# Iterative methods

(Difference between revisions)
 Revision as of 07:04, 16 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 22:32, 17 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 10: Line 10: :$A\varepsilon ^{(k)} = r^{(k)}$
:$A\varepsilon ^{(k)} = r^{(k)}$
the purpose of iterations is to drive this residual to zero. the purpose of iterations is to drive this residual to zero. + + ===Stationary Iterative Methods=== + 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.

## 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.

### Stationary Iterative Methods

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.