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].
322
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Fast Fourier Transform on Meshes
I have a (closed, manifold, oriented) triangular mesh for which I build a matrix $L\in\mathbb{R}^{n\times n}$ discretising the negated Laplace-Beltrami operator. The matrix $L$ is symmetric positive ...
- 1,271
0
votes
1
answer
179
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Problems on the algebraic theory of sparse matrices
I have finished testing basic large densely parallel matrix multiplication on 4 gpu's ,and have done work on TSLU and TSQR on cpu's based on mpi. I am going to continue working on the theory of ...
0
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0
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129
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Right blocked linear equation solver on Dense Algebra and Sparse Algebra
I have implemented 1D mesh parallel QR decomposition and LU decomposition,I would like to ask if a linear equation Ax=b,b is a large matrix and I need to shard b or Shard A,b at the same time. Is ...
4
votes
1
answer
300
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How to efficient solve $e^{-tA} x =b$, where A is a very sparse matrix
I am going to solve an equation containing an exponential matrix $e^{tA} x =b$, which can be obtained naturally through $x=e^{-tA} b$. A is a 1million $\times$ 1 million matrix with stores 7.15 ...
- 81
1
vote
2
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125
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Are there good block sparse matrix solver libraries?
There are some great libraries with linear solvers for sparse matrices - SuiteSparse is the obvious one. The methods work on sparse matrices with scalar entries.
However, often in optimization ...
- 111
2
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0
answers
40
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Diagonalization of large sparse matrix, computational programme recommendation and methods
According to this link, All eigenpairs of large sparse symmetric matrix. The guy @Baranas seems to have given a very confident answer about solving the whole Eigen spectrum. May I know if anyone has ...
- 21
2
votes
2
answers
155
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Solving Poisson's equation without a Dirichlet boundary condition
Some context of what I am trying to do:
I am trying to implement a function that uses the heat method to calculate geodesic distances in a tetrahedral mesh, and I want to calculate the distances for ...
- 145
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101
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What is the difference between approximations of mixed derivative and how to implement it
currently I am solving 2D nonlinear second order differential equation containing mixed derivative. I started searching how to descretisize it and found two formulas for 4th order approximation. First ...
- 31
6
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1
answer
206
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Accelerating the computation of scipy.sparse.linalg.expm_multiply
I have a tridiagonal antiHermitian matrix ($-i*Hami*t$) with nonzero elements only along the upper diagonal and lower diagonal, and the goal is to know the action of exponential of such matrix on a ...
- 63
0
votes
1
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148
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How to efficiently fill in, in parallel, a PETSc matrix from a COO sparse matrix?
Considering the following COO sparse matrix format, with repeated indices:
...
1
vote
1
answer
147
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Crank Nicolson Method with closed boundary conditions
I want to simulate 1D diffusion with a constant diffusion coefficient using the Crank-Nicolson method.
$$\frac{\partial u (x,t)}{\partial t} = D \frac{\partial^2 u(x,t)}{\partial x^2}.$$
I take an ...
- 111
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0
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44
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Assembling a sparse matrix for the PaStiX solver
I've been searching the whole afternoon for some documentation or code sample showing how to assemble a sparse matrix of a problem to be solved with PaStiX, but couldn't find any. The relevant module ...
- 131
2
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1
answer
112
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How to numerically solve differential equations involving sines, cosines and inverses of the unknown function?
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find that the main idea is to ...
- 35
1
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0
answers
129
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Solving 2D Poisson equation with mixed boundary conditions in Python
I am trying to numerically solve the Poisson's equation
$$
u_{xx} + u_{yy} = - \cos(x) \quad \text{if} - \pi/2 \leq x \leq \pi/2 \quad \text{0 otherwise}
$$
The domain is the rectangle with vertices ...
- 119
2
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92
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Parameter choice rules for L1 regularization?
I am solving an L1 regularized least squares of the form like:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| \boldsymbol{x} \...
- 121
0
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1
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158
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Assign and print the results of CHOLMOD package
I am trying to solve a simple working example, a linear system $Ax=b$, where $A$ is sparse SPD and $b$ is dense, using CHOLMOD.
...
2
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0
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100
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How to save multiplication computation time between a dense vector and a not that sparse matrix?
I am trying to compute $\mathbf{X}\mathbf{u}$ for many times in my algorithm, where $\mathbf{X}\in \mathbb{R}^{n\times m}$ and $\mathbf{u} \in \mathbb{R}^{m}$. The problem is that, during the ...
- 21
3
votes
2
answers
211
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Recover sparse matrix from linear operator
For many linear operators, it is easy to express them as a function. For example, to blur an image, simply compute a weighted sum of neighboring pixels (or even easier, import ...
- 41
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1
answer
151
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How to speed up sparse matrix index operation in Matlab?
I need to create spare matrices with variable elements. Unfortunately, sparse matrix index operations are very slow.
Is there any way to speed up the process? Maybe there are some tricks that I don't ...
- 21
1
vote
1
answer
85
views
Consider only triangular part of symmetric matrix for operations
It seems common practice to store only the upper or lower triangular part of a sparse symmetric matrix to save memory. Now I am wondering how, e.g. the SpMV kernel in CSR format is designed and how ...
- 840
2
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65
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Sparse generalized symmetric eigensystem solver
Can anyone recommend a good software for solving generalized symmetric eigenvalue problems of the form,
$$
A x = \lambda B x
$$
where $A,B$ are symmetric and sparse, and $B$ is positive definite?
I ...
- 1,048
2
votes
0
answers
52
views
Is there a permutation used in sparse QR factorizations that better locates small elements on the diagonal of R?
Is there a permutation used in sparse QR factorizations that better locates small elements on the diagonal of R to the end of the diagonal? As an example, consider the following snippet of MATLAB ...
- 757
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45
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Lowest eigenvalues of a matrix with divergent entries
In high energy physics we oftentimes encounter the following problem. For a given parametrized matrix $\{H_{ij}(\Lambda)\}$, we know that in the limit $\Lambda\to\infty$ some of its entries become ...
- 151
1
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1
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122
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Efficiency of developing PDE solvers using sparse matrices versus loops
I am new to solving PDEs, but have been looking at different implementations of finite difference and finite volume schemes. One thing I have noticed in different implementations is that some ...
- 287
2
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0
answers
43
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Iterative methods for underestimate of smallest eigenvalue for large sparse matrices
I recently read the paper "EUCLIDEAN-NORM ERROR BOUNDS FOR SYMMLQ AND CG" by Estrin et al. and there they use an underestimate (i.e. something in $(0,\lambda_{min}]$) of the smallest ...
- 1,271
2
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84
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How to find eigvals/eigvecs of huge symmetric tridiagonal matrix using multiple processes or threads?
In python, there is a function in scipy (scipy.linalg.eigh_tridiagonal) which is extremely efficient for an exact diagonalization of a symmetric tridiagonal matrix (...
- 21
2
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0
answers
198
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solve Ax=b for outrigger A matrix python
I implement Crank-Nicolson 2D finite-difference method.
I get a matrix A which is banded with 1 band above and below the main diagonal, but also contains 2 additional bands , further apart from the ...
- 131
1
vote
1
answer
133
views
Sparse direct solver that works inside opened omp parellel region?
Does anyone know a library that implements sparse direct solver working in already open omp parallel region? The only library that I know that works with this requirement is Pardiso7.2 worth ~8K USD ...
7
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135
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Can we sparse solve a few eigenvalues specified by index range?
I need to solve a few eigenvalues of a large sparse matrix specified by their index range. These indices are according to the whole eigenspectrum sorted in algebraic (not absolute value) ascending ...
- 303
4
votes
2
answers
220
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nnz-preserving sparse matrix multiplication
Let A and B be sparse matrices in $ \mathbb{R}^{m\times m}$ with (roughly) the same density $p$. I want to efficiently compute a matrix $C$ that in some sense is "closest" to $AB$ while ...
- 41
0
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2
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136
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Where can I find matrices and it's preconditioner for testing?
I want to find some kinds of matrices for testing my code such as GMRES , MINRES and so on. But I can't find some testing matrices and corresponding preconditioner to verify my program.
I know some ...
- 3
4
votes
1
answer
109
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Sparse least squares with a (black-box) ill-conditioned operator
It was suggested on math.stackexchange.com that I try to ask this question here.
Consider a bounded linear operator $A : U \to V$ where $U$ is finite dimensional and where $V$ is a separable Hilbert ...
- 149
3
votes
1
answer
505
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A method for finding the number of eigenvectors with a given, known eigenvalue
Is there a method for finding the number of eigenvectors with a given eigenvalue? I do not need the eigenvectors themselves, and to find the eigenvectors seems quite tough, given the comments on the ...
- 133
1
vote
1
answer
914
views
Givens rotation algorithm without matrix-matrix multiplication
I would like to implement a givenRotation algorithm without having matrix-matrix multiplication. Matrix-vector is fine or just for looping. I am to decompose a rectangular (m+1)xm Hessenberg matrix.
I ...
- 201
1
vote
1
answer
158
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What is the difference between Adittive Schwarz as a preprocessor and a solver?
As we all know, the Additive Schwarz approach can be used as either solver or preconditioner, however, my question is, what is the difference between the two? In other words, how to use AS as solver, ...
- 13
1
vote
1
answer
817
views
What is the conventional approach for sparse matrix multiplication?
When you're multiplying sparse matrices against other sparse matrices or dense matrices, what is the conventional approach for each? How are the sparse matrices stored? What does matrix multiplication ...
- 281
2
votes
1
answer
163
views
Bounds for the optimal bandwidth of 2D/3D FEM stiffness matrices
is anyone here aware of whether there exist bounds on the optimal bandwidths of 2D/3D FEM stiffness matrices?
Edit: more specifically, I would like to have bounds on the minimum bandwidth after ...
- 91
6
votes
1
answer
249
views
Algorithm for solving systems which are nearly symmetric/adjoint?
I am familiar with Cholesky decomposition and LU factorization for solving systems of linear equations.
I have a problem where I have large sparse matrices (say, 1000x1000 or larger) where only one or ...
- 173
0
votes
1
answer
176
views
Best search algorithm for optimal weight factor in SOR method
I had written an algorithm that searches for the optimal weight parameter to be implemented in the successive-over relaxation (SOR) method which worked cleanly by vectorizing the interval and for ...
- 169
1
vote
1
answer
690
views
Efficiency of scipy.sparse.linalg.expm_multiply with sparse vs unsparse vectors
From the package scipy.sparse.linalg in Python, calling expm_multiply(X, v) allows you to compute the vector ...
- 185
3
votes
0
answers
145
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$$
where
I is an Identity Matrix,D is a Diagonal ...
- 31
1
vote
0
answers
54
views
Tolerance is not satisfied in eigsh function
I am diagonalizing a sparse matrix of size $153600\times 153600$ via eigsh function. I did not set the tolerance to specific value, so it would be the default value ...
- 171
1
vote
0
answers
62
views
Viable algorithms for efficiently solving a block matrix system with non-uniform sparsity structure
I am trying to use Newton's method to get a stationary solution for a system of equations of the following form:
$$
\begin{Bmatrix}
\frac{\partial x}{\partial t} \\
0
\end{Bmatrix} = \begin{Bmatrix}
f(...
- 91
3
votes
1
answer
400
views
How can I extract the banded or block diagonal part of a sparse matrix in MATLAB?
Given a large sparse (square) matrix in MATLAB, how can I extract the banded or the block-diagonal parts (of fixed size) of it efficiently?
These are useful operations when prototyping and testing ...
- 2,411
3
votes
0
answers
100
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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 ...
- 297
3
votes
1
answer
241
views
`eigsh` (Lanczos algorithm) slows down for degenerate eigenvalues
I have a complex Hermitian matrix of size about $70000\times 70000$. I want about 100 eigenvalues near 0. However, I know that every eigenvalues are two-fold degenerate. I found out that the running ...
- 171
1
vote
1
answer
135
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RCM better than Nested dissection? (For FEM discretizations in 2D and 3D)
I realize this might be a too general question but here goes nothing:
I am trying different re-ordering strategies and checking the fill-in of $A=LU$.
I have 2D ($p=1$, $h=1/40$ on $\Omega = [-1,1]^2$)...
- 91
3
votes
0
answers
119
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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 ...
- 171
2
votes
0
answers
110
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)?
- 21
0
votes
3
answers
671
views
How to apply the boundary condition when global stiffness matrix is stored in csr format? [duplicate]
I am solving the poisson equation and I constructed the global stiffness matrix in compressed row storage format. Then I wrote the preconditioned conjugate gradient solver for solving the system of ...
- 29