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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. |
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It isn't clear here whether you're really interested in just the effects of floating point round off error within a computation or the effects of uncertain/noisy inputs to a computation. In most practical situations, assuming that you've been reasonably careful in selecting an appropriate numerical method, and assuming that the problem you're trying to solve isn't inherently ill-conditioned/ill-posed, the effects of uncertain or noisy inputs to the model are much larger than the effects of floating point round off. There's a burgeoning field of "uncertainty quantification" which attempts to deal with this question. 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. Recently there has been a lot of interest in "polynomial chaos expansions", a non Monte Carlo approach to analyzing the uncertainty in differential equation models of dynamical systems. |
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