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Questions tagged [dense-matrix]

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3
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0answers
55 views

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 ...
4
votes
1answer
107 views

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 ...
8
votes
3answers
199 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 ...
6
votes
2answers
229 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-...
6
votes
2answers
502 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 ...
9
votes
2answers
291 views

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 ...
1
vote
1answer
50 views

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} \...
1
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0answers
51 views

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. ...
1
vote
1answer
157 views

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....
0
votes
1answer
88 views

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) ...
2
votes
2answers
182 views

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 ...
1
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2answers
109 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'...
5
votes
2answers
514 views

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 ...
2
votes
2answers
524 views

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. ...
0
votes
1answer
92 views

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}{...
1
vote
1answer
340 views

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}-\...
11
votes
2answers
5k views

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 ...
1
vote
2answers
693 views

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 ...
8
votes
6answers
9k views

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 ...
1
vote
2answers
1k views

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'...
0
votes
1answer
112 views

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) ...
3
votes
3answers
599 views

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'...
6
votes
1answer
491 views

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. ...
3
votes
1answer
860 views

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 ...
5
votes
2answers
381 views

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, ...
1
vote
2answers
141 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 ...
2
votes
3answers
234 views

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?
5
votes
1answer
128 views

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 ...
1
vote
0answers
87 views

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 ...
2
votes
0answers
58 views

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\...
1
vote
1answer
1k views

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 ...
7
votes
1answer
1k views

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 ...
4
votes
1answer
343 views

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 ...
3
votes
3answers
131 views

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-...
5
votes
2answers
409 views

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?
3
votes
2answers
118 views

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
votes
3answers
5k views

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 ...
6
votes
4answers
2k views

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 ...
11
votes
1answer
533 views

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=...
10
votes
2answers
650 views

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 ...
5
votes
2answers
730 views

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 ...