Suppose I have a stochastic matrix $M$ (with thousands or millions of stochastic column vectors), which I split into two matrices: $D$ containing only the diagonal entries of $M$, and $R$ containing the remaining entries of $M$. What would be a fast way to compute $D(I-R)^{-1}$ (or a good approximation)? Instead of low computational complexity, I'm looking for fast practical performance. Thanks. Michelle
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1$\begingroup$ probably you care about the operation $Mx$ or $M^{-1}x$, rather than the inverse itself, because if $M$ is huge, how do you store it? suppose it is sparse, then its inverse will be almost always dense, so a decent approximation might end up being dense too, and thus difficult to store. however, the crude approximation $(I-R)^{-1}=\sum_k R^k$ can be "quickly" applied to a vector $x$... $\endgroup$– SuvritCommented Jul 12, 2012 at 9:21
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2$\begingroup$ I suggest asking on scicomp.stackexchange.com. $\endgroup$– David KetchesonCommented Jul 12, 2012 at 10:16
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1$\begingroup$ In general, so-called "black-box" linear algebra techniques offer very good theoretical and practical performance for sparse or structured matrix operations (which I presume covers your problem). The basic idea is to treat matrix multiplication as an oracle: in the sparse or structured case, this will have subquadratic complexity and can be leveraged to accelerate more complex operations. You can combine this thinking with Krylov methods. $\endgroup$– Steve HuntsmanCommented Jul 12, 2012 at 14:40
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$\begingroup$ Suvrit, Thanks. You're right. M is sparse, as it represents some links in a web graph. However, instead of $M^{-1}x$, I care about $D(I-R)^{-1}x$ for a continuous stream of $x$'s, though $M$ may also change (both nodes and edges), but less often. Looks similar to PageRank, but still quite different. Can we do better than this crude approximation? In either case, what would be a fast implementation? Thanks. - Michelle $\endgroup$– Michelle DollyCommented Jul 12, 2012 at 14:51
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$\begingroup$ as the crudest baseline, i'd approximately solve the linear system $(I-R)z=x$ for each $x$ that comes in. If the $x$'s are not too different from each other, you can "warm-start" your linear system solvers (GMRES, PCG, etc.), and run those solvers to only a few iterations to save time---for a large sparse problem, this should give a reasonable approximation. $\endgroup$– SuvritCommented Jul 12, 2012 at 16:54
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