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Iterative methods

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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.
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===Stationary Iterative Methods===
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Iterative methods that can be expressed in the simple form: <br>
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:<math>
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x^{(k)}  = Bx^{(k)}  + c
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</math> <br>
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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.

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