All Questions
16 questions with no upvoted or accepted answers
8
votes
0
answers
481
views
Problems where Conjugate gradient works much better than GMRES
I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
- 489
5
votes
0
answers
392
views
Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix
I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
- 500
5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 922
4
votes
0
answers
233
views
Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same
I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, i.e.,...
- 1,517
3
votes
0
answers
122
views
Preconditioners for $Ax=y$ that rely on hierarchical statistical modeling
Solving $Ax=y$ exactly can be done as:
fit a linear autoregressive model by treating rows of $A$ as data
apply this model to $A^T y$
Imperfect predictive model corresponds to an approximate inverse ...
- 2,252
3
votes
0
answers
130
views
Computing the norm of the columns of an implicitly defined matrix
I have an $n \times n$ matrix $M = \Sigma W$ where $\Sigma$ is diagonal and $W$ orthogonal. $W$ is implicitly defined, i.e. I can only perform matrix-vector products (but I also have access to $W^T$).
...
2
votes
0
answers
618
views
block diagonal approximation of (SPD) matrix
I am interested in approximating a symmetric matrix in a block diagonal form, i.e. compute just some entries of the matrix located in blocks around the diagonal. Are there any theoretical guarantees ...
- 335
2
votes
0
answers
184
views
Checking for error in conjugate gradient algorithm
What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
- 489
1
vote
0
answers
95
views
Vandermonde-type factorization of moment matrix?
Consider $n,d \in \mathbb{N}_{>0}$, there are many functions $y:\mathbb{N}^{n} \to \mathbb{R}$. Now for simplicity, we denote $y(\alpha)$ to be $y_{\alpha}$. Let $|\alpha| = \sum_{i=1}^{n}\alpha_{i}...
- 275
1
vote
0
answers
198
views
Complexity of singular value decomposition using matrix multiplication oracles
Suppose I have an $n\times m$ real matrix $A$, $n\ll m$ with full row rank $(\mathrm{rank}(A) = n)$. I have an oracle that can compute $Ax$ or $A^T y$ for any $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
- 662
1
vote
0
answers
125
views
smallest singular value over invertible sub-matrices
Consider the matrix $M = \begin{bmatrix} A & A B \end{bmatrix} \in R^{n \times (n+m)}$, with $A \in R^{n\times n}$, $B \in R^{n \times m}$, $m < n$, $m > 1$, $A$ symmetric positive definite.
...
- 303
0
votes
0
answers
67
views
Concentration of bilinear forms
This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
- 6,962
0
votes
0
answers
228
views
Decomposition of symmetric block matrix
I came across this question and got really interested about it. There, the OP asks whether is possible to decompose a $2n \times 2n$ block matrix:
$$ \begin{pmatrix}
X & I \\
I & Y
\end{...
0
votes
0
answers
57
views
Numerically finding matrix approximation by lower-dimensional "pseudo-similar" matrix
Consider an $N\times N$ (real or complex) matrix $A$, and some $n<N$. Is there a good numerical algorithm that finds the set consisting of an $n\times n$ matrix $B$, an $n\times N$ matrix $I$, and ...
- 3,001
0
votes
0
answers
159
views
How to solve a non-local self-consistent equation
I have been struggling lately with solving numerically an equation of the form:
$$ g(x\pm x_{0}) = F[ g(x) ] $$
where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...
- 123
0
votes
0
answers
83
views
Bits of precision matrix reconstruction
We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
- 13.9k