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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15
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2answers
1k views

Estimation of condition numbers for very large matrices

Which approaches are used in practice for estimating the condition number of large sparse matrices?
16
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4answers
2k views

Why can't Householder reflections diagonalize a matrix?

When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you ...
27
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4answers
16k views

Dealing with the inverse of a positive definite symmetric (covariance) matrix?

In statistics and its various applications, we often calculate the covariance matrix, which is positive definite (in the cases considered) and symmetric, for various uses. Sometimes, we need the ...
16
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2answers
941 views

Practical example of why it is not good to invert a matrix

I am aware about that inverting a matrix to solve a linear system is not a good idea, since it is not as accurate and as efficient as directly solving the system or using LU, Cholesky or QR ...
6
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2answers
319 views

What guidelines should I use when choosing a scalable linear solver?

There are many different linear solvers, some which work best for diagonally dominant matrices, some for symmetric, some for positive definite ones, some for banded matrices, etc... There are direct ...
12
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3answers
2k views

What's the current state of the art regarding algorithms for the singular value decomposition?

I'm working on a header-only matrix library to provide some reasonable degree of linear algebra capability in as simple a package as possible, and I'm trying to survey what the current state of the ...
15
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3answers
2k views

Why isn't my Matrix-Vector Multiplication Scaling?

Sorry for the long post but I wanted to include everything that I thought was relevant in the first go. What I want I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices....
5
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2answers
509 views

Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$ where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...
6
votes
1answer
250 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 ...
11
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2answers
2k views

How is the SVD of a matrix computed in practice

How does MATLAB, for instance, calculate the SVD of a given matrix? I assume the answer probably involves computing the eigenvectors and eigenvalues of A*A'. If ...
5
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3answers
1k views

Performance optimization or tuning possible for Scalapack Gemm?

I'm comparing the performance of distributed gemm, using Scalapack over OpenBLAS, with threaded gemm, using OpenBLAS. It seems quite hard for me to get scalapack to give better results than ...
4
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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 ...
17
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5answers
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What is the best way to determine the number of non zeros in sparse matrix multiplication?

I was wondering whether there is a fast and efficient method to find the number of non zeros in advance for sparse matrix multiplication operation assuming both matrices are in CSC or CSR format. I ...
8
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1answer
8k views

Matlab solution for implicit finite difference heat equation with kinetic reactions

I am trying to model heat conduction within a wood cylinder using implicit finite difference methods. The general heat equation that I'm using for cylindrical and spherical shapes is: Where p is the ...
7
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1answer
865 views

Sparse hermitian eigensystems: are there better techniques than Arpack or TRLan?

As a part of other work I need to solve relatively large (~1E5x1E5) and sparse (~100 non-zero elements in each raw in few blocks) hermitian eigensystems. Usually only few eigenvalues+vectors are ...
4
votes
1answer
270 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 ...
15
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3answers
6k views

Efficient computation of the matrix square root inverse

A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this? I came across some literature (...
9
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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 ...
9
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3answers
4k views

Fastest algorithm to compute the condition number of a large matrix in Matlab/Octave

From the definition of condition number it seems that a matrix inversion is needed to compute it, I'm wondering if for a generic square matrix (or better if symmetric positive definite) is possible to ...
9
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2answers
1k views

Safe application of iterative methods on diagonally dominant matrices

Suppose the following linear system is given $$Lx=c,\tag1$$ where $L$ is the weighted Laplacian known to be positive $semi-$definite with a one dimensional null space spanned by $1_n=(1,\dots,1)\in\...
8
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1answer
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Solving system of linear equations with cyclic tridiagonal matrix

I have this problem in my textbook: Suggest efficient algorithm for solving system of linear equations with cyclic three-diagonal matrix, that is of the form: \begin{bmatrix} a_1&b_1&0&...
7
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1answer
347 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\...
6
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1answer
532 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 ...
4
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2answers
2k views

Reordering sparse matrices in computational science

On page 3 of this document, there are some matrix forms for sparse matrices. I wonder if there are other forms used in computational problems encountered in physics, chemistry, etc., so that ...
4
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3answers
4k views

debugging a rotation matrix for elastic constants

I have a transformation matrix which takes in the elastic constants from the local $rtl$ coordinates and then converts the elastic constants to the global $xyz$ coordinates via a rotation about the $z$...
4
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1answer
652 views

How to improve this double-shift QR algorithm for non-symmetric matrices?

I've implemented a version of the double-shift QR algorithm featured in this report from ETH Zurich (Begins on page 77). The algorithm takes advantage of the Implicit Q theorem by applying an ...
4
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3answers
329 views

Inverse of ill-conditioned symmetric matrix

I've got a matrix K, with dimensions $(n, n)$ where each element is computed using the following equation: $$K_{i, j} = \exp(-\alpha t_i^2 -\gamma(t_i - t_j)^2 - \...
3
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2answers
1k views

Efficently invert tiny matrix in Fortran

I have a piece of code in Fortran90 in which I have to solve both a non-linear (with the Newton-Raphson method, for which I have to invert the Jacobian matrix) and a linear system of equations. When I ...
3
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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 ...
3
votes
1answer
867 views

Spectral decomposition with eigenvalue shift

Suppose a square, real and symmetric matrix $G\in\mathbb{R}^{n\times n}$ is given, and it is known to have one zero eigenvalue associated with all ones eigenvector, $1_n$. I'm aware that the (possibly)...
0
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1answer
53 views

Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
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4answers
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Finding the square root of a Laplacian matrix

Suppose the following matrix $A$ is given $$ \left[\begin{array}{ccc} 0.500 & -0.333 & -0.167\\ -0.500 & 0.667 & -0.167\\ -0.500 & -0.333 & 0.833\end{array}\right]$$ with ...
4
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1answer
129 views

Constructing the origin position by transforming distance information

Suppose a set of $n$ points, $n\in M$, is given in some $d-$dimensional space, $X\in\mathbb{R}^{n\times d}$. Among these $n$ points, some $k\in K$ are selected, so $k<n$, and the distances from ...
4
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1answer
156 views

Factorization for reweighted least squares

I am solving a problem using an iteratively-reweighted least squares method: http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares Essentially this requires solving a number of least-...
4
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2answers
354 views

LAPACK - singular matrices - what does the positive integer info mean?

please can you help me with my code - I use Lapack to solve complex matrix (quite biq) and do it in two steps: I call zgetrf (LU factorization) and then ...
4
votes
2answers
1k views

Computational complexity and implementation of UDU Modified Cholesky Rank 1 Update

I am attempting to increase the performance of a legacy Kalman Filter implementation. The state covariance is factored in terms of UDU, i.e. $\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^T$. Many ...
4
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2answers
199 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 ...
4
votes
1answer
53 views

Configuration shift for determination of a true dimensionality

What would then be the way to determine a true dimensionality of a configuration of points $X\in\mathbb{R}^{n\times k}$ based on its Gram matrix $G=XX^T$? The "true" dimensionality refers to the ...
3
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1answer
856 views

Is a checkerboard block decomposition of a matrix useful when solving a linear system with a parallel conjugate gradient method?

According to these lecture notes, a checkerboard block decomposition should exhibit better scalability when applied to parallel matrix-vector multiplication (presumably because of greater cache hit ...
3
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0answers
76 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 ...
3
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1answer
437 views

How to find the nearest/a near positive definite from a given matrix?

I'm given a matrix. How do I find the nearest (or a near) positive definite from it? The matrix can have complex eigenvalues, not be symmetric, etc. However, all its entries are real valued. The ...
3
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0answers
367 views

On solution of a class of discrete-time Lyapunov equation for systems with multiplicaitve noise

Let's consider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment $F=[F_{1}...F_{p}]$ ...
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3answers
407 views

Real eigenvalues finding

I have a question about matrix diagonalization. I don't know if this is the right forum... Is there a way to compute the smallest real eigenvalue (and eigenvector if possible) of a general real nxn ...
0
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1answer
249 views

Linear solve using CHLOMOD in C

I am using the open-source CHLOMOD (as here http://faculty.cse.tamu.edu/davis/suitesparse.html) in order to solve a linear system Ax=b (performing A/b=x) in my domain decomposition code but I am ...
0
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1answer
503 views

ZGETRF and ZGETRS from MKL - zgetrf fails and still zgetrs works?

I have a large system of equations $$Ax=b$$ and I know matrix $A$ and right-hand side vector $b$. I'm using MKL to solve this system. The matrices are complex. I have used the general solver ...
0
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1answer
181 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 ...
0
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2answers
335 views

closed form approximation of matrix inverse with special properties

I'm trying to find some theory to help me explicitly express the inverse of a matrix (or a close approximation of the inverse). My matrix has the following properties: invertible positive definite ...