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Error Estimate for Linear Systems

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If $ \tilde{x}$ is a numerical solution of a non-singular linear system $ Ax=b$ and $ A \tilde{x}= \tilde{b}$, then the error $ \Delta x= \tilde{x}-x$ satisfies

   cond$\displaystyle (A)^{-1} \frac{\Vert\Delta b\Vert}{\Vert b\Vert} \le \frac{\Vert\Delta x\Vert}{\Vert x\Vert} \le$   cond$\displaystyle (A) \frac{\Vert\Delta b\Vert}{\Vert b\Vert} $

with cond$ (A) = \left\Vert A \right\Vert \left\Vert A^{-1} \right\Vert $ denoting the condition of the matrix $ A$.

Both inequalities are sharp.

Example:

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  automatically generated 3/ 8/2007