Propogation of error in a matrix inversion

I'm trying to find the deterministic error bounds for some parameters calculated through distance geometry. The equation can be simplified to the following form:

$\left[\begin{matrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{matrix}\right] = D^{-1}\left[\begin{matrix} 0\\0\\0\\1\end{matrix}\right]$

Where $D^{-1}$ is $4 \times 4$ matrix. $D$ contains some distances indirectly obtained from another set of distances. I am able to calculate an error for each element of $D$ and I would like to figure out how those errors affect the solution. The uncertainty of $x_4$ is of particular interest to me.

Some online searching lead me to the concept of a matrix condition number which would seem ideal for such purposes. However, the vector on the right hand side ($\left[\begin{matrix} 0 & 0 & 0 & 1\end{matrix}\right]^{T}$) is exact and will have no errors. As such I cannot see how I could use the condition number for $D$ to find uncertainties in $x$.

Another thing that might be worth noting is that the final column and the final row of D are also exact (no uncertainty) and they are equal to $\left[\begin{matrix} 0 & 0 & 0 & 1\end{matrix}\right]^{T}$ and $\left[\begin{matrix} 0 & 0 & 0 & 1\end{matrix}\right]$ respectively.

• Condition number of a matrix shows how much the error will amplify during calculations. You don't need the vector on the right hand side to have any errors, $D$ itself already introduces errors. If the errors of distance are additive (that is, each element is measured as true distance + error), then you should be able to use condition numbers to analyse – Artem Sobolev Mar 7 '15 at 9:30
• Hm, doesn't it follow immediately from the last paragraph that $x_4 = 1$? Then, indeed, $x_4$ should have no errors. – Artem Sobolev Mar 7 '15 at 9:35
• Finally, even if you have to measurement errors, there is precision loss induced by out digital floating points. Not every number can be represented exactly, which gives some (extremely) small error, roughly, for each number you calculate. If your condition number is extremely huge, you can suffer a major loss of precision even if you assume your numbers are exact. – Artem Sobolev Mar 7 '15 at 9:42
• The errors of my distance are additive, and their is a distinct error term for each element of $D$. So each element of $D^{-1}$ should also have an error. That is what I'm trying to find. Yet as you mentioned yourself, because of the exactness of the vector on the right, using a condition number would not let me determine an error in $x$. This cannot be correct though. This would mean no matter how inaccurate my measurement devices are, my final result will have no error. – somerandomdude Mar 8 '15 at 19:30
• No, I didn't say that having measured right hand vector exactly would give you exact result, neither it would prevent you from estimating errors introduced by uncertainty in $D$ using condition number. And, your numerical procedure could indeed come up with a noised solution for $x_4$. Yet, if you know the last row and column of $D$ exactly, you can (seemingly) write out an exact solution for $x_4$ with no uncertainty. – Artem Sobolev Mar 9 '15 at 0:45