Iterative methods
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(Difference between revisions)
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:<math> A\varepsilon ^{(k)} = r^{(k)} </math> <br> | :<math> A\varepsilon ^{(k)} = r^{(k)} </math> <br> | ||
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: <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. |
Revision as of 22:32, 17 September 2005
For solving a set of linear equations, we seek the solution to the problem:
After k iterations we obtain an approaximation to the solution as:
where is the residual after k iterations.
Defining:
as the difference between the exact and approaximate solution.
we obtain :
the purpose of iterations is to drive this residual to zero.
Stationary Iterative Methods
Iterative methods that can be expressed in the simple form:
When neither B nor c depend upon the iteration count (k), the iterative method is called stationary iterative method.