# Questions tagged [numerical-linear-algebra]

{numerical-linear-algebra} questions involving algorithms for linear algebra computations.

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### Compute Frobenius inner product of two tensor-trains in terms of tensor contractions

Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...
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### Computational complexity in linear solvers [migrated]

I have recently been trying out methods of coding for solving systems of linear equations on Python. Of course, I first used the inbuilt function $\mathit{inv}$ under certain if-conditions to obtain ...
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### Action of square root of tridiagonal matrix product on vector

Assume nonsymmetric, tridiagonal matrices $A, B \in \mathbb{R}^{n\times n}$ (where $n$ is in the order of 1000) and $A, B, AB$ are diagonalizable and have positive eigenvalues. How do you efficiently ...
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### Obtain a sparse solution for a bad conditioned linear system with either or constraints

What is the best way to obtain a sparse solution for a linear system $\mathbf{A}\vec{x}=\vec{b}$ with $x_n \in \mathbb{R}$? The linear system is special, because I know that: some columns $\vec{c}_n$ ...
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### Sparse perturbation

Let $x, x_0\in\mathbb{R}^n$ be two vectors satisfying $$\frac{||x||_1}{||x||_2}\leq\frac{||x_0||_1}{||x_0||_2}.$$ $|| \cdot||_1$ and $|| \cdot||_2$ are the $\ell_1$ and $\ell_2$ norm in $\mathbb{R}^n$,...
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