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3

Another approach, which might be of interest to you is randomized sampling. This is of particular interest if you can quickly compute matrix-vector products $x\rightarrow Ax$ and $x\rightarrow A^* x$. The core idea is to form a small sampling matrix $S = A\Omega$, where $\Omega$ is a Gaussian random matrix. If the sampling matrix is large enough, $S$ will ...

2

Usually, not always but usually, even if both $A$ and $x$ are sparse, $Ax$ is not. Even when it is, it is denser than $x$. If you consider something like $e^Ax$, which can be rewritten as $(I+A+A^2/2+\dots)x$, there is no guarantee that it will have the same sparsity pattern or the same number of nonzeros as $x$. Which means that either scipy would have to ...

0

You probably know that matrix norms can be defined by the vector norms in the following way: \begin{equation} ||A||:= \max_{x\neq 0} \frac{||Ax||}{||x||} \end{equation} for a matrix $A$. So you just look up the definition for the infinity or 2-norm and plug it into the expression above. When doing this you will further realize that the following equalities ...

2

I am not sure about your application -- and we say the $L^2$ norm of a function and not a system. But for simplicity I will explain the concepts for real valued functions. Consider an open domain $\Omega$ and a function $f:\Omega \to \mathbb{R}$. We say that $f \in L^2(\Omega)$ if $||f||_{L^2(\Omega)} < \infty$ where \begin{equation} ||f||^2_{L^2(\Omega)} ...

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