Introduction for (numerical) linear algebra of random variables

I am in search of an introduction into numerical linear algebra - or, at least, pure linear algebra - that treats the case when the input data are random variables.

A typical application would be to derive an estimate for the expected numerical error when the input variables are subject to random noise. Note that this neither implies that algorithms has access to a perfect coin, nor that the errors in inexact arithmetic obey any probability distribution.

• Ouch, how do you know errors do not obey any probability distribution? Have you plotted the errors? Mostly these books are called "Introductory Statistics"... There's also the good ol' Nicholas J. Higham "Accuracy and stability of numerical algorithms" (SIAM, 2002). – Deer Hunter Jan 19 '13 at 19:36
• I'm sure your arithmetic obeys some probability distribution. You may simply not be able to say which one by giving a particular formula for it. But it would likely be obtainable experimentally, and you can give it a symbol, compute mean values and other statistics as necessary. – Wolfgang Bangerth Jan 20 '13 at 17:05

Some simple cases are quite easy. For example, if $x$ has a multivariate normal distribution, a matrix $A$ is known exactly, and $y=Ax$, then $y$ also has a multivariate normal distribution, with $E[y]=AE[x]$, and $\mbox{cov}(y)=A\mbox{cov}(x)A^{T}$. If $y=f(X)$, where $f$ is nonlinear, then $y$ won't have a multivariate normal distrubiton, but as long as $f$ isn't strongly nonlinear and the uncertainty in $x$ is small, it may be possible to use a linear approximation to $f$ to get an approximate multivariate normal distribution for $y$.
A very general technique is Monte Carlo simulation. If $x$ is an uncertain input with known distribution, generate lots of random values of the $x$ vector, compute $y=f(x)$ for each of these random vectors, and then look at the empirical distribution of $y$. There's a whole literature on methods such as "importance sampling" for making this process more efficient.