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Linear systems of 10,000 equations in 10,000 unknowns can easily be solved in a few seconds using double precision floating point arithmetic on typical consumer grade PC's and even laptop computers. …

Being able to get elements of the matrix isn't very useful (particularly if you don't know where the nonzero elements of the matrix are without checking.) Iterative methods can be useful if you hav …

Yes, this is more or less true. See for example Timothy Davis's book, "Direct Methods for Sparse Linear Systems." You can work out in advance the locations of all possible non-zero entries in $L$ …

This is hopeless without further assumptions, because no numerical procedure with a finite number of function evaluations can ever rule out a lack of injectivity in parts of the domain where the funct …

You're really asking the wrong question here... Let's back up a bit. You're attempting to estimate some parameters here, either by a maximum likelihood method or more likely by $\chi^2$ minimization …

Computing $L_{*}^{-1}$ from $L_{*}$ takes only $O(n^2)$ time. If you can afford the two matrix-matrix multiplications (which are $O(n^3)$ but parallelize and use cache very efficiently), then that …

It's worth pointing out that many inverse problems in the functional analytic setting go beyond ill-conditioning to ill-posedness. That is, a small change in the data (noise) can lead to an aribtrari …

You haven't really told us much about the problem. Are you working in conventional single or double precision floating point arithmetic, or are you working in some obscure field? Are the element …

Now that you've provided some more information, I think I can make some useful suggestions. First, a quick review of linear transformations of multivariate normal random vectors. If $z$ is an MVN …

There are a number of widely used primal--dual interior point codes for SDP, including SeDuMi, SDPA, SDPT3, and CSDP. Of these, SeDuMi approaches this problem by using the self dual embedding, while …

In practice, on typical desktop computers and server class machines using the x86-64 architecture, matrix-vector multiplication is limited more by memory bandwidth than floating point operations. Thi …

You haven't said so, but I'm assuming that $\psi$ and $\phi$ are vectors. These could more generally be functions in some function space, and you would typically discretize those functions to work wi …

If $Ax=b$ has a unique solution $x^{*}$, then $x_{m}=x^{*}$. If $Ax=b$ has infinitely many solutions, then $x_{m}$ will be one of these solutions. In particular, it will be the solution with the s …

The connection between random walks, diffusion, and the heat equation is an amazing example of "the unreasonable effectiveness of mathematics." However, it's important to understand that this doesn't …

I agree that this is a better question for scicomp.stackexchange.com. Maybe. It depends on the sparsity structure of your particular matrix and the actual numerical values of the nonzero elements …