Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrix]

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

1
vote
1answer
155 views

solve linear system of equation of a large sparse symetric positive definite matrix

I want to invert large matrices ($10^4 \times 10^4$ to $10^6 \times 10^6$) but sparse (less than $100$ non-zero entries per line) on clusters with $16$ to $48$ processors per node. I'm looking for an ...
3
votes
1answer
70 views

Is there an efficient $O(n^2)$ way to estimate in MATLAB a matrix condition number given its LDL decomposition?

Since evaluating a matrix condition number usually takes $O(n^3)$, I wonder whether there is an efficient $O(n^2)$ way to estimate in MATLAB a matrix condition number given its LDL decomposition. ...
2
votes
1answer
336 views

Recommendation for C/C++ library which offers Schur complement functions?

I need to find C/C++ libraries which offer function for computing Schur complement. I know about MUMPS and Pastix, but I need more of them to compare them in my research. Do you have any experience ...
0
votes
1answer
87 views

Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
3
votes
2answers
1k views

SVD of large block-hankel matrix

I am trying to do SVD of a large block-hankel matrix for model order reduction (Low rank approximation). However, I quickly run into memory issues in forming the large Block-Hankel matrix and CPU ...
1
vote
1answer
211 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$ \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C $$ Method: I ...
1
vote
1answer
409 views

ADR equation implicit solution: Penta-diagonal matrix for a 2D $N\times N$ system

Objective: I am trying to simulate the following advection-diffusion-reaction equation in 2D space (x,y) and time. $$\begin{align} \text{ADR Equation: }\frac{\partial C}{\partial t} + \nabla\left(v.C ...
4
votes
1answer
1k views

What is wrong with this matrix multiplication?

I am attempting to write a matrix multiplication routine because I need to do some analysis in CUDA and I want to validate it with CPU code. I am trying to use <...
1
vote
1answer
246 views

Fast algorithms for computing only the generalized singular values (but not the vectors)

I am interested in computing only the generalized singular values, and was wondering if this was faster (and by how much?) than computing the full GSVD. In particular, I was wondering what the ...
3
votes
3answers
504 views

Exact analytical matrix inversion of sparse 100x100 matrices in C++

I need to invert a matrix. Of course, I'm not the first person in this situation, and I know that there's a wealth of powerful libraries out there, of which I only know a couple. That being said, ...
1
vote
1answer
112 views

Is this the correct procedure for calculating matrix spectrum?

I am not sure if my question is on topic but I have a piece of Fortran code that is used to perform successive over relaxation. Prior to performing successive over relaxation the author is calculating ...
3
votes
2answers
2k views

BLAS, LAPACK or ATLAS for Matrix Multiplication in C

I am trying to find the most optimized way to perform Matrix Multiplication of very large sizes in C language and under Windows 7 or Ubuntu 14.04. And searching led me to BLAS, LAPACK and ATLAS. ...
3
votes
1answer
279 views

Sparse matrix vector product using PETSC

I am trying to do a simple parallel sparse matrix vector multiplications using PETSC. My sparse matrix is a simple tridiagonal laplacian matrix, which is distributed over multiple processors using ...
1
vote
1answer
655 views

Convert Image of Map to 2D Grid in Python

I have this map showing the geography of Europe (below), and I wish to convert it to a matrix in python that would be a 2D approximation of this image where 0's would represent the ocean and 1's would ...
1
vote
0answers
252 views

How can I efficiently solve $Ax$=$b$ given $A$ is symmetric and contains very small (even negative) eigenvalues using EIGEN

Currently I am using the EIGEN C++ library to try to solve $x$ from the equation $Ax$ = $b$. One problem I encountered is that the matrix $A$ is a correlation matrix with size > 5000 and can ...
2
votes
1answer
102 views

How do I make sparse solvers to accept custom matvec function insted of matrix?

I have tried it with Lis, Intel mkl and PETSc. Everywhere you need to pass an actual matrix ...
3
votes
0answers
122 views

What is the computational cost of using complex numbers in contrast to real numbers in matrix operations, e.g. $LU$ or $LDL^T$ factorizations? [duplicate]

I am curious about how much one loses in terms of computational cost, when complex numbers are used instead of real numbers? I guess the number of floating point operations and memory doubles, but I ...
1
vote
2answers
1k views

How can I prove that two eigenvectors are orthogonal?

I obtained 6 eigenpairs of a matrix using eigs of Matlab. How can I demonstrate that these eigenvectors are orthogonal to each other? I am almost sure that I ...
4
votes
2answers
2k views

Optimization with matrix determinant as constraint

I'm solving a constrained optimization for matrix $\mathbf{A}$ with dimension 6x6, where one of the constraints is $\mathrm{det}(\mathbf{A})>0$. I use the NLopt package to solve my problem and ...
-1
votes
1answer
173 views

Solving large system of equations, is linear programming best option? [closed]

I have a problem where I am trying to solve many systems of equations, that have very few variables per equation, but a lot of equations. For example potentially 10 variables max in a single equation,...
1
vote
2answers
3k views

How to use the basic Sparse matrix operations (multiplication, .etc) in PyCUDA

I try to use sparse matrix operations in GPU in Python and now try to use PyCUDA with theano. But I can't find how to do sparse matrix and vector multiplication. I only got an example showing how to ...
4
votes
1answer
437 views

Numerically stable approach for calculating x in Ax=b

I have an equation $Ax=b$ for which I need to solve for numerous $x$ matrices given $b$. Both $x$ and $b$ are nx1 matrices. Unfortunately, $A$ is a 32x32 matrix and inversion gives highly unstable ...
2
votes
1answer
133 views

Is my matrix symmetric?

I obtained a mass matrix through Finite Elements discretization. Now, I want to check if it is symmetric. To do that I subtract to my matrix $M$ its transposed $M^T$. The result is another matrix of ...
9
votes
1answer
25k views

What is the fastest algorithm for computing the inverse matrix and its determinant for positive definite symmetric matrices?

Given a positive definite symmetric matrix, what is the fastest algorithm for computing the inverse matrix and its determinant? For problems I am interested in, the matrix dimension is 30 or less. ...
1
vote
2answers
761 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 ...
6
votes
0answers
110 views

How does an unpivoted QR fail to reveal rank?

An unpivoted QR factorization produces a triangular factor $R$. A rank-revealing QR factorization is typically done with column pivoting. My question is, how does an unpivoted QR factorization fail to ...
8
votes
6answers
10k 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 ...
2
votes
1answer
741 views

Solve Ax=B where B is a matrix in parallell

I try to solve the problem $Ax=B$ where $A$ is a large sparse $n\times n$ matrix, and $B$ is a dense $n\times m$ matrix (here $n=754850$ and $m=182$). The backslash operator yields correct solution (<...
0
votes
1answer
248 views

Standard Algorithms for Permuting CCS or CRS Sparse Matrices

I need to permute the degrees of freedom of a system and apply this permutation to a few sparse matrices in CCS (or CRS) format. I could construct a permutation matrix and perform sparse matrix-matrix ...
7
votes
1answer
202 views

Compute eigenvectors of a matrix with known eigenvalue spectrum

If I have already accurately known the eigenvalue spectrum (i.e. all eigenvalues) of a matrix, is there any efficient numerical algorithm to compute all the eigenvectors corresponding to these ...
4
votes
0answers
195 views

Alternative to (costly) matrix multiplication

Consider the integral $$2\pi T = -\frac{1}{2} \int_0^{2\pi} A\frac{\partial B}{\partial \xi} \ d\xi$$ where $$A= \sum_{-N}^N i \ sign(n) \ B_n e^{-in\xi}; \quad \quad B= \sum_{-N}^N \ B_n e^{-in\xi}...
0
votes
2answers
2k views

How to obtain a convergent solution iteratively for a linear system of equations? [closed]

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} &...
1
vote
0answers
128 views

Estimating the second largest eigenvalue

I am currently dealing with the following problem. I am given a matrix $A$ of order $n \times n$ where $n \leq 20.$ The principal $(n-1) \times (n-1)$ matrix of $A$ is symmetric and contains only $\{...
6
votes
1answer
522 views

Is there a faster method to compute the geometric series of a matrix?

I want to calculate the geometric series of a matrix $A$: $$S=I+A+A^2+\dots+A^n$$ and then apply to a vector $v$, $Sv$. I've done it in Matlab with a loop and I think it's quite efficient applying ...
1
vote
0answers
56 views

Library for calculating determinants with Kronecker products

I need to calculate a determinant consisting of vectors, using the Kronecker product as product. As an example I would need to be able to calculate: $\left| \begin{array}{cc} \left(\begin{array}{c}1\\...
3
votes
1answer
141 views

Simple way to derive transpose of a vectorized operation

In a program I'm writing, I have a sub-routine that does some vectorized linear operations (specifically differentiation). Say for convenience I have defined the following inline function, which ...
0
votes
1answer
1k views

Difference between eigendecomposition and singular value decomposition for Hermitian matrices

Let consider the following Hermitian matrix ...
0
votes
1answer
163 views

Solve eigenvalue problem using finite differences without vectorization

I am interested in solving the problem $-A u = \lambda u$ on a finite differences grid on a square. In my case, the operator $A$ is of the type $-\Delta + \mu I$, where $\mu$ is there to impose some ...
4
votes
0answers
142 views

How big a matrix can we row reduce in reasonable time?

I have very large matrices that I would like to row reduce (I need to keep track of the steps and find a basis of the kernel/image, not just find the rank). The good news is that I work mod 2 and the ...
5
votes
1answer
678 views

What is a good way to take fractional powers of a matrix in MATLAB?

I am working on a problem that involves taking fractional powers of particular matrices. For the matrix A with 2 on the main diagonal and -1 on the sub and super diagonal (the finite difference ...
4
votes
1answer
2k views

Why is my MATLAB code for back-substitution slower than the backslash operator?

I wrote the code below to invert an upper triangular matrix, avoiding any possible multiplication/subtraction by zero. It just uses $\frac{1}{6}n^3+\ldots$ flops instead of $n^3+\ldots$ flops. ...
1
vote
0answers
332 views

Optimizing rank computation for very large sparse matrices

I have a sparse matrix such as ...
1
vote
0answers
89 views

smallest eigenvalues for linear elasticity

I want to compute a few tens of the smallest eigenvalues of a linear system which is a discretization of a linear elasticity. In the presence of additional constraints like Dirichlet boundary ...
1
vote
2answers
516 views

Are there any popular (paralleled) implementations of Lanczos methods for SVD/eigendecomposion?

I want to use it in Matlab or Java. Will these two languages be much slower for computing the algorithm compared to C, C++, in case efficiency is an important factor? I'm aware of that there's a ...
-4
votes
1answer
194 views

Can anyone give me some suggestions about optimize my Matlab codes?

recently, I try to write a Matlab codes to implement a sparse approximation inverse factorization method proposed by M. Benzi in his paper http://www.mathcs.emory.edu/~benzi/Web_papers/ainv.pdf this ...
5
votes
1answer
110 views

Are there any algorithms “incrementally remove part of data (esp., old data)” from the existing SVD model of a data?

Sometimes it is meaningful to remove the influence of some old data from a SVD-based model, so as to reflect the most updated trends and provide more accurate results. I've seen there're incremental ...
1
vote
3answers
2k views

Minimize quadratic form with equality constraints

I want to minimize function: $f(x) = x^T \cdot A \cdot x + b \cdot x$ given constraints: $B \cdot x = 0$. Here: $x$ is a vector ($x \in \mathbb{R}^n$), $A$ is a matrix of size $n \times n$, $b$ ...
3
votes
2answers
610 views

Get symmetric Finite Difference matrix in non Laplacian settings

I would like to solve a system of differential equations $u+\nabla(\nabla\cdot u)=f$ or in more detail $a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$ $b+\partial_x^2b+\partial_x\...
7
votes
1answer
327 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\...
3
votes
3answers
642 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'...