Gauss-Seidel Iteration Method

In the Jacobi method, the iterated vector of the (k + l)th iteration is obtained based entirely on the vector of the previous iteration, that is, The Gauss-Seidel iteration method is similar to the Jacobi method, except that the component for = 1, 2,1 are used immediately in the calculation of the component The iteration equation for the Gauss-Seidel 

Like the Jacobi method, the Gauss-Seidel method requires diagonal dominance for the convergence of iterated solutions. 

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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. 

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). 

Rework Prob. 3-43, using the Gauss-Seidel iteration method. 

Jacobi and Gauss-Seidel iterative methods are easy to implement in simple computer programs, but they can be slow to converge when the system of equations is large. Hence they are not considered suitable for CFD simulations. 

Solving a system of N linear equations using the Gauss-Seidel iterative method, we can rearrange the rows so that the diagonal elements have larger absolute value than the sum of the absolute values of the other coefficients in the same row. This is defined as... 

The Gauss-Seidel iterative method requires an initial approximation of the values of the unknowns X, to X. We use these values in Equation 1-69 to start calculation of new estimates of X s. Each newly calculated Xj replaces its previous value in subsequent calculations. The iteration continues until all the newly calculated X s converge to a... 

An initial guess of C, C 2 3 and C 4 = 0.5 is used to start the Gauss-Seidel iterative method. The method converges after five iterations to the solutions ... 

The quantity o> in (4.11) is called the relaxation factor. We observe that, for CO = 1, this iterative method reduces to the Gauss-Seidel iterative method of (4.8)-(4.8 ). For reasons of brevity, we shall say that a matrix G, which is cyclic of index 2, is consistently ordered [52] if it is the form of (4.10). With the concept of a consistent ordering. Young [52] established the following general relationship between the eigenvalues A of the successive overrelaxation matrix... 

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. 

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. 

When the number of nodes is very large, an iterative technique may frequently yield a more efficient solution to the nodal equations than a direct matrix inversion. One such method is called the Gauss-Seidel iteration and is applied in the following way. From Eq. (3-31) we may solve for the temperature T, in terms of the resistances and temperatures of the adjoining nodes 7 as... 

This means that now a new value of,+,, T cannot be calculated only from values at time i as in Eq. (5.4-2) but that all the new values of T at t -(- At at all points must be calculated simultaneously. To do this an equation is written similar to Eq. (5.4-26) for each of the internal points. Each of these equations and the boundary equations are linear algebraic equations. These then can be solved simultaneously by the standard methods used, such as the Gauss-Seidel iteration technique, matrix inversion technique, and so on (G1,K1). 

For an evaluation of the two methods one has to take into account the computational costs for the single iteration steps (3.4), (3.5). Using the Gauss-Seidel iteration the number of F- and G-iterations is the same whereas it differs within the lANM method. The latter is advantageous in our example since the discretization of (3.4) leads to a symmetric linear system of equations whereas the discretization of (3.5) leads to a non-symmetric one. Thus the costs for one G-iteration step are more than twice the costs for one F-iteration step. Finally, the expense for one 02-evaluation (2.4) cannot be neglected since it is at least half of the expense for one F-iteration. 

Gauss-Seidel iterative matrix and Eigenvalue method are used to calculate the relative weights of elements. The biggest eigenvalue is obtained from following equation [21],... 

In this study, the life cycle assessment of microwave hot air systems were developed using the analytic hierarchy process and the fuzzy comprehensive evaluation. The hierarchical structure consists of assessment aspect and assessment objective were established. The fuzzy assessment matrices of the assessment mathematical model were calculated using eigenvalue method and Gauss-Seidel iterative matrix. The life cycle assessment results show that the microwave hot air systems have a good green degree. 

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. 

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. 

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 ... 

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