ITERATIVE METHODS FOR SOLUTION OF SYSTEMS OF LINEAR EQUATIONS
In general, these methods are based on fixed point method and the process is iterated so substitute in a formula.
Iterative methods compared with direct, we do not guarantee a better approach, however, are more efficient when working with large matrices.
- JACOBI METHOD
The basis of the method is to construct a convergent sequence defined iteratively. The limit of this sequence is precisely the solution of the system. For practical purposes if the algorithm stops after a finite number of steps leads to an approximation of the value of x in the solution of the system.
- GAUSS SEIDEL METHOD
The methods of Gauss and Cholesky methods are part of direct or finite. After a number of operations nito, in the absence of errors rounding, we get x solution of the system Ax = b. The Gauss-Seidel method is part of the so-called indirect methods or iterative. They start with x0 = (x01, X02,:::; x0 n), an approximation initial solution. Since x0 is building a new approach of the solution, x1 = (x11, x12;:::; x1n).
From built x1 x2 (here the superscript indicates the iteration and does not indicate a power). So on construye fxkg a sequence of vectors, with the aim, not always guaranteed to quelimk! 1xk = x:Generally, indirect thods are a good option when the matrix is very large and dispersed or sparse (sparse), ie when the number of onzero elements is small compared to n2, total number of elements A.
In these cases you must use an appropriate data structure lets you store only the nonzero elements. In each iteration of the Gauss-Seidel method, there are n subiteraciones. In ca first subiteracion be amended only x1. The other coordinates x2, x3, ..., xn are not modified can.