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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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1answer
61 views

Thomas Algorithm Kernel OpenCL

I am trying to implement the Thomas algorithm using OpenCL. ...
2
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2answers
284 views

Lost on Matrix Inversion

I try to implement some big matrix inversion. My system configuration is Hardware:- Memory: 62.8GiB, Processor: Intel Xeon(R)CPU E5-2670 v3 @2.30GHZ*48 To implement matrix inversion I am using ...
3
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0answers
36 views

Reweighted least squares factorization

This is a continuation of the question asked here. I want to solve numerous least squares systems of the form $$ D_i A x \approx D_i b $$ where $D_i$ are $m \times m$ diagonal matrices with positive ...
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0answers
141 views

An invertible matrix that minimizes the norm of the product with a given matrix

Given a fat matrix $B \in \mathbb{C}^{n \times m}$ (where $m > n$) with full row rank, I would like to find (numerically) a full-rank matrix $A$ that minimizes the Frobenius norm of the product $A ...
2
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1answer
255 views

Finding probability vectors from an implicit equation

I have $q$ $n$-dimensional vectors $\vec y_i$ and a matrix $\hat B$ of shape $n\times m$. I'm looking for $q$ $m$-dimensional vectors $\vec x_i$ such that: $\vec y_i=\hat B \vec x_i$ each vector $\...
4
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2answers
195 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 ...
3
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1answer
110 views

Compute nearly degenerated eigen values and vectors

In a quantum physics calculation, I have to deal with a matrix that has a lot of eigenvalues really close to each other ($10^{-6}$ relative difference for example) and I can't manage to obtain ...
5
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1answer
131 views

Generate approximately semi-orthogonal tall matrix approximately satisfying constraints

I have a set of matrices $\{(A_i,D_i)\}$ for $i\in\{1,\ldots,n\}$, where: Each $D_j\in\mathbb{R}^{S\times S}$ is diagonal, and every entry on the main diagonal is non-negative. Each $A_j\in\mathbb{R}^...
5
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1answer
49 views

Transform from linear index of a packed triangular matrix to dense indices

Given indices $i,j$ s.t. $0\leq i \leq j <n$, the function $f(i,j)=i+j(j+1)/2$ maps 2d indices to linear indices in column major order. What is the fastest way to invert this function? My first ...
7
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2answers
448 views

Is large condition number good measure of nearness to singularity for a matrix?

I am new to numerical linear algebra, so i came to know that condition number in 2-norm case will be ratio of largest to smallest singular value. Another concept "Nearness To Singularity" is measured ...
5
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1answer
162 views

How can I apply Euler's Method to predict a point in time rotating around multiple axis'

I am xposting this from my original stackoverflow question where I was presented with a coding challenge that I have been able to narrow down extensively and I think it lies with Euler's Method. Here'...
6
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2answers
657 views

Matrix multiplication accuracy Matlab vs Python

I am translating some Matlab code into Python and I having some problems regarding matrix multiplication accuracy. Assuming we have following data: ...
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2answers
58 views

Discrete-time input matrix when one of the eigenvalues of the system matrix is zero

If we have a continuous time state-equation, $$ \dot{x}(t) = A x(t) + B u(t)$$ where $A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^{n\times1}, B \in \mathbb{R}^{n \times m}, u \in \mathbb{R}^{m \...
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2answers
743 views

Does armadillo library slow down the execution of matrix operations?

I've converted a MATLAB code to C++ to speed it up, using the Armadillo library to handle matrix operations in C++, but surprisingly it is 10 times slower than the MATLAB code! So I test the ...
3
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2answers
183 views

Image hash similarity matching possible?

I have the following question: We have two face image files (JPEG), a Matrix of $128\times 128$ with values between 0-255. We would like to hash both image files using a function $f(x, key)$. Where I ...
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0answers
54 views

Dense decomposition of very non-square matrices

I have inherited code that solves Eigen::Matrix problems using the code shown on this page: ...
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2answers
493 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 ...
3
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1answer
102 views

Solve $A^{-1} b$ when one column is replaced

Given square matrix $A_0$, vector $b$, vector $A_0^{-1}b$ and matrices $A_1, A_2, \dots, A_k$, in which each $A_i$ is generated from $A_{i-1}$ by replacing one single column, I would like to find an ...
2
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0answers
80 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) $$x=(I-A)^{-1}b,$$...
8
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2answers
104 views

Exploiting patterns in matrix for efficient matrix-vector multiplication

I have the following situation: I have a sequence of vectors $x_1, x_2,.. $ and for each I want to compute the product $Ax_i$ where $A$ is fixed at the outset. Although there is no information about ...
4
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1answer
260 views

What is the fastest method to invert millions of matrices?

My project involves large simulation and estimation. For each simulation I need to solve 600,000 systems of nonlinear equations. Currently I am using Newton's method to find the solutions. That ...
3
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1answer
109 views

Normalizing a density matrix at each iteration

I need to numerically evolve a density matrix using this formula(Actually I have more terms but right nows I am starting with this and facing problems): $$\dot\rho(t) = -i[H(t), \rho(t)]$$ $H(t)$ is ...
0
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1answer
96 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 ...
7
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0answers
265 views

Speed and accuracy of Strassen vs Winograd matrix multiplication algorithms

I am doing work which requires as fast matrix multiplication as possible and just want to double-check with this community that the Winograd variant of Strassen's MM algorithm is the fastest practical ...
4
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2answers
158 views

MATLAB Matrix Multiply Efficiency

I am using MATLAB to prototype a few matrix multiply techniques and compare efficiency. Eventually, I will move the prototype codes to C. It is for a homework assignment where we need to write an ...
3
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0answers
75 views

Backward stable algorithm to get orthogonal projection onto the column space of a matrix

I have to find the orthogonal projection of a vector $b$ onto the matrix $A$ of size $m \times n$. In my application, I don't have the luxury of calculating the QR factorization. All I have are ...
2
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3answers
262 views

Matrix vector multiplication performance

I have been learning about the impact of cache size on code performance. I wrote a small code to see how using a column major loop in MATLAB would be better than using a row major loop, since MATLAB ...
1
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1answer
53 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} \...
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0answers
201 views

Multithreading determinant computation of a Big Matrix

I've been trying to design an algorithm for computing determinant of Big square Matrices (N <= 1000). I am allowed to use multithreading, but still, I can't design the algorithm. The assignment ...
6
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1answer
173 views

Fast multiplication of highly structured matrix

I want to compute a fast matrix-vector product using a matrix $T$ which has a peculiar quasi-Hankel structure. For example, \begin{equation} T_2= \left( \begin{array}{c|ccc|cccccc} a & b & c &...
0
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1answer
74 views

Which are some good algorithms and heuristics to calculate the similarity between two matrices?

Say I have a matrix like this: \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ \end{bmatrix} And this one: \begin{bmatrix} ...
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0answers
95 views

Low memory algorithm for matrix diagonalisation

I'm trying to find the largest eigenvalues of very large $N \times N$ matrices ($N = 10^{10}$ and larger). The matrices are not sparse but the multiplication operation is fast. For now, I'm using ...
1
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2answers
179 views

Inverting big symmetric and singular matrices

In this post I found a very similar probem to the one I have, but not a satisfactory answer for my purposes. I have a set of matrices $C_\ell$. They are exactly symmetric by construction. ...
7
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0answers
259 views

Eigenvalue with largest imaginary part

Iterative eigensolvers such as ARPACK, give the option to find a subset of the eigenvalues which have the largest imaginary part. My question is how do these algorithms work. As I understand it, ...
6
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1answer
249 views

Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
3
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1answer
270 views

How to compute the Frobenius norm of matrices whose entries are either too large or too small?

While implementing in Matlab the Frobenius norm of a matrix $$\| A\|_{\text F} := \sqrt{ \sum_{i=1}^m \sum_{j=1}^n a_{ij}^2 },$$ a problem arises when numbers are too big or too small: If a number ...
3
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1answer
869 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
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1answer
193 views

Methods for solving $Ax=b$, small and sparse A

I'm trying to solve/implement a system of linear equations of the following form/structure: $$ Ax=b$$ $$A = \begin{bmatrix} * & * & 0 & * & -1 & 0 & 0 & 0 \cr * & * &...
1
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0answers
33 views

Deterministic method to compute “Process noise covariance matrix, Q” for a Kalman filter when parameter variations of the model is known apriori

I am implementing a Kalman filter (for a linear ODE system for now). My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
1
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1answer
518 views

Kronecker product among multiple arrays

I was wondering whether there is a smart and efficient way in Matlab to compute the kronecker product of several 1D arrays. What I mean is something like this ...
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0answers
150 views

Numerical Precision in Matrix Inversion Routines

Let's say I have a matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$, and its associated inverse, $\mathbf{A}^{-1}$. The elements of $\mathbf{A}$ are given in IEEE single precision, i.e. 23-bit mantissa, ...
4
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1answer
126 views

Using GSL for basic operations

I am learning C/C++ for Scientific Computing and I have a question regarding the usage of scientific libraries for basic operations. Suppose I have to write a small program in C for a bioinformatics ...
1
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1answer
163 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....
1
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1answer
379 views

LAPACK sorting eigenvalues differently each time

I'm using LAPACK zgeev routine to get eigenvalues and eigenvectors of a symmetric matrix in C++. Problem is zgeev is being called in a loop but it sorts eigenvalues (and eigenvectors) differently ...
3
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1answer
2k views

Computing the Cholesky decomposition based of the QR decomposition

Let A be a n×n positive-definite Hermitian matrix. I already have the QR decomposition of A. Is there an efficient way to utilize this knowledge to speed up the Cholesky decomposition of A?
1
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1answer
93 views

Efficiency of Array Slicing

I have large arrays of data organized so that it can be processed efficiently using array processing libraries. However, there are times when I only need to process slices of the arrays where a slice ...
1
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0answers
378 views

Vectorizing Matrix Multiplication

I would like to do the following operation: I have a "4D" matrix A and a "3D" matrix B. Both A and B are actually 2D matrices, where for A, each element is a 2D matrix, and for B, each element is a 1D ...
3
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2answers
128 views

Sharing a matrix with the math community

What would currently be the best way to share a matrix with the math community? (I'm aware of Matrix Market but it seems the last update was in 2007...)
0
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1answer
75 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 ...
13
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4answers
273 views

Rapidly determining whether or not a dense matrix is of low rank

In a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, ...