Questions tagged [dense-matrix]
Questions about working with or solving equations involving matrices which are known or suspected to have many nonzero elements. Contrast with sparse matrix.
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Complexity of matrix inversion in numpy
I am solving differential equations that require to invert dense square matrices. This matrix inversion consumes the most of my computation time, so I was wondering if I am using the fastest algorithm ...
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13
votes
3
answers
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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 ...
- 143
11
votes
1
answer
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Solving huge dense linear system?
Is there any hope in solving the following linear system efficiently with an iterative method?
$A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^n, b \in \mathbb{R}^n \text{, with } n > 10^6$
$Ax=...
- 213
11
votes
2
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Matrix exponential of a Hamiltonian matrix
Let $A, G, Q$ be real, square, dense matrices. $G$ and $Q$ are symmetric. Let
$$H = \begin{bmatrix} A & -G \\ -Q &-A^T \end{bmatrix}$$
be a Hamiltonian matrix. I want to compute the matrix ...
- 168
10
votes
1
answer
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full rank update to cholesky decomposition
Let $A$ be a real, symmetric, positive definite matrix. It has at least 500 rows, possibly much more. I compute its Cholesky decomposition, which allows me to calculate
$det(A)$
$A^{-1}X$ for some ...
- 375
10
votes
2
answers
820
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Diagonalization of Dense Ill Conditioned Matrices
I am trying to diagonalize some dense, ill-conditioned matrices. In machine precision, results are inaccurate (returning negative eigenvalues, eigenvectors do not have the expected symmetries). I ...
- 101
9
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6
answers
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Super C++ optimization of matrix multiplication with Armadillo
I'm using Armadillo to do very intensive matrix multiplications with side lengths $2^n$, where $n$ can be up to 20 or even more. I'm using Armadillo with OpenBLAS for matrix multiplication, which ...
7
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2
answers
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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 ...
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7
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2
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Is there an iterative solver for dense matrices with possible zero diagonal entries?
Is there an iterative solver that can handle potentially zero entries on the central diagonal? I am implementing a polynomial fitting algorithm (up to $10^{th}$-order) and my matrix is a "...
- 213
7
votes
4
answers
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computing the determinant of a dense nonsymmetric 100x100 matrix having very big and very small eigenvalues
The motivation for my question is the following: in one of Project Euler questions there is a need to count the spanning trees of a rectangular grid graph of dimension 100x500. By the Matrix-Tree ...
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6
votes
3
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Algorithm for Principal Eigenvector of a Real Symmetric 3x3 Matrix
I have a 3x3 covariance matrix (so, real, symmetric, dense, 3x3), I would like it's principal eigenvector, and speed is a concern. Is there a fast algorithm for this specific problem? I've seen ...
- 203
6
votes
2
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300
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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-...
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6
votes
2
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What are the most common dense matrix storage formats?
I'm looking to write some code to read in a dense matrix from a file, and I was wondering what are the most common storage formats that my code should support?
- 69
6
votes
1
answer
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Efficient RQ decomposition
I have an upper trapezoidal matrix stored in column major format. That is, my matrix looks like this:
I'd like to RQ decompose it, and store Q in the rectangular part of my upper trapezoidal matrix. ...
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5
votes
2
answers
753
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Largest invertible dense matrix with standard solvers such as Lapack
I have a matrix which is complex symmetric. It is around 50,000 elements per side. It is a Method of Moments matrix. Is it feasible to use a standard direct solver such as Lapack to do a matrix ...
- 1,320
5
votes
2
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Spectral decomposition of symmetric matrix
What is a good direct method to compute the spectral decomposition / Schur decomposition / singular decomposition of a symmetric matrix?
"Direct" means as in LU decomposition, Cholesky decomposition, ...
- 3,580
5
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2
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Smart way to multiply 3 matrices
I have a quantum mechanics simulation where I need to multiply three matrices that look like this:
$$\rho(t_1)=U^\dagger \rho(t_0) \, U$$
where $U^\dagger$ is the hermitian conjugate of $U$. This ...
5
votes
1
answer
170
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Computing eigendecomposition of a Hermitian matrix that is almost unitary
I have a dense Hermitian matrix that is approximately unitary, so it has eigenvalues that are $\sim \pm1$. I would like to compute all the eigenvectors corresponding to the $+1$ eigenvalue (not ...
- 4,460
4
votes
1
answer
975
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LU Decomposition of PSD Matrix + Diagonal Matrix
If I have a psd, symmetric matrix $\mathbf{A}$ and I need to do LU decomps on $\mathbf{B_i}= \mathbf{A} + \mathbf{D_i}$ (where $\mathbf{D_i}$ is a diagonal psd matrix, where $\mathbf{D_i}$ changes ...
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4
votes
1
answer
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Fast matrix multiplication with matrix elements computed on-the-fly (without forming the matrix)
Is there any library or routine for high-performance matrix-matrix product, where the matrix elements are computed on-the-fly using a given function of $i$ and $j$?
More specifically, in the problem ...
- 141
4
votes
1
answer
155
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Trace of inverse from LU decomposition
Given an LU decomposition of $A\in \mathbb{R}^{n\times n}$, is there a way to compute $\operatorname{trace}(A^{-1})$ with lower complexity than that of the inversion ($O(n^3)$ in practice)?
This ...
- 10.1k
4
votes
0
answers
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Block matrix and DSYRK
I want to compute the matrix
$$
A = \sum_{i=1}^N v_i v_i^T
$$
where each $v_i$ is a given vector of length $2500$, so that $A$ is $2500 \times 2500$, and my $N$ is about 2 million. Rather than call ...
- 1,016
3
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3
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Large overdetermined system of linear equations
I'm looking for a method to solve a large overdetermined system of linear equations in a least squares sense. The matrix is dense.
I'd like to use a method that works even with limited memory (we can'...
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3
votes
1
answer
1k
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Fast way to compute all eigenvalues of a dense Hermitian matrix
I am finding the eigenvalues of dense NxN Hermitian matrix which is calculated from a density operator in quantum physics. All the eigenvalues are needed as I need to calculate the sum of the absolute ...
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3
votes
3
answers
137
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Smoothly varying dense matrices arising from computational science
I have written an algorithm to solve a dense system with smoothly varying entries. This means I assume there is no large jump from any entry to its neighbors.
I would love to use real-application-...
- 41
3
votes
2
answers
155
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Computing sparse matrix products into a dense result
I need to assemble a matrix (in dense form, of moderate size, say dimension 1000) which is most easily expressed as the product of several (4) sparse matrices. These matrices are most easily expressed ...
- 4,460
2
votes
2
answers
2k
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Solve large dense positive-definite linear system
Which method should I choose to solve a large (~20 000 variables) dense symmetric positive-definite, possibly ill-conditioned, system of linear equations?
The system will be solved for two vectors. I'...
- 21
2
votes
3
answers
286
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Dense distributed matrix
A dense matrix is distributed for parallel computation column-wise, then multiplied from left & right by sparse matrices. What would be appropriate c++ libraries for these tasks?
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2
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1
answer
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Solve linear system for only part of the solution vector
I am using the ScaLAPACK PDGESV routine to solve large dense linear systems distributed over many supercomputer nodes, but ultimately I only need a small portion of the solution vector (e.g. the first ...
- 21
2
votes
0
answers
93
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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 ...
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0
answers
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Are the eigenvalues of the product matrix of two real symmetric square matrices also real values?
Suppose $A,B \in \mathbb{R}^{n\times n}; A=A^T, B=B^T$, let $C = AB, D =BA$,
If we have all the real eigenvalues of $A$ and $B$, e.g. the eigenvalue decomposition of them:
$A=P\Lambda_1 P^T$,
$B=Q\...
- 719
1
vote
2
answers
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Compute all eigenvectors and eigenvalues of small symmetric matrices
My problem is to compute eigenvectors and eigenvalues of a lot of small (n < 30) symetric, positive definite matrices.
So far I am using LAPACK's DSYEV.
The priority is speed more than accuracy. ...
- 111
1
vote
2
answers
1k
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Parallelization of element-wise matrix multiplication
I use Armadillo as an interface to OpenBLAS. In my current program, I have a loop, in which I do multiplications of the form
...
1
vote
2
answers
249
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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'...
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1
vote
1
answer
63
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Optimal algorithm choice for mixed diagonal/dense problem
$$
\text{Let}\\
A, B \in \mathbb{C}^{n \times n} \text{ and } \hat{\alpha}, \hat{\beta} \in \mathbb{C}^{n}, \hat{f} \in \mathbb{C}^{2n}
\\
\text{Find }\\
\underline{\mathbf{x}} \in \mathbb{C}^{2n} \...
- 11
1
vote
2
answers
1k
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Comparison between two matrices
I have a large dense matrix $A$. For simplicity in simulation purposes and application demand, I induced sparsity by replacing lower/insignificant values by zero and reordering it in block diagonal ...
- 19
1
vote
1
answer
748
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Preconditioning of two step iteration for dense matrices
I would like to solve a dense linear system the form in python
$$
L\left(\boldsymbol{x}\right):=\left[\gamma^+\left[\boldsymbol{A}+\frac{1}{2}\boldsymbol{B}^{-1}\right]
+\gamma^-\left[\boldsymbol{A}-\...
- 297
1
vote
2
answers
220
views
Rearrange a dense distance matrix to a 2x2 non-perfect block diagonal form
I have a distance matrix (square, symmetrical, non-negative, dense).
I want to split the objects into two well-connected groups.
Mathematically speaking, I want to group (re-arrange) the rows/columns ...
- 131
1
vote
1
answer
1k
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How can I reuse the SVD of matrix A to solve LS problems for both A and its transpose via Eigen C++?
If $A\in R^{m\times n}, b\in R^m, c\in R^n$,
if I need to solve the least square problems via SVD of $A$ and $A^T$, i.e.
I need to solve the least square solutions to following linear systems via ...
- 719
1
vote
0
answers
167
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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 ...
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1
vote
0
answers
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Parallel dense solve with submatrices from mesh refinement with Petsc
For a Bounday Element Method problem I require the solution of a system of linear equations with multiple right-hand sides. Though this is a dense system, I still want to do it via Petsc in parallel. ...
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1
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0
answers
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Is it possible to construct such a symmetric matrix with desired eigenvalues?
Suppose a real, dense and asymmetric square matrix $A\in\mathbb{R}^{n\times n}$,
all its eigenvalues $\lambda_i \in \mathbb R$
Is it possible to construct a symmetric matrix $B\in\mathbb{R}^{n\times ...
- 719
1
vote
1
answer
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GPU libraries for integer matmul | overflow tolerated
Are there any high performance integer BLAS libraries that implement matrix multiplication i.e. i32gemm and i64gemm ? I need to use them for a cryptographic application and can tolerate overflows, i.e....
- 287
0
votes
1
answer
63
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Solving a linear system whose coefficient matrix is dense but symmetric
For solving a linear system,
$Ax = b$.
If $A$ is a dense but symmetric $n \times n$ matrix, how much memory is required?
$A$ is symmetric, which means only the upper (or lower) triangular part of $n \...
0
votes
1
answer
140
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Library to solve dense linear system with GMRES
I have a fortran 90 code and I want to solve a dense linear system with GMRES. I would prefer the restarted GMRES with preconditioning. Is there some library that you know of that I could use? Now I ...
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0
votes
1
answer
128
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Discretized matrix from the integral kernel function
Recently, I read a paper [1] and then I want to handle the two-dimensional linear integro-differential equation
\begin{equation*}
-\triangle u + q\Big(\frac{\partial u}{\partial x} + \frac{\partial u}{...
- 129
0
votes
1
answer
169
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Help me analyze the computational cost of two kinds of operations
everyone, I have a question about computational costs for a algorithm. That is:
I have two vectors $u_n,\ v_n\in \mathbb{C}^N$, a matrix $A\in \mathbb{C}^{N\times N}$ (can be both sparse and dense) ...
- 129
0
votes
1
answer
109
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Need clarification on a piece of book excerpt about spectral element method!
I am reading "Using MPI (3rd edition)" from William Gropp, where in chapter 4 application section 4.13, it introduces an MPI application Nek5000/NekCEM which is based on spectral element method (SEM) ...
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