Gauss-Seidel iteration

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 

The Gauss-Seidel iteration makes use of the difference equations expressed in the form of Eq. (3-32) through the following procedure. 

An initial set of values for the F,is assumed. This initial assumption can be obtained through any expedient method. For a large number of nodes to be solved on a computer the 7, s are frequently assigned a zero value to start the calculation. 

the new values of the nodal temperatures 7, are calculated according to Eq. (3-32), always using the most recent values of the Tj. 

The process is repeated until successive calculations differ by a sufficiently small amount. In terms of a computer program, this means that a test will be inserted to stop the calculations when 

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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. 

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. 

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). 

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. 

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.

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

Physical situation (second equation in situation is in form for Gauss-Seidel iteration)... 

We choose to solve the set of equations by the Gauss-Seidel iteration technique and thus write them in the form 7, = /(7 ). The solution was set up on a computer with all initial values for the 7, s taken as zero. The results of the computations are shown in the following table. 

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

From a practical point of view, a Gauss-Seidel iteration scheme may be the most efficient numerical procedure to follow in solving the set of equations for the J/ s. For the Gauss-Seidel scheme the above equations must be organized in explicit form for Jh Solving for J, in Eq. (8-107) and breaking out the Fu term gives... 

Solve the equations for the J, s. If a Gauss-Seidel iteration is performed, use the following steps ... 

Solution of the differential equations was by Gauss-Seidel iteration (with the fluid property values given in Table II) on an IBM 370 digital computer using implicit difference equations of the Crank-Nicholson type. The program was convergent and stable for all conditions tested. 

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 ... 

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... 

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