Questions tagged [matrix]

Matrix is a rectangular array of elements (e.q. numbers, symbols, or expressions), arranged in columns and rows.

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8
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1answer
372 views

Least-squares for a diagonal matrix

This is a follow-up to a different question I asked with more detail. For $v\in\mathbb{R}^n$, denote $D_v\in\mathbb{R}^n$ as the diagonal matrix with elements in $v$. Given a "tall" matrix $B\in\...
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3answers
718 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'...
8
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1answer
252 views

Are there improved method of computing the following expression?

given a symmetric matrix $Y \in \mathbb{R}^{n \times n}$, and an arbitrary matrix $X \in \mathbb{R}^{n \times n}$, and a vector $v \in \mathbb{R}^{n \times 1}$, is it possible to compute the following ...
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0answers
542 views

Does it matter if I use principal component analysis on the transpose instead of the original matrix?

My data set is a 60x10 matrix. I performed principal component analysis of this matrix with matlab using the princomp(AdjustedData) after I adjusting my original data set by subtracting the mean of ...
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2answers
202 views

Azimuthal average in Fortran? Find indexes in Fortran?

I am working on an eigenvalue problem in fortran. I have used Lapack to solve the problem and get the eigenvalues and eigenvectors. This is done for $201\times101$ wavenumbers, only half the wavespace ...
3
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1answer
2k views

Translate a 3D point along a heading

I need to translate a point (P1) in 3D a certain amount, call it stepSize, along a vector described by a heading composed of ...
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1answer
132 views

Writing a 3D array from Petsc

I am trying to do something fairly simple somehow I have made it hard. Is there an example of how to send a 3D array to a binary file?
2
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1answer
1k views

Constructing matrix from eigenvalues, eigenvectors (Inconsistency with Matlab's eig())

Let $F_1$, $F_2$ be the foci points of an ellipse $\mathcal{E}\colon \mathbf{x}^TA\mathbf{x}=1$, $\mathbf{x}\in\mathbb{R}^2$, $A\in\mathbb{S}_{++}^{2}$. Let also $a$, $b$ be the semi-axes of $\mathcal{...
2
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1answer
624 views

OpenCL C Matrix Multiplication over Multiple Runs

I am attempting to translate a dgemm / MPI matrix multiplier onto the GPU through OpenCL C. My issue is that the code below gives the correct output for a 9x6 matrix being multiplied by a 6x450 ...
3
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1answer
241 views

How does matrix-matrix product scale with multiple CPUs?

These days, one can have 64 cores in a single node. I wonder how well the dense matrix-matrix product (SGEMM and DGEMM) scales ...
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2answers
435 views

Recommendations for symmetric preconditioner

Given symmetric, positive-definite matrix $A$ and its preconditioner $M^{-1}$. What's kind of preconditioner $M^{-1}$ which preserve the symmetry of $M^{-1}A$?
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1answer
755 views

Overcoming floating point issues when Inverse does not exist but determinant provides nonzero result

I have an expression (let's say determinant of matrix A) expressed in symbolic form in terms of 2 decision variables x1, x2 and 2 parameters q1 and q2. I'm minimizing this using fmincon for different ...
5
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1answer
647 views

Sparse matrix ordering in Python

I would like to implement custom, domain-specific algorithms for sparse matrix orderings. I am looking for Python packages for ordering sparse matrices. It would be nice to have: The underlying ...
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1answer
451 views

Algorithm to calculate the exponential of an Hessenberg matrix

I am interested in computing the solution of a lage system of ODEs using a krylov method as in [1]. Such method involve functions related to the exponential (the so-called $\varphi$-functions). It ...
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2answers
541 views

How to decrease computation time for symmetric matrices?

We all know the problem that computation time explodes when simulating systems with big matrices. I got just this problem, but I have the advantage that I know that my matrices are symmetric. My ...
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3answers
2k views

Fast vector - “diagonal” matrix multiplication

Let $\mathbf{1}\in\mathbb{R}^d$ be a vector with all elements equal to $1$. Define: $$\mathbf{D} = \mathrm{diag}(\mathbf{1}^\top,\mathbf{1}^\top,\ldots,\mathbf{1}^\top) = \begin{bmatrix} 1 \cdots 1 &...
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2answers
350 views

Computing element stiffness matrices with variable coefficients

I am trying to implement a simple FEM approach, using p1 triangular elements, for solving the diffusion equation with variable nodal diffusivities and I was wondering how to incorporate the variable ...
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1answer
184 views

GMRES: Making the matrix square without solving for boundaries

How do we define the matrix for GMRES, if we do not want to solve the boundary elements but only the interior ones. I am using pentagonal elements so in a row there are 6 elements (cell itself + 5 ...
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2answers
166 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 ...
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0answers
381 views

Implementing the transition matrix for page rank

I'm trying to implement PageRank. I'm reading the description here: http://nlp.stanford.edu/IR-book/html/htmledition/markov-chains-1.html Everything is very clear to me, however I'm concerned about ...
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1answer
313 views

Most memory-efficient way to store a list of numbers

My problem deals with a large $n \times m$ matrix from which I extract and store several square $k \times k$ submatrices. The original matrix may be very large, and I may need to store many thousands ...
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1answer
290 views

Error propagation on GSL eigenvalues computation

The problem comes from the need to estimate the error propagation of the spectral norm computation of a square matrix $A$ of which I know the components' $A_{ij}$ absolute error. The fundamental step ...
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3answers
2k views

fortran code-algorithm for qr decomposition of non-square matrix

I am trying to implement QR factorization of a non-square matrix in FORTRAN. I have the algorithm for a square matrix but not for a non-square. I use Housholder matrices. Do you know where I could ...
2
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1answer
317 views

LSA, SVD and the Frobenius norm

In Latent Semantic Analysis one uses the SVD to perform a dimensional reduction of the term-document matrix, via the Eckart-Young theorem. Now, the rank $k$ approximation obtained by E-Y is proven to ...
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4answers
2k views

Efficient solver for a symmetric tridiagonal system where the upper/lower diagonals are offset

I'm looking for an efficient way to solve a symmetric tridiagonal system $Mx = d$, where the upper and lower diagonals of $M$ are offset from the main diagonal by $k$ rows/columns: $$ \begin{bmatrix} ...
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1answer
198 views

oSVD and cSVD terms

In one article I faced with such terms as sSVD, cSVD and oSVD. As I understand sSVD - standart SVD, cSVD - svd for block-circulant matrices, but I can't find what is oSVD. 1) What is oSVD? 2) Can ...
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3answers
258 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?
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0answers
919 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 ...
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3answers
323 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
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1answer
493 views

SVD and HITS Algorithm Power Iterations

As we know, computing the authority (or hub) score of HITS ranking method, means to use the following matrix equation: $$ \textbf{a}^{k}=A^T A\textbf{a}^{(k-1)} $$ and apply the power iteration ...
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1answer
1k views

computing the inverse of a large block diagonal sparse matrix in r

I would like to compute the inverse of some large block diagonal sparse matrix. The number of rows and columns is somewhat over 50,000. The blocks are 12 by 12 and are sparse (27 non zero elements). ...
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242 views

Wanted: sequences of linear systems for recycling Krylov solver analysis

In the solution of sequences of linear systems $$A_ix_i=b_i\quad\text{for}\quad i=1,2,\dots$$ with Krylov subspace methods, data can be recycled from already solved linear systems in order to speed up ...
3
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1answer
2k views

Sparse matrix - matrix multiplication

How can a sparse matrix - matrix product be calculated? I know the 'classic' / mathematical way of doing it, but it seems pretty inefficient. I thought about storing the first matrix in CSR form and ...
0
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1answer
334 views

Gilbert-Peierls algorithm for LU Decomposition

I searched for Gilbert-Peierls algorithm, but I haven't found anything useful (well, I found this, but it's not working as it should). I think the problem is the second part, and also that those lines:...
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0answers
194 views

Givens method for sparse matrix

I have a (very) large sparse matrix in CSC form and I'm supposed to factorize it. I've read that between Givens and Householder transformations, Givens is better for a sparse matrix. The problem is ...
3
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2answers
1k views

Finding eigenvalues of a complex symmetric tridiagonal matrix

I am trying to find specific eigenvalues and -vectors of a large complex symmetric tridiagonal matrix (at least 10000x10000, and ideally larger). I know roughly which eigenvalues I am looking for, so ...
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2answers
1k views

What to do with singular (non-invertible) rotation matrix

I have an orthotropic material with a (6x1) stress vector known in the global coordinate system and yield surfaces known in a local coordinate system. So far I have only needed to convert from local ...
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1answer
50 views

detecting special $2 \times 2$ matrices in a large array of zeros and ones

I have a large array of zeros and ones and I need to count instances of 0 1 1 0 0 0 1 1 0 1, 1 0, 1 1, 0 0 And I would like to exclude all ...
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0answers
99 views

Tracking for two meshes

I'm dealing with physical simulation (position based dynamic). Now I'm trying to realize tracking for two meshes. In order to explain what does it mean, let's assume that we have two similar meshes: ...
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0answers
56 views

Dominant contributions of a quadratic form

Let $\Sigma$ be a covariance matrix (e.g. symmetric positive definite). For arbitrary vectors $\epsilon$, I need to compute $\chi^2 \equiv \epsilon^\top\Sigma^{-1}\epsilon$, which I do using a ...
3
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2answers
1k views

Solving Newton-Raphson step with ill-conditioned sparse matrix

I am trying to build a complex simulator for the transport of mass & heat in porous media. I am currently following coarsely the algorithm laid out in an older software and have got the simulator ...
4
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1answer
162 views

Updating an approximate solution to a linear system in response to a small change

This question was original posted on SO but it was suggested that I post it here. I'm working on a program in which I have a banded matrix M and a vector b, and I want to maintain an approximate ...
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0answers
245 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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1answer
343 views

Problem with convergence of Jacobi iterative algorithm

I'm dealing with Jacobi iterative method for solving sparse system of linear equations. For small matrices it works well and gives right answers even if matrix is not strictly diagonal dominant, ...
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0answers
95 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 ...
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0answers
136 views

Efficient way to do congruent transformation using matrix inverse?

I know a square self-adjoint matrix $S_{vv}$ and I want to find: $S_{rr} = HS_{vv}H^{\dagger}$ where $\dagger$ denotes conjugate transpose. I do not know $H$ but I do know $H^{-1}$. What is the ...
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4answers
7k views

cartesian products in numPy

Have two arrays, in my case $X = \{1,2,\dots, n\}$ or X = np.arange(n). How do I get $Y = X \times X = \{ [i,j]: 1 \leq i,j \leq n \}$ as a 2D array in numPy? ...
3
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0answers
64 views

Matrix completion when the eigenvectors are a tensor product?

Suppose we have incomplete observations of the square matrix $X$. Most matrix completion algorithms assume the matrix is low-rank. What if instead we assume the matrix of eigenvectors is a tensor ...
4
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1answer
1k views

Nullspace algorithm for a sparse matrix

I am dealing with large, sparse, rational matrices that I need to determine the nullspace of. Currently, I have one that is about 12000x12000 (but not square), where one in every 2000ish elements is ...

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