Questions tagged [sparse-matrix]

Questions related to storage, assembly, operations, and other aspects of dealing with sparse matrices, for which only non-zero elements are stored. Questions that do not with sparse matrices directly, but other means of using sparsity should be tagged with [sparse-operator].

53 questions with no upvoted or accepted answers
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6
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293 views

Compute sparsity pattern of $A^2$

Suppose we have a sparse matrix $A$. Is there any way to compute just the sparsity pattern of $A^2 = A \cdot A$ (I do not actually need to know what exactly the nonzero value are) faster than to ...
5
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464 views

Symmetric sparse direct solvers in scipy

scipy.linalg.solve, in its newer versions, has a parameter assume_a that can be used to specify that the matrix $A$ is symmetric ...
5
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83 views

How to construct a eigensolver targeting a specific type of matrix

I need to diagonalize such kind of matrix during research: it's n-by-n, with it's upper-left (n-1)-by-(n-1) corner be diagonal while the nth row & column are dense. It's observed that the self ...
5
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0answers
435 views

Optimisation of matrix exponential

I have a 7000x7000 sparse matrix (scipy), which I want to exponentiate. I've tried using scipy.sparse.linalg.expm, which works quite well for smaller matrices (takes a few seconds for a 1000x1000 ...
5
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0answers
199 views

Preconditioning technique for large sparse non-hermitian matrix

I am attempting to solve a computational acoustics problem that involves solving an underlying sparse matrix. The size of the problem varies with grid size (3D) and fill-in's obviously make direct ...
5
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713 views

Sparse matrix format and sparse-matrix sparse-matrix multiplication

I'm having some performance problems with my code dealing with the multiplication of big sparse matrices (stiffness and aerodynamic influence coefficient matrices). Mainly I have to multiply such ...
5
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0answers
225 views

Adaptive mesh data structure for Fast Marching Method to overcome RAM limit

On an uniform mesh of positions in space $\ (x_i,y_j,z_k)$: $$\ x_i = x_0 + i\Delta x,\quad i=0,\ldots,n_x$$ $$\ y_j = y_0 + j\Delta y,\quad j=0,\ldots,n_y$$ $$\ z_k = z_0 + k\Delta z,\quad k=0,\...
3
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0answers
109 views

Invert a huge sparse operator;

please help me with this question, I want to invert a huge sparse (non-circulant) this below in a $Ax=y$ equation: $$(\lambda I+ \beta D+ \sigma C)x=y$$ where I is an Identity Matrix,D is a Diagonal ...
3
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70 views

Solving PDEs: What is the best way to deal with non-banded/dense jacobians?

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
3
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0answers
62 views

Does shift-invert method has invertibility issue?

Please note that I have nearly zero background on numerical methods. I understand the shift invert method as described in SciPy Tutorial The main argument of the above link is as follows. Suppose we ...
3
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1answer
99 views

Choice of iterative solver for a sparse asymmetric matrix with symmetric structure

I have a sparse $nxn$ matrix A with pretty interesting structure. It has a block structure with symmetric structure but asymmetric blocks. Expressed mathematically the block $A_{jk} = A_{kj}$ but $A_{...
3
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200 views

Compute Nullspace of Sparse Matrix

I am computing the nullspace of a sparse rectangular $m$ x $n$ matrix $A$, where $m$ << $n$. I do this by computing the QR decomposition of $A^T$ and extract the $n-m$ right-most columns of the ...
3
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0answers
293 views

What is the algorithm to convert an adjacency matrix to a block diagonal structure?

I'm a little at loss as to what my supervisor meant by the "hierarchical block decomposition" of a matrix, but the goal is to put a sparse symmetric adjacency matrix into a block diagonal structure to ...
3
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72 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 ...
3
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99 views

Why the MIRACLE of Lanczos/CG-like?

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
3
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217 views

Left eigenvectors using ARPACK

I'm trying to find both the dominant $k$ left and right eigenvectors, that is, $$V_L\mathcal{A} = \Lambda V_L\\ \mathcal{A}V_R = V_R\Lambda\\ V_LV_R = I_{k\times k}$$ $V_L$ being the $k\times N$ ...
3
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0answers
320 views

Efficient assembly of finite element matrix(coupled equations case)

I noticed this post, where spalloc and sparse are recommended for efficient assembly in Matlab. I personally use sparse ...
2
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0answers
59 views

Software for solving large systems of linear equations over gf(2)

What available solvers are there for linear equation solver over GF(2) (Boolean), capable of dealing with large sparse systems (in the 10k - 100k variables range)?
2
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0answers
76 views

How to determine the finite difference coefficient matrix in 2D with periodic BC?

I'm solving a PDE in matlab using ode15s, and since the spatial dimension is 2, and number of variables grow large very quickly, I need to supply the structure of ...
2
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0answers
102 views

What is the state-of-the-art in parallel sparse matrix and dense vector multiplication?

For sure, there has been many highly optimized library on this. But I am working on a matrix-free context since the problem size does not allow explicit storage of sparse matrix elements. I'd love ...
2
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0answers
24 views

Barrier algorithm Gurobi and interior-point quadprog; what kind of matrices can it handle the best (sparse or dense, large or small problems)?

I am trying to solve a QP problem. Does anybody know the differences between the interior-point-convex algorithm of quadprog and the barrier method of Gurobi in terms which kind of matrices can the ...
2
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0answers
101 views

Acceleration of matrix geometric series

Suppose we want to find $x$ such that: $$x=b+Ax$$ where $A$ is a large sparse square matrix with eigenvalues in the unit circle. There are two representations of the solution: 1) $$x=(I-A)^{-1}b,$$...
2
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0answers
302 views

How to reuse permutation-orderings within scipy's SuperLU-wrapper?

i'm solving sparse linear equations within scipy 0.18 which internally resorts to SuperLU (after umfpack got removed due to license-issues). Current, i'm doing a complete re-factorization in each ...
2
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0answers
70 views

Fortran solver for the Sparse LSE problem

I was wondering if there is a Fortran library that contains a solver for the Sparse LSE(linear equality-constrained least squares) problem $$ min_{x}\|Cx-d\|^2 \text{ subject to } Ax=b $$ where $A$ ...
2
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0answers
311 views

Sparse Linear Algebra vs Dense Linear Algebra

I am interested in a reference in the literature that discusses the performance of Dense Linear Algebra (blas routines) and dense linear algebra (sparse blas routines). I am interested in knowing for ...
2
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0answers
149 views

Solvers for stiff initial value ODEs with sparse Jacobian

What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite ...
2
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0answers
994 views

How to solve singular non symmetric poisson equation with Neumann boundary condtions?

I am trying to solve 2D Poisson equations with Neumann boundary conditions. When the mesh is uniform, Poisson equation is singular and symmetric, so the method listed in Null Space Projection for ...
2
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0answers
103 views

Preconditioner for large size hermitian eigenvalue problems

Basically I try to compute several smallest eigenvalues of some sparse 50k*50k eigenvalue problems using matlab. $$Ax = \lambda Bx$$ With matlab eigs, it's not as fast as I expected. So I tried some ...
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0answers
35 views

Tolerance is not satisfied in eigsh function

I am diagonalizing a sparse matrix of size $153600\times 153600$ via eigsh function. I did not set the tolerance to specific value, so it would be the default value ...
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0answers
48 views

Viable algorithms for efficiently solving a block matrix system with non-uniform sparsity structure

I am trying to use Newton's method to get a stationary solution for a system of equations of the following form: $$ \begin{Bmatrix} \frac{\partial x}{\partial t} \\ 0 \end{Bmatrix} = \begin{Bmatrix} f(...
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0answers
78 views

C.S.R method in finite element matrix assembly

I have solved the 2D Poisson equation using finite element method with simplex triangular element in MATLAB. First, I generated the triangular mesh using pdetool in ...
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0answers
51 views

Nonsymmetric permutations for LU factorisation of symmetric matrix

Let $A$ be a symmetric matrix. It is then well known that computing the LU factorisation of $PAP^T$ instead of $A$ for a suitably chosen permutation matrix $P$ can greatly reduce fill-in. My question ...
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0answers
80 views

performance comparison between PETSc and SLATE

We want to start a new project to solve a large-scale inverse problem (O(10^6) number of parameters) to invert for subsurface wave speeds. We will use FEM to solve forward and adjoint PDEs. In our ...
1
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0answers
60 views

Numerical methods. MDF (ILU) implementation

I am trying to implement Minimum Discarded Fill (MDF) Ordering algorithm for incomplete matrix factorization. The algorithm description is here on page 60 Preconditioning Techniques for a Newton–...
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0answers
109 views

understanding Domain Decomposition with example

I am new in Domain Decomposition method. I am started to solve $-\Delta u = f$ in $\Omega$ and $u = 0$ on $\partial\Omega$. From that I get in $\Omega _1$ $$\begin{bmatrix}4&-1\\-1&4\end{...
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0answers
84 views

Hybrid Ellpack-Itpack (ELL) + COO Sparse Matrix Representation decomposition threshold

Hybrid ELL-COO sparse matrix representation can be done as in the picture, I was looking intensively, however I couldn't find out what is the threshold of decomposing the original matrix into ELL part ...
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0answers
45 views

How to use matlab **fft** function or other fast methods to solve a convection-diffusion system quickly?

for a convection-diffusion equation with Dirichlet boundary conditions as follows: $$-u''+qu'=f$$ Using centered difference for $u''$ and $u'$, we get a linear system $$Ax=b$$where matrix $A$ is ...
1
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1answer
312 views

Robin Boundary Condition with Implicit Upwind - Finite Difference Method for 2D Convection-Diffusion Equation

I am trying to solve a problem with 2D Convection-Diffusion equation with U = Concentration ($mg/m^{2}$) using Implicit Upwind Finite Difference Method like this $$ \frac{\partial U}{\partial t} + ...
1
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0answers
78 views

Fast calculation of $A^T B$

I need to compute a matrix-matrix product, $A^T B$, where $A$ is $n \times r$ sparse, and $B$ is $n \times q$ dense. The number of rows $n$ is far larger than both $r$ and $q$. In fact $n$ is so large ...
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0answers
496 views

Pseudoinverse of a large sparse matrix in r

This question was moved from Cross-Validated: https://stats.stackexchange.com/questions/274042/pseudoinverse-of-large-sparse-matrix-in-r I am trying to calculate the pseudoinverse of a large sparse ...
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0answers
60 views

Reduce large sparse linear operators to memory efficient loops?

I'm dealing a lot with large sparse linear operators these days and I'm quite new to them. A lot of the matrices I deal with originate with only a few unique integers, however, there are lots of them. ...
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0answers
255 views

How to reshape matrix into row-major order for MKL DSS?

I would like to use MKL to solve a sparse linear system. I chose the DSS (Direct Sparse Solver) interface, which implements the following steps: ...
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0answers
209 views

All eigenpairs of large sparse symmetric matrix

In advance I am sorry for my noobish question. I am a physics PHD student and basically I use python for my math/physics problems. But now I have a problem which requires more computing capacity and ...
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0answers
145 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. ...
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0answers
650 views

Implementation of a direct solver in Fortran 90?

My question may be elementary, but it is quite essential as I am getting confused. Here I am supposed to solve the following equation: $Ax=B$ From my understanding I have options of using either ...
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0answers
98 views

Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components

I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
1
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0answers
265 views

Markowitz Pivoting to reduce size of a dense integer system

I am dealing with a large sparse integer matrix that I need to find the nullspace of. I've seen Markowitz Pivoting come up in several places discussing similar problems such as here: http://www....
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0answers
71 views

How to make a directed graph symmetric?

Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...
0
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0answers
73 views

Derivative-free ill-conditioned non-linear least squares

I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
0
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0answers
68 views

Fast algorithm for computing lower mode shapes and natural frequencies in MATLAB using sparse stiffness and mass matrices

I am looking for a fast algorithm for computing eigenvalues and eigenvectors from sparse stiffness and mass matrices in MATLAB. The eig(K, M) doesn't work with ...