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The question is:

Consider the linear system $\left( {\begin{array}{*{20}{c}} 1&\alpha \\ a&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right)$, what is the condition number for computing $x$.

I know the condition number of a matrix ($\kappa(A) = \left\Vert A^{-1} \right\Vert \cdot \left\Vert A \right\Vert$), but I don't know what does "condition number for computing $x$" mean. Anyone can help with this? Thank you!

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  • $\begingroup$ If you solve the system of $2$ equations in $2$ variables "$1x+ay=1$" and "$ax+1y=0$", then you get "$x=1/(a^2+1)$". $\endgroup$ Commented Feb 3, 2016 at 4:56
  • $\begingroup$ @barakmanos Thanks for reply. But what is the condition number? $\endgroup$
    – Tony
    Commented Feb 3, 2016 at 5:17

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In general, condition numbers measure the relative sensitity of a function to small relative changes in the argument. Here we have a linear system for which the solution is \begin{equation} \begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{1 - \alpha a} \begin{pmatrix} 1 & -\alpha \\ -a & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{equation} In particular, we have \begin{equation} x = \frac{1}{1 - a \alpha} \end{equation} which means that we must treat $x$ as function of two variables, rather than one. We are interested in sensitity of $x = x(a,\alpha)$ to changes in the arguments. By Taylor's formula, \begin{equation} \Delta x = x(a+\Delta a, \alpha + \Delta \alpha) - x(a,\alpha) \approx \frac{\partial x}{\partial a} \Delta a + \frac{\partial x}{\partial \alpha} \Delta \alpha \end{equation} In our case we have \begin{equation} \frac{\partial x}{\partial a} = \frac{ \alpha}{(1 - a \alpha)^2} = \alpha x^2 , \quad \quad \frac{\partial x}{\partial \alpha} = \frac{a}{(1 - a \alpha)^2} = ax^2 \end{equation} In situations such as this were there are no specific applications present, one is primarily interested in controling the relative error, rather than the error itself. Therefore we study $\frac{\Delta x}{x}$ in terms of the $\frac{\Delta a}{a}$, and $\frac{\Delta \alpha}{ \alpha}$. We have \begin{equation} \frac{\Delta x}{x} = a \alpha x \frac{\Delta a}{a} + \alpha a x \frac{\Delta \alpha}{\alpha} \end{equation} In this context, a reasonable first assumption is that $a$ and $\alpha$ have been entered into the computer directly and are not the result of a lengthy computation during which a significant error has accumulated. Therefore \begin{equation} \left| \frac{\Delta a}{a} \right| \leq u , \quad \text{and} \quad \left|\frac{\Delta \alpha}{\alpha}\right| \leq u, \end{equation} where $u$ is the unit round off error. It follows that \begin{equation} \left | \frac{\Delta x}{x} \right | \leq 2|a \alpha x | u \end{equation} from which we conclude that it is reasonable to identify \begin{equation} \kappa = 2|a \alpha x| \end{equation} as the condition number of $x = x(a,\alpha)$. We notice to our satisfaction that the condition number reflect the extreme sensitivity of $x$ in the vicinity of the singularity where $a \alpha = 1$.

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