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Questions tagged [sparse-matrices]

Use this tag for questions regarding sparse matrices, that is matrices with relatively few entries compared to their size. Related: [numerical-methods] and [numerical-linear-algebra].

Use this tag for sparse matrices, that is matrices with relatively few entries compared to their size.

198 questions
18
votes
3 answers

Inverse of sparse matrix is not generally sparse

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the inverse of the sparse matrix is not usually sparse and…
Sungmin
  • 505
18
votes
3 answers

On computing the largest eigenvalue of a very large sparse matrix

I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter w, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of the corresponding matrix. For small values of w,…
Gadi A
  • 19,839
10
votes
2 answers

Sparse basis for linear subspace

Suppose I have a linear subspace of some vector space, e.g., described as the column space of some big matrix. How would I algorithmically find a basis of that same subspace where the basis matrix is sparse, i.e., most entries in most basis vectors…
MvG
  • 44,006
9
votes
1 answer

Fast algorithm for approximating eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16} \times 2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather the approximate number of eigenvalues in,…
Dominik
  • 444
7
votes
1 answer

Sparse PCA vs Orthogonal Matching Pursuit

Can't wrap my head around the difference between Sparse PCA and OMP. Both try to find a sparse linear combination. Of course, the optimization criteria is different. In Sparse PCA we have: \begin{aligned} \max & x^{T} \Sigma x \\ \text { subject to…
Natan ZB
  • 95
7
votes
1 answer

In common tongue, what is the differences between sparse and dense matrices?

What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level. Obviously everyone knows about the dictionary definition of sparse and dense matrices (a definition based on…
AlanSTACK
  • 4,185
6
votes
1 answer

What does LSQR stand for

One of the most popular and efficient iterative methods to solve large sparse systems of equations in the least squares sense is LSQR. It is related to CGLS (Conjugate Gradient Least Squares) in that it has the same iterates (mathematically, not…
Jens Renders
  • 4,464
6
votes
1 answer

COLAMD matrix reordering algorithm in MATLAB

Background I'm dealing with some variable size square sparse matrices resulting from a FEM analysis, and my next step is optimizing the system solving in terms of speed. This is a visualization of some aspects of a typical matrix using MATLAB's…
Marcelo
  • 153
6
votes
1 answer

Fast way to Invert ADA' when D is a diagonal matrix that changes each iteration?

So I have a statistical learning algorithm in which D is a diagonal matrix that changes each iteration while A stays the same. I'm looking for a fast way to invert ADA' each iteration which ends up being a .9 million by .9 million sized matrix. A is…
Charles Hernandez
  • 61
5
votes
0 answers

Parallel rank computation of huge sparse matrices over the binary field

I am looking for a parallel algorithm to compute ranks of huge, sparse binary matrices over $\Bbb F_2$, say, $10^5 \times 10^5$ with $10^7$ ones in total. Currently, I am doing this by packing $64$ bits in a long and applying Gaussian elimination…
user1111929
  • 481
5
votes
1 answer

Sparse Cholesky decomposition of factorized matrix

I want the diagonal of a matrix $Y^TA^{-1}Y$ where $A=X^TX$ and $X$ is very sparse with dimensions ~1e6 x ~1e5 (so $A$ is 1e5 by 1e5). $Y$ is something like 1e5 by 1e4 (also sparse). Currently I'm explicitly calculating $A$ then getting its Cholesky…
daknowles
  • 153
  • 4
5
votes
1 answer

Is there an efficient way to solve a system of linear equations with an almost tridiagonal sparse matrix?

I have a problem where I need to solve a linear system of equation $Ax = b$ where the matrix A is almost tridiagonal, except for elements on the last two columns (see below). I need to solve such a system at each time step and the matrix changes…
Benji
  • 71
  • 4
5
votes
1 answer

Why does MINRES converge in 3 iterations on matrices of specific form?

The MINRES algorithm for solving $Ax = b$ for symmetric $A$ can be described as follows: The $k$-th iterate of the algorithm is $$x_k = \arg\min_{K_k(A)} \lVert Ax-b \rVert_2$$ where $K_k(A)=\text{span}\{A^ib\mid i < k\}$ is the $k$-th Krylov…
Jens Renders
  • 4,464
5
votes
2 answers

General formula for $f(n)$

Let for $n\geq 3, C_n$ denote the $(2n) \times (2n)$ matrix such that all entries along the diagonal are $2$, all entries along the sub- and super-diagonal are $1$, all entries along the antidiagonal are $1$, all entries along the diagonals…
james99
  • 121
5
votes
1 answer

Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A? I guess it has to be its Jordan normal form but I am not sure. Remarks: A matrix is sparser than other if it has less nonzero entries. Two…
Nacho Garcia Marco
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