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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 …

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 …

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 …

You haven't said whether $K$, $\Sigma$, or $L_{1}$ are sparse or otherwise specially structured (e.g. Toeplitz.) This could be hugely important in finding the most efficient way to do this. You als …

Groebner basis methods have already been mentioned as an approach to exactly solving this kind of system of equations. They aren't a way to find the best least squares solution in a case where there …

The variance of the mean of $n$ random variables is $\mbox{Var}(\bar{x})=\mbox{Var}(\sum_{i=1}^{n} x_{i}/n)$ $\mbox{Var}(\bar{x})=\sum_{i=1}^{n} (1/n)^{2} \mbox{Var}(x_{i})$ $\mbox{Var}(\bar{x})=n …

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 …

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. …

I'm assuming that by $\| A^{1/2}b-z \|$, you're referring to the 2-norm and that by $A^{1/2}$, you're referring to the unique symmetric matrix square root of $A$. If you can precompute $A^{1/2}z$, …

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 …

A lot depends on $\hat{\eta}(\xi)$. When you convolve this with $\hat{u}$ and $l(\xi)$, you will lose the sparsity if $\hat{\eta}(\xi)$ has broad support. Just how much do you know about the spect …

In using least squares, you normally want the residuals from the various equations to be indpendent of each other. If your $q_{i}$ vectors are imprecise measurements, then this will introduce correla …

The real issue here is the constraint $\sum_{i} x_{i}1_{x_{i}>a} < b $ whose left hand side has horrible discontinuities. Rather than using a solver designed for problems with continuous variable …

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 …

Unfortunately you can't. With any orthogonal factorization (e.g. QR, LQ, or SVD) you have the problem that because some of the columns of the orthogonal matrix have to span a particular subspace, and …