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.
62
questions
0
votes
0
answers
38
views
Preconditioner Implementation with matrix-free methods (sparse iterative solvers)
How can I define preconditioners (SPILU,SPAI, etc.) for sparse iterative methods (TFQMR,GMRES,CGS, etc.) for the matrix-free left-hand side? I defined Ax=b using matrix-free A (with LinearOperator and ...
- 1
5
votes
2
answers
476
views
Solving a generalised eigenvalue problem with non-square matrices
I need to solve a generalised eigenvalue problem of the form
$$A\mathbf{x}=\lambda B \mathbf{x}$$
where $A$ and $B$ are $m \times n$ complex matrices, that are not symmetric with $m>n$.
I am aware ...
- 51
0
votes
1
answer
59
views
How to properly use ARPACK's dsaupd and dseupd?
In Rust, I am trying to solve an eigendecomposition problem through ARPACK. I made the following subroutine for this purpose:
...
- 273
2
votes
1
answer
69
views
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-...
- 2,695
10
votes
0
answers
169
views
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 ...
- 435
0
votes
1
answer
81
views
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 $...
- 127
4
votes
1
answer
103
views
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 ...
- 227
2
votes
0
answers
86
views
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 ...
- 23
1
vote
1
answer
191
views
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 $...
- 35
2
votes
2
answers
84
views
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 ...
- 21
1
vote
0
answers
53
views
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),...
- 1,077
2
votes
0
answers
92
views
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-...
- 21
1
vote
1
answer
117
views
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 ...
- 113
5
votes
2
answers
156
views
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 $B$...
4
votes
1
answer
142
views
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 $A$...
- 317