Iterative or approximated Methods

Iterative or approximated Methods----Iterative or approximated Methods----Iterative or approximated Methods----Iterative or approximated Methods----Iterative or approximated Methods

Two techniques similar to obtain the roots of a single equation are the iterative methods. The approaches consisted of guessing a value and then using a systematic method to obtain a refined estimated of the root.

Gauss-Seidel
Assume some equations set:

If the diagonal elements are all non zero, the equation can be solved for:

The process begin guesses x’s values, the easiest way to obtain initial guesses is to assume all x’s values in zero. The zeros are used in the first equation to obtain a new x1 value, them will be use in the second equation where x3 will be supposed (zero is the suggestion); with x1 and x2 values can be obtained x3. By the iterations, use the new x1, x2 and x3 to substitute the equations again until the solution perform:

For all i, where j and j-1 are the present and previous iterations.

Example:

5X-Y+Z=10
8Y+2X-Z=11
-X+Y+4Z=3



First, solve each of the equations:


For the second iteration, take the x1, x2, and x3 values from the first iteration and calculated the error as the solution converge on the solution:

Gauss-Seidel method is similar to the fixed-point iteration technique to solve the roots of a single equation so, this method have to problems to: sometimes the solution no converge and when it converge, the process is very slowly.

The Gauss-Seidel algorithm can be expressed as:

The partial derivates of these equations can be evaluated with respect to each of the unknowns as:

This can be substituted in the convergence fixed-point criteria to obtain:

This criterion is not necessary for convergence.
--Rremember, the convergence fixed-point criteria are:



RELAXATION-SOR (Simultaneous over-relaxation)

Relaxation is a modification of Gauss-Seidel method. The x values obtained in the Gauss-Seidel iteration equation are computed in the equation:

When λ is a weighting factor that is assigned a value between 0 and 2. * If λ 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 under-relaxation. It is typically employed to make a non-convergent system converge or to hasten convergence by dampening out oscillations. For values of λ 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 in the correct direction toward the true solution but at too slow rate. Thus, the added weight of λ is intended to improve the estimate by pushing it closer to the truth. Hence, this type of modification which is called over-relation is designed to accelerate the convergence of an already convergent system*.


REFERENCES
 CHAPRA, Steven C. y CANALE, Raymond P.: “Numerical Methods for engineers”. McGraw Hill, fifth edition, 2006.*
 www.lawebdelprogramador.com