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Questions tagged [matrix-factorization]

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2
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0answers
60 views

Numerical analysis, pivoting and incomplete LU decomposition

When doing LU decomposition, the algorithm will break down if any of the diagonal element $x_{ii}$ is zero. Therefore, we can use pivoting on the matrix such that $x_{ii}$ is no longer zero. That is ...
4
votes
1answer
85 views

Numerically find the nearest positive semi definite matrix to a symmetric matrix

I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
2
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0answers
89 views

Factorize laplacian in terms of first derivative matrix

I am trying to factorize the following Laplacian matrix in terms of $ D^TD$, D is the first derivative matrix. The tridiagonal form of the secon derivative matrix using Neumann boundary condition is ...
-4
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1answer
84 views

what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works ...
3
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2answers
118 views

Factoring the sum of two matrices

Given \begin{equation} A_i=B+C_i \end{equation} where $A_i$,$B$ and $C_i$, $i=1,\dotsc,N$ are large square matrices, $B$ is symmetric, $C_i$ are zero matrices aside for a square block on the diagonal. ...
4
votes
2answers
98 views

Term for the typical “linear in the larger dimension, quadratic in the smaller” cost for linear algebra

Many dense linear algebra decompositions (QR, SVD...) on an $m\times n$ matrix have cost $$ O(\max(m,n)\min(m,n)^2) $$ when implemented in practice on a computer. Is there a colloquial name or a more ...
3
votes
0answers
34 views

Reweighted least squares factorization

This is a continuation of the question asked here. I want to solve numerous least squares systems of the form $$ D_i A x \approx D_i b $$ where $D_i$ are $m \times m$ diagonal matrices with positive ...
1
vote
1answer
188 views

Ways to solve $Ax=b$ for a sparse (banded) $A$ with updates

I want to solve the time-dependent Schrodinger Equation using the Crank-Nicolson scheme. I end up with the following matrix equation ...
6
votes
2answers
240 views

inertia count sparse matrix with dense low-rank perturbation

I would like to determine the number of negative eigenvalues (inertia count) of the $(N \times N)$ symmetric real matrix $K - \sigma M$, with $K$ a positive-definite sparse matrix and $M$ a positive-...
7
votes
1answer
165 views

Least Squares with Dense-Block Diagonal Structure

I need to solve a least squares problem that takes the following form: $$p = \arg \min_{x}\Vert J V x - y \Vert_2, $$ where $J \in \mathbb{R}^{N \times N}$ is a general dense matrix, and $V \in \...
2
votes
1answer
101 views

How we can use CUR decomposition in place of SVD decomposition?

I have understood how CUR and SVD works, but have not been able to understand the following. How can we use CUR in place of the SVD decomposition? Do the $C$ and $R$ matrices in the CUR follow the ...
1
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0answers
58 views

Stable Method of orthogonal projection onto a subspace with the help of Moore-Penrose inverse,

Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v $. Below is ...
0
votes
1answer
75 views

Matrix factorization empty rows and columns

I would like to do non negative mf and I wanna ask a question about my main matrix. The question is should I include rows and columns that have no non-zero entry in them. I think if there is not a ...
1
vote
1answer
110 views

Supernodal vs multifrontal factorizations

What are the differences between supernodal and multifrontal matrix factorizations? Can you provide a few references or high-level points about the approaches?
13
votes
4answers
234 views

Rapidly determining whether or not a dense matrix is of low rank

In a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, ...
1
vote
1answer
143 views

Matrix Decomposition of Conics

I was reading about ellipse-ellipse intersection and I came across this article: https://math.stackexchange.com/questions/679622/intersection-between-conic-and-line-in-homogeneous-space/867428#...
2
votes
1answer
290 views

Fastest way to solve a sparse unsymmetric system many times

I have to solve a system $Ax^{(n)} = b^{(n)}$ many times, $A$ being a sparse (pentadiagonal in most part of its structure), unsymmetric, constant matrix. Currently, I am performing the LU ...
1
vote
1answer
37 views

Householder QR for abstract vectors

Assume I have a sequence of vectors $v_k \in V$ in some abstract vector space $V$ not necessarily equal to $\mathbb{C}^n$. Can I still use the Householder QR decomposition in this case, even though ...
1
vote
0answers
109 views

Updating factorization of Laplacian (add/remove edges)

For a graph $G=(V,E)$, recall that the unweighted Laplacian is $L:=D^\top D$, where $D\in\{-1,0,1\}^{|E|\times|V|}$ is the graph "gradient" operator that subtracts adjacent vertex values onto edges. ...
0
votes
1answer
78 views

$LU$ Factorization of a nonsingular matrix with a particular pattern

Consider $S\in\mathbb{R}^{n\times n}$ whose nonzero elements have the following pattern for $n = 8$: $$\begin{pmatrix} 1 & 0 & 0 & 0 & \mu_1 & 0 & 0 & 0\\ 0 & 1 &...
6
votes
2answers
257 views

Solving $A=B+AB$ without matrix inverse

I have a linear system of equations that can be expressed $$A=B+AB,$$ where $A$ and $B$ are real, symmetric matrices. I would like to solve for $A$ given $B$. At present, I solve for $A$ directly via $...
10
votes
1answer
222 views

The fast, and The Backward-Stable (left) $3\times 3$ matrix inverse

I need to compute a lot of $3\times3$ matrix inverses (for Newton iteration polar decomposition), with very small number of degenerate cases ($<0.1\%$). Explicit inverse (via matrix minors divided ...
5
votes
1answer
332 views

Code to update dense QR and Cholesky factorizations

I am looking for some production-ready code to update dense QR and/or Cholesky factorizations (by adding / removing rows and columns or making small-rank updates -- yes, I need all these cases). I ...
24
votes
10answers
6k views

Robust algorithm for $2 \times 2$ SVD

What is a simple algorithm for computing the SVD of $2 \times 2$ matrices? Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...