Questions tagged [linear-algebra]
Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.
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How to properly use ARAPCK's dsaupd and dseupd?
In rust, I am trying to solve an eigendecomposition problem through ARPACK. I made the following subroutine for this purpose:
...
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When does linear system have linearly growing singular values?
Suppose $W$ is a large matrix where $i$th smallest singular value grows as $O(i)$. What kind of matrix can $W$ be?
For instance, this appears to hold for random matrix with IID entries and for lower-...
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What's the most computationally efficient implementation of Kalman Filter
I know there are many formulations of the Kalman Filter. A few I can name are:
Classical Covariance Form
Informational Filter Form
Square-Root Form or Factor Form
But somehow it's hard for me to ...
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optimize this python code that involves matrix inversion
So I have this line of code that involves a matrix inversion
X = A @ B @ np.linalg.pinv(S)
$A$ is an $n$ by $n$ matrix, $B$ is an $n$ by $m$ matrix and $S$ is an $...
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How can I efficiently find an anti-symmetric generator of a special orthogonal matrix?
Given a special orthogonal matrix $O$ (i.e: $OO^T = 1$ and $\det(O) = 1$), I am trying to efficiently find a matrix $X$ such that $O = e^X$ and $X = -X^T$ using Python (NumPy & SciPy).
One obvious ...
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What is fastest method for finding the minimum and maximum eigenvalues of a (possibly very large) symmetric matrix?
What is the best way to find the extreme eigenvalues - in order to find the spectral radius - of a general real dense symmetric matrix?
Looking at similar questions e.g.: What's the most efficient ...
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Solving linear system of equations with constraints on unknowns
I would like to solve a system of linear equations $y=Uh$ for an unknown vector $h$, where I have a few constraints on some of the elements of $h$. The matrix $U$ is composed of a vector $u$ (length $...
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Getting singular matrices for lid driven cavity problem
I was trying to solve the lid driven cavity problem using the galerkin method with SUPG stabilization. I was using GMRES method as my solver and I am also getting a solution. And the solution looks ...
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Particular linear systems: sparse matrix + column
I am trying to understand a limitation in a routine in the interval arithmetic software Intlab. From matrices starting from a given size (in my particular problems),...
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An alternative to Levenberg–Marquardt algorithm
When trying to solve for a (over)determined non-linear least square method:
$$\underset{x}{\min}||f(x)||^2_2, f: \mathbb{R}^n \rightarrow \mathbb{R}^m, x\in \mathbb{R}^n, m\geq n$$
we use the Gauss-...
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On the calculation of the first m generalized eigenvectors
This is a classic generalized eigenvalue/eigenvector problem:
$$
A\,\vec{x}=\lambda\,B\,\vec{x}
$$
which, however, is characterized by:
$A,B$ are real, symmetric and positive definite matrices of ...
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Optimized Lanczos method for finding eigenvalues of $A \otimes B$
Recently my supervisor told me about an efficient way to calculate eigenvalues and eigenvectors of matrix $A \otimes B$ with $a_{1} \times a_{2}$ as dimensions of $A$ and $b_{1} \times b_{2}$ is of $B$...
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Products of the Householder matrices during QR decomposition
It is often said that there is no need to form the Householder matrix during QR decomposition, however I fail to see how to "manage" the product of $n$ Householder matrixes and the matrix $A$...
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Matrix Diagonalization and Computational Requirements
I have some questions about diagonalizing matrices. My interest lies in computing all eigenvalues of a given matrix. To avoid wasting time and improve my research efficiency, I want to understand the ...
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Is AMG supposed to work with discontinuous Galerkin discretizations?
As the question says, are algebraic multigrid methods well suited to be used as preconditioners for problems discretised with Discontinuous Galerkin methods (say $p=1$)?
I've always used AMG (actually,...
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