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Questions tagged [sparse]

Problems in which an operator or function can be represented with asymptotically less data than the naive representation. Not limited to sparse matrices.

7
votes
2answers
97 views

Integer operations vs floating point operations

I have been working with an algorithm, which uses additions of floating point vectors, (sparse matrix of floats)x(dense vector of floats) dot products I recently found out that I can get the same ...
8
votes
3answers
166 views

Rule of thumb for sparse vs dense matrix storage

Suppose I know the expected sparsity of a matrix (i.e. the number of non-zeros / total possible number of non-zeros). Is there a rule of thumb (perhaps approximate) for deciding whether to use sparse ...
3
votes
2answers
98 views

Moore-Penrose pseudoinverse of singular rank degenerate matrix

I am trying to attain the Moore-Penrose pseudoinverse of a very large, very sparse, rank-degenerate, singular, and square matrix. ($75000 \times 75000$, near rank). The matrix is a graph Laplacian and ...
2
votes
0answers
60 views

Which SciPy nonlinear solver when Jacobian is analytically known and sparse?

I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
4
votes
1answer
65 views

Method to Efficiently Solve “Centered” Least Squares without centering “A”

Suppose I want to solve $$\text{arg min}_x \frac{1}{2}\|\tilde{A}x - b\|_2^2 + \frac{1}{2}\|x - c\|_2^2$$ where $A$ is a wide sparse matrix and $\tilde{A} = A C_n = A (I - \mathbf{1}\, \mathbf{1}^T/...
1
vote
1answer
101 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
222 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-...
4
votes
1answer
82 views

Level scheduling of triangular sparse matrices

Assume one has a triangular sparse matrix and want to solve $Lx=b$ where $b$ and $L$ are known. This can be done easily by using forward substitution when $L$ is a lower triangular matrix. Forward ...
2
votes
1answer
64 views

How can a CG solver solve a non positive definite sparse matrix

I am using the CUSP CG solver and I ran it on couple of sparse matrices from the University of Florida sparse matrix collection. The solver was able to solve non positive definite sparse matrices. My ...
6
votes
1answer
133 views

Solving linear system of the form $ABx=b$

I would like to solve a linear system of the form $ABx=b$ in parallel, where $A$ and $B$ are large, sparse matrices. Currently, I am forming the system matrix $AB$ explicitly, but I would like to ...
4
votes
2answers
125 views

How “sparse” should a sparse matrix be to see benefits?

I have a matrix, whose size scales as $2^N$ (assume even $N$). In each row of the matrix, only about $2^{N/2}$ of the entries are filled ($N$ can be somewhere between 10 and 40, depending on what's ...
2
votes
1answer
40 views

FFT of “implicitly” uniform data

I am trying to take a Fourier transform of a density field estimated from mock galaxy survey catalogs. Basically, you start with a list of galaxy positions, then you bin these positions over some ...
5
votes
2answers
470 views

Why does sparse linear algebra have a low arithmetic intensity?

I often see the terms "low arithmetic intensity" and "memory-bound" associated with sparse matrix operations. However, my intuition is that a sparse matrix operation should be less memory-bound, if ...
1
vote
1answer
68 views

Reordering algorithm for minimization of ram usage of a skyline matrix

The stiffness matrix of $Ax=B$ system of linear equations, where $A$ is an $n\times n$ symmetric matrix stored in the form of symmetric skyline matrix, that is associated with a finite element model ...
3
votes
1answer
35 views

Partial diagonalisation of large symmetric positive-definite band-diagonal matrices

I want to partially diagonalise real sparse symmetric positive-definite matrices, that are of dimension $n = 10^5$ and I need on the order of $k = 500$ of the smallest eigenvalues and eigenvectors. ...
7
votes
1answer
97 views

bit-packing and compression of data structures in scientific computing

I recently came across this paper, which describes a scheme for saving memory in representing sparse matrices. A 32-bit integer can store numbers up to ~4 billion. But when you solve a PDE you've gone ...
2
votes
0answers
62 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) $$x=(I-A)^{-1}b,$$...
3
votes
0answers
52 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 $k\times N$ ...
5
votes
1answer
114 views

Can redundant variables be beneficial for root-finding convergence

Suppose I have $n$ generally nonlinear equations for $n$ variables, like e.g. for $n=2$ the system $F(x,y)=0$ $$ \begin{aligned} x^2+2y-4&=0\\ \sqrt{8}x+y^2-5&=0 \end{aligned} $$ By ...
1
vote
2answers
224 views

Which python library for GPU sparse linear system solver library

I have a fluid dynamic solver written in python which I want to accelerate by moving the most expensive computations to the GPU. Ideally all arrays and sparse matrices used in my code should remain on ...
0
votes
1answer
54 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 ...
4
votes
2answers
124 views

Solve $Ax=b$ repeatedly where $A$ is a sparse weighted Laplacian matrix with changing weights

In the problem I am dealing with, I require to repeatedly solve $Ax=b$ where $A$ is a weighted Laplacian matrix of a sparse graph. The right-hand side remains constant. However each time I solve the ...
2
votes
2answers
189 views

Knapsack problem with fixed number of elements?

I am looking at an optimization problem that looks like this: $$ \text{minimize}\;\; \mathbf{x}^TQ\mathbf x \;\;, \; \mathbf x \in \mathbb R^n, x_i \in \lbrace 0, 1 \rbrace\\ \text{subject to}\;\; ||...
2
votes
1answer
102 views

MA57 vs HSL_MA57: symmetric indefinite solvers

What are the differences between MA57 and HSL_MA57 solvers? I'm in an optimization class that will make use of symmetric indefinite factorizations, and I'm trying to learn about the distinction ...
0
votes
1answer
51 views

Matlab backslash reordering algorithm

For the linear system $\mathbf A \mathbf x = \mathbf b$ generated from 2D Poisson equation using the standard central finite difference method, $$ \mathbf A = \begin{bmatrix} \mathbf K & -\mathbf ...
1
vote
1answer
98 views

More efficient matrix multiplication with diagonal matrix

A sparse matrix $\mathbf{L}$ is formed by $\mathbf{L} = \mathbf{DKG}$. $\mathbf{D}$ is a sparse matrix of size $m \times n$. $\mathbf{G}$ is a sparse matrix of size $n \times m$. $\mathbf{K}$ is a ...
3
votes
1answer
426 views

How can I speed up this code for sparse matrix-vector multiplication?

I've written a C++ function that multiplies a sparse matrix (stored in CSR format) by a dense vector. Here's the code: ...
1
vote
1answer
80 views

Data structure for efficient high dimensional histogramming

What data structure (or C++ library implementing it) is most suitable for efficient high dimensional histogramming? I have an application where I need to compute something similar to a histogram in a ...
5
votes
0answers
74 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 ...
1
vote
1answer
93 views

Leveraging scipy for matrix free finite elements

This will be a very general question. I have a 3D finite element code in Python which I would like to extend to handle "large" problems (~10^8 unknowns in the global system). Right now I am using the ...
0
votes
1answer
71 views

How to do matrix operations with this way of storing sparse matrices

I read the way of storing the five or seven point laplace matrix for some poisson problem but I don't understand how can i multiply, add and subtract this stored sparse matrix by a vector or another ...
0
votes
1answer
72 views

Sparse matrix inverse with reduced bandwidth

I have a sparse symmetric matrix of dimension 1393x1393 (8308 no zero elements), with bandwidth 1380. By Cuthill–McKee algorithm, I could achieve a new matrix with ...
4
votes
1answer
204 views

Why is the speed of the parts of the LU-decomposition so different?

I know that an easy way to solve the matrix problem $$A\cdot x=b$$ is the LU decomposition $$\begin{split} L,\,U&=\text{lu}(A)\\ y&=\text{solve}(L,\,b)\\ x&=\text{solve}(U,\,y) \end{split}$...
1
vote
5answers
3k views

Fast c++ library to solve very big sparse systems

I am working on a project with electrical circuits, where I am trying to compute the voltages at all the nodes of an electrical circuit. I know that the electrical circuit is a perfect grid, so each ...
2
votes
2answers
1k views

Eigen - Max and minimum eigenvalues of a sparse matrix

I am working with an Eigen::SparseMatrix matrix of type double. I would like to find the largest and the smallest eigenvalues. A solution of the problem is to convert it to dense and then find its ...
1
vote
0answers
260 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 ...
2
votes
1answer
383 views

Which is the best subroutine available for solving sparse linear system of equations [closed]

I am trying to solve the system of linear equations: $AX=B$. For this currently I am using Intel MKL Pardiso solver. It works well when the order of $A$ is around $13500\times13500$ and below. Above ...
4
votes
2answers
358 views

Solving Ax = b with sparse A and sparse b

Let's suppose I'm numerically solving the Poisson equation for a delta function source: $$ \nabla^2 f(x) = \delta(x-x') $$ I can represent the Laplacian $\nabla^2$ using the finite difference method ...
1
vote
1answer
229 views

Compressed sensing: $\ell_0$ “norm” vs $\ell_1$ norm

Suppose we have a very efficient way to perform $\ell_0$ "norm" compressed vs $\ell_1$ norm compressed sensing. Specifically, $\ell_0$ "norm" compressed sensing is $$\eqalign{ & \min \quad {x^T}...
1
vote
0answers
53 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. ...
2
votes
1answer
224 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
0answers
139 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: ...
3
votes
0answers
192 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 ...
3
votes
1answer
1k views

Iteratively solving 3D Poisson equation in MATLAB

I have written a function that sets up a sparse matrix A and RHS b for the 3D Poisson equation in a relatively efficient way. The set-up is nothing fancy: I have extended the 2D 5-point stencil to an ...
0
votes
1answer
84 views

Iterative single variable solutions in large linear systems

I have a system where $A$ is a large $n\times n$ marix with fast MVMs. It may have many nonzero entries (albeit in a structured way so as to allow fast MVMs), and is not necessarily diagonally ...
4
votes
1answer
169 views

Test matrices for large sparse overdetermined system of linear equations

I'm working on some c++ code to solve (conjugate gradient, least squares conjugate gradient, LSQR,..) large sparse overdetermined systems of linear equations. There is a twist to my matrices and the ...
1
vote
0answers
135 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 ...
1
vote
1answer
129 views

How to complete sparse matrix?

I am trying to solve an equation by a meshfree method. Shape functions (and their derivatives) have a compact support domain. So, over a given integration cell used for numerical integration, only a ...
5
votes
0answers
113 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 ...
1
vote
2answers
106 views

How is the dense system usually dealt with in spectral method?

Unlike finite element (FEM) or finite difference methods (FDM), where the original PDE is transformed into a sparse linear system, spectral methods return a dense linear system. For a large system, it'...