Iteratives Methods

GAUSS SEIDEL
The Gauss- Seidel method is the most commonly used iterative method.Assume that we are given a set of n equations:

[A]{X}={B}

Suppose that for concisensess we limit ourselves to a 3x3 set of equations. If the diagonal elements are all nonzero, the first equation can be solved for x1, the second for x2, and the third for x3. Now, we can start the solution prcess by choosing guesses for the x's. A simple way to obtain initial guesses is to assume that they are all zero. these zeros can be susbtituid into first equation, which can be used to calculate a new value for x1. Then, we substitute this new value for x1 along with the previous guess of zero for x3 into next equation to compute a new value for x2. The Process is repeated to calculate a new estimate for x3. Then we return to the firs equation an repeat the entire procedure until our solution converges closely enough to the true values.

As each new x value is computed for the Gauss- Seidel method it is immediately used in the next equation to determine another x value. Thus, if the solution is converging, the best available estimates will be employed. An alternative approach, called JACOBI ITERATION ( Excel) utilizes a somewhat different tactic. Rather than using the latest available x's this technique estimates a set of new x's on the basis of a set of old x's. Thus, as new values are generated, they are not inmediately used but rather are retained for the next iteration.

Improvement of convergence using relaxation
Relaxation represents a slight modification of the Gauss- Seidel method and is designed to enhance convergence. After each new value of x is computed, that value is modified by a weighted average of the results of the previous and the present iterations:

Xi(new)= w*xi(new) + (1-w) xi(old)
where w is a weighting factor that is assigned a value between 0 and 2.
If w=1, (1-w) is equal to 0 and the result is unmodified. However, if w is set at a value between 0 and 1, the result is a weighted average of the present and the previous results. This type of modification is called underrelaxation. It is typically employed to make a nonconvergent system or to hasten convergence by dampening out oscilations.
For values of w from 1 to 2, extra weight is placed on the present value. In this instance, there is an implicit assumption that the new value is moving int he correct direction toward the true solution but at too slow a rate. Thus, the added weight of w is intented to improve the estimate by pushing it closer to the truth.Hence, this type of modification, which is called overrelaxation, is designed to accelarate the convergence of an already convergent system. The approach is also called successive or simultaneous overrelaxation, or SOR.

Graphical depiction

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Exercise Gauss Seidel - SOR Excel