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
See more
Exercise Gauss Seidel - SOR Excel
Direct Methods for Solving Linear Equations Systems
Matrix Notation
A matrix is a rectangular array of elements represented by a single simbols. A set of horizontal elements called line ( or row ) and vertical , column.
The First subscript i always designates the number of the row in which is the element. The Second subscript j designates the column. ViewPDF
Matrix Operating Rules
Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are :
A matrix is a rectangular array of elements represented by a single simbols. A set of horizontal elements called line ( or row ) and vertical , column.
The First subscript i always designates the number of the row in which is the element. The Second subscript j designates the column. ViewPDF
Matrix Operating Rules
Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are :
cij= aij + bij
Similarly, the Substraction of two matrices, say, [E] minus [F] is obtain by subtracting corresponding terms:
dij= eij-fij
The addition and Subtraction can be performed only between matrices having the same dimensions.
Both Addition and subtraction are Conmutative:
[A] + [B] = [B] + [A]
Addition and Subtraction are also associative, that is,
([A] + [B]) + [C] = [A] + ([B] +[C])
NAIVE GAUSS ELIMINATION
Elimination of unknows
The Procedure:
1. The Ecuations were manipulated to eliminate one of the unknows from the ecuations.The result of this elimination step was that we had one equation with one unknown.
2. Consequently, this equation could be solved directly and the result back- substituted into one of the original equations to solve for the reaming unknown.
This basic approuch can be extended to large sets of equation by developing a systematic scheme or algorithm to eliminate unknows and to back- sustitute. Gauss Elimination is the most basic of these schemes ( The first phase is designed to reduce the set of equations to an upper triangular system).
PITFALLS OF ELIMINATION METHODS
Some pitfalls:
* Division by Zero. The primary reason that the foregoing technique is called " naive" is that
during both the elimination and the back-substitution phases, it is possible that a division by zero can occur.
* ill-Conditioned Systems. Well conditional systems are those where a small change in one or more of the coefficients results in a similar small change in the solution. Ill- Conditional systems are those where small changes in coefficients result in large changes in the solution.
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