method... 

Gauss-Seidel iteration is faster than Jacobi, because it uses new information from the already improved points, i.e., the points to the left and below i, j... 

Figure 8.11 Schematic representation of the Gauss-Seidel iterative scheme.

SOR introduces a relaxation parameter, uj, into the iteration process. The correct selection of this parameter can improve the convergence up to 30 times when compared with Gauss-Seidel. SOR uses new information as well and starts as follows,... 

All three methods eventually arrived at the same result however, each used different number of iterations to achieve an accurate solution. Figure 8.13 shows a comparison between the convergence process between Jacobi and Gauss-Seidel. The error plotted in in the Fig. 8.13 was computed using,... 

Figure 8.13 Convergence for the Jacobi and Gauss-Seidel iterative solution schemes for the FD compression molding problem.

The optimization can be carried out by several methods of linear and nonlinear regression. The mathematical methods must be chosen with criteria to fit the calculation of the applied objective functions. The most widely applied methods of nonlinear regression can be separated into two categories methods with or without using partial derivatives of the objective function to the model parameters. The most widely employed nonderivative methods are zero order, such as the methods of direct search and the Simplex (Himmelblau, 1972). The most widely used derivative methods are first order, such as the method of indirect search, Gauss-Seidel or Newton, gradient method, and the Marquardt method. 

Iterative methods (like Gauss-Seidel, Successive over relaxation and conjugate gradient) have often been preferred to the... 

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. 

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