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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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5
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1answer
1k views

Why do libraries need hand-vectorized code instead of compiler auto vectorization

C++ eigen library does vectorization for different architecture, like SSE, NEON etc. In their documentation they mentioned that, Eigen vectorization is not compiler dependent. But most modern ...
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1answer
100 views

Mapping from n x n complex symmetric tridiagonal to 2n x 2n real symmetric tridiagonal

In my program I have a complex symmetric tridiagonal matrix. In order to perform some algorithmic optimizations I am searching for a (ideally linear) mapping from $n\times n$ complex symmetric ...
6
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1answer
744 views

Effects of Lumping Mass Matrix

I've recently finished an introductory course on the finite element method from a more mathematical perspective (following Brenner and Scott) and we were introduced to the finite element mass matrix ...
5
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0answers
456 views

Sparse matrix format and sparse-matrix sparse-matrix multiplication

I'm having some performance problems with my code dealing with the multiplication of big sparse matrices (stiffness and aerodynamic influence coefficient matrices). Mainly I have to multiply such ...
0
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1answer
68 views

Show the symmetric Gauss-Seidel converges for any $x_0$

Let $A\in\mathbb{R}^{n\times n}$ is symmetric positive definite and consider solving linear system $Ax = b$. Show that the symmetric Gauss-Seidel iteration converges for any $x_0$. Solution - Since $...
4
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1answer
181 views

Efficient algorithm for a matrix product

Recall that a unit lower triangular matrix $L\in\mathbb{R}^{n\times n}$ is a lower triangular matrix with diagonal elements $e_i^{T}L e_i = \lambda_{ii} = 1$. An elementary unit lower triangular ...
2
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1answer
237 views

Determine a sufficient condition for a Hessenberg matrix to be nonsingular

Consider $A\in\mathbb{R}^{n\times n}$ whose nonzero elements are restricted to the main diagonal the strict upper triangular part, and the first subdiagonal. For $n = 8$ the locations that must be ...
2
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1answer
196 views

Schur complement of a matrix $A$

Let $A\in\mathbb{R}^{n\times n}$ and its inverse be partitioned $$A = \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\\ \end{pmatrix},\:\: A^{-1} = \begin{pmatrix} \tilde{A_{11}} & \...
0
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1answer
167 views

Parallel linear algebra without OpenMP

I have searched through the archives without success. Apparently, the question is simple: What linear algebra library can I use that is parallel (shared memory) but without OpenMP? As far as I've ...
6
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0answers
272 views

Compute sparsity pattern of $A^2$

Suppose we have a sparse matrix $A$. Is there any way to compute just the sparsity pattern of $A^2 = A \cdot A$ (I do not actually need to know what exactly the nonzero value are) faster than to ...
3
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0answers
446 views

Fixing a near singular covariance matrix

Given a near singular covariance matrix, the standard method of 'fixing' it seems to be to add a small damping coefficient $c>0$ to the diagonal, which serves to bump all the eigenvalues up by this ...
1
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1answer
54 views

Use double index in matrix multiplication

I want to run a simulation which involves rates between different states. Each state is label by a pair of indices $(m,n)$, so that a certain rate $R_{(m,n)\rightarrow(m',n')}$ requires four indices ...
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2answers
1k views

Stabilizing a 3x3 real symmetric matrix eigenvalue calculation

I have many 3x3 real symmetric matrices for which I need to determine the eigenvalues. Wikipedia gives a nice non-iterative algorithm for this case, which I have translated into C++: ...
3
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1answer
290 views

Mobile robot path following using model predictive control (MPC)

I'am trying to implement a path following algorithm based on MPC (Model Predictive Control), found in this paper : Path Following Mobile Robot in the Presence of Velocity Constraints Principle: ...
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1answer
566 views

Armadillo Multi-threaded Linear Solve Yielding Different Answers

I'm working on some problems that ultimately boil down into a simple assembly of an overdetermined system of equations, $Ax=b$, where $A$ is $m \times n$ for $m \gg n$. I'm leveraging Armadillo's C++ ...
2
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2answers
692 views

Sparse Matrix Matrix multiplication terminology (SpGEMM or SpMM?)

I have seen sparse matrix-matrix multiplication commonly referred to as SpGEMM, which means general/generalised sparse matrix-matrix multiplication. I've seen it once or twice (forgot where) as SpMM. ...
0
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1answer
76 views

Construct tridiagonal matrix from eigenvalues

I have a sort of reverse problem, and I'm not sure if there is a simple solution. I have a tridiagonal Hermitian matrix: $$ A = \begin{bmatrix} 0 & a_1 & 0 & 0 & 0 \\ a_1 & 0 &...
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1answer
70 views

Creating a matrix that saves storage

Suppose $A\in\mathbb{R}^{n\times n}$ is a banded matrix, i.e., a matrix with all of its nonzero elements on the main diagonal, i.e., $\alpha_{i,i}\neq 0$, the first superdiagonal, i.e., $\alpha_{i,i+1}...
2
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1answer
95 views

Condition Number of Rectangular Matrices

The 2-norm condition number can be easily extended to rectangular matrices. I'm wondering if the inequality for the product of matrices still holds in that case, i.e., $\operatorname{cond}(AB) \leq \...
3
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1answer
197 views

Algorithm to decompose a sparse unitary matrix into a Kronecker product of smaller unitary matricies

Given some sparse unitary square matrix $A$ ($dim=2^n$ if it matters), is there an algorithm to decompose $A$ into a Kronecker/tensor product of smaller unitary matrices? In other words: decompose ...
1
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1answer
735 views

Matrix transpose multiplication

In CVX, I encounter a problem. I want to multiply a Matrix of 2x4 with its transpose. I know the result must be positive definite. However, it couldn't let me do the multiplication directly. Says: ...
2
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2answers
1k views

Methods for fast approximation of convolution

What are the state of the art methods for fast 2D convolution approximation? I'm familiar with SVD based multiplication and cross approximation approaches, but would be thankful to get additional ...
9
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1answer
242 views

The fast, and The Backward-Stable (left) $3\times 3$ matrix inverse

I need to compute a lot of $3\times3$ matrix inverses (for Newton iteration polar decomposition), with very small number of degenerate cases ($<0.1\%$). Explicit inverse (via matrix minors divided ...
4
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1answer
130 views

Efficient way to generate a list of possible matrices (all integer components) with a determinant $V$

I have an interesting problem from my research that I have been struggling to solve. One part of the problem involves generating all possible matrices, where each set contains three integer vectors, ...
0
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1answer
164 views

Fast computation of square root inverse of matrix, matrix being determined from Ax=b form

I have an equation of the form $J^Te=f$, where $e$ and $f$ are known vectors and $J$ is an unknown matrix. How can I efficiently compute $J^T(JJ^T)^{-1/2}e$ ? My motivation to address this problem ...
0
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1answer
180 views

Is there any rapid way to calculate the determinant of NXN covariance matrix?

I searched the web and found some C code for calculating the determinant of a $n\times n$ matrix. This code however seems timing complexity, and run pretty slow especially when handling a larger ...
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0answers
37 views

Optimal distribution of zeros and ones over matrix

I have the following problem: Given a matrix with n rows and m columns. Some elements of the matrix are unavailable. For each column, you have a set containing a number of zeros and ones which must ...
3
votes
1answer
441 views

How to read a Harwell-Boeing Matrix file format into a compressed sparse row format in a C program?

I have to write a program where I have to perform matrix-vector multiplication and the matrix is sparse matrix. Most sparse matrices available online are in Harwell-Boeing format and they have to be ...
2
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1answer
674 views

strassen algorithm vs. standard multiplication for matrices

I am trying to figure out at exactly what dimension it is better to use Strassen algorithm rather than the standard multiplication. I know that there is 18 addition and subtraction in Strassen ...
2
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0answers
81 views

Solve for $D$ in $R^{T}DSDR = Id$

Given that $R$ is a rectangular matrix, $D$ is a diagonal, square matrix and $S$ being a square matrix along with the fact that both $D$ and $S$ are invertible. $S$ in this specific case can be ...
3
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1answer
176 views

Resources for solving mixed left and right matrix equations

I'm looking to solve a matrix equation and not sure where to start looking for resources. The equation is $$AX + XB = C\,,$$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, $C\in\...
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1answer
989 views

Extending the Frobenius inner product to all matrix inner products

So in ${\bf R}^{n\times p}$ we have the Frobenius inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$ which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. My ...
2
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2answers
534 views

Solve $AX=B$ where $A$ is a skyline matrix

Solve a matrix equation of the type $AX=B$, where $A$ is an $n \times n$ symmetric matrix stored in the form of symmetric skyline matrix. With the solution given by Bill and some more research on ...
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0answers
214 views

Extracting sinograms from tomography projections

my name is Lorenzo and I am a postdoc researcher in Italy. My job is related to tomography imaging using synchrotron radiation. This is a new field for me and I start to face with Matlab to write some ...
7
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3answers
2k views

How to compute the rank of a large sparse matrix in MATLAB

I am interested in computing the ranks of fairly large, the largest being of magnitude $10^6$ x $10^6$, sparse matrices whose entires are all 0, 1, or -1. I have been trying to use Matlab to ...
5
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1answer
260 views

What category is this problem?

My first question, please excuse me if its too basic. I have a matrix of evenly spaced geographical points; say 10 x 10, which I will call seed points. Each seed ...
4
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1answer
105 views

Finding the matrix inverse given a solver for the matrix equation $Ax=b$

So I'm given a solver that can solve for $x$ in the matrix equation $\underset{=}{A} \underline{x} = \underline{b}$ where $b$ can be anything we specify. (NB: A is an NxN matrix). I now want to find ...
0
votes
1answer
265 views

Calculating adjacency matrix of platonic solids

I need to devise a algorithm (in Python) that calculates adjacency matrices for the platonic solids. Inputted into the algorythm needs to be the number of polygons meeting at each vertex and the ...
5
votes
1answer
139 views

Finding all binary vectors with given A-length

I am given a $n \times n$ matrix $A$ with real entries and define the inner product $$\langle x,y\rangle = x^T A y.$$ I am also given an integer $k$ and need to find all binary vectors $x$ such that $...
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2answers
118 views

Inverse of “diagonally not dominant matrix”

I want to frame a higher order Central difference scheme of about $20^{th}$ order for first derivative. I'm using $20^{th}$ order because I need one scheme with good modified wave number. To find the ...
2
votes
1answer
1k views

Armadillo library appears slow

I have been experimenting in building a C++ project for a FDTDS system for Electro-Magnetics. I have implemented a class [see below] which I called mesh using the Armadillo library. The 3D matrices {...
1
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1answer
38 views

How to use compiled python packages for matrix initialization

Assume I have an expression for an matrix initialization, for example the following: A[i,i-2*j+k] = B[i-k] * C[i] * D[i+j+k] In order to execute such a process, I could loop over all i, j and k. ...
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2answers
7k views

In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...
2
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1answer
143 views

MATLAB : Does the qr algorithm and the DGEMM used in MATLAB take into account if the input matrix is tridigonal and optimize accordingly?

Let's say we want to solve for the eigen-values of a symmetric matrix of size $n$ x $n$. In the Phase 1 of the computation, the matrix is reduced to a tridigonal form using Householder/Arnoldi's ...
2
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1answer
283 views

How to compute the Jacobi matrix (tridiagonal matrix) of a polynomial with a recurrence relationship?

I am looking at Trefethen and Bau Exercise 37.1: I have two normalizations of the Legendre polynomials with corresponding recurrence relations: $$P_n(1)=1$$ which follows $$P_n(x) = \frac{2n-1}{n} x ...
8
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1answer
265 views

Fast computation of component-wise $\exp(-XY^T)G$ for random $G$

I have the following question: Suppose I have two matrices $X, Y$ both of size $m\times p$ and a random i.i.d Gaussian matrix $G$ of size $m \times k$, $m\gg p>k$. Is there a fast way to compute $...
0
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2answers
134 views

Matlab Vectorization of columns in a 2D matrix and single element multiplication

MATLAB. I am trying to vectorize a loop in which each column vector of a 2D matrix (n-by-n) is found by multiplying each single element in a diagonal matrix with a column vector in another n-by-n 2D ...
3
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2answers
239 views

Optimized parallel routine for $X' W X$ with $W$ diagonal

$X$ is a dense matrix of real doubles, typically of size 20 million rows and 500 columns, and $W$ is a diagonal matrix of real, non-negative doubles stored as a vector. I'm working in C and have ...
3
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1answer
118 views

Get a matrix with absolute values in PETSc [closed]

Is there any function to create or change a matrix, to have $A_{ij} = \text{abs}(A_{ij})$ in PETSc? If possible it should work with MPIAIJ matrices, not only local.
6
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1answer
389 views

Fast algorithm for computing matrix square root using randomized linear algebra?

Is there a fast algorithm for computing the matrix square root of a real symmetric matrix using random matrices or randomized algorithms?