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I know that there are many methods that solve linear equations $Ax = b$ iteratively. But I'm curious if there's one that in each iteration tells us a bound on the solution. Specifically, assume $Ax^* = b$, and our iterations are $x^1, x^2, x^3, \ldots, x^i, \ldots, x^*$(the superscripted $i$ are counters of iterations). Can we from $x^i$ obtain that $l^i \leq x^* \leq u^i$ where $\leq$ should be understand element-wise, or $\|x^*\| \le \varepsilon$?

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Iterative solutions require an initial guess $x_0$. If the bounds on $x_0-x$ are known, then the bounds that you are asking for a calculable. Otherwise, I don't think it is possible. –  dexter04 Nov 18 '12 at 17:56

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