Iterative methods

(Difference between revisions)

Revision as of 22:32, 17 September 2005

For solving a set of linear equations, we seek the solution to the problem:

$AX = Q$

After k iterations we obtain an approaximation to the solution as:

$Ax^{(k)} = Q - r^{(k)}$

where $r^{(k)}$ is the residual after k iterations.
Defining:

$\varepsilon ^{(k)} = x - x^{(k)}$

as the difference between the exact and approaximate solution.
we obtain :

$A\varepsilon ^{(k)} = r^{(k)}$

the purpose of iterations is to drive this residual to zero.

Iterative methods that can be expressed in the simple form:

$x^{(k)} = Bx^{(k)} + c$

When neither B nor c depend upon the iteration count (k), the iterative method is called stationary iterative method.

@@ Line 10: / Line 10: @@
 :<math> A\varepsilon ^{(k)}  = r^{(k)}  </math> <br>
 the purpose of iterations is to drive this residual to zero.
+===Stationary Iterative Methods===
+Iterative methods that can be expressed in the simple form: <br>
+:<math>
+x^{(k)}  = Bx^{(k)}  + c
+</math> <br>
+When  neither '''B''' nor '''c''' depend upon the iteration count (k), the iterative method is called stationary iterative method.