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Iterative methods

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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.

 \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.

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