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?
17
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2answers
2k 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 ...
16
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4answers
3k 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 ...
5
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1answer
626 views

Fast algorithm for computing cofactor matrix

I wonder if there is a fast algorithm, say ($\mathcal O(n^3)$) for computing the cofactor matrix (or conjugate matrix) of an $N\times N$ square matrix. And yes, one could first compute its determinant ...
29
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4answers
18k 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 ...
28
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11answers
9k views

Robust algorithm for $2 \times 2$ SVD

What is a simple algorithm for computing the SVD of $2 \times 2$ matrices? Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...
13
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2answers
10k 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 ...
6
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2answers
381 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 ...
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 ...
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
3k 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. ...
4
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3answers
5k 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 $z$...
5
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2answers
784 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
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1answer
425 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 ...
17
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5answers
7k views

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 ...
11
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2answers
3k 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 ...
8
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2answers
12k views

Rearrange an ordinary matrix to block diagonal form

Is there an algorithm to rearrange a matrix into block diagonal form, given that the matrix is block diagonal in nature but randomized with an unwise choice of basis? In particular, are there any ...
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 ...
17
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3answers
8k 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 (...
8
votes
1answer
10k 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 ...
8
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3answers
4k views

Calculating the log-determinant of a large sparse matrix

I need to calculate $\log(\det (\mathbf M_i))$ where the $\mathbf M_i$'s are large sparse matrices, which are real, symmetric and positive semi-definite. I hope to have between $10$ and $100$ of ...
8
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1answer
440 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\...
7
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1answer
951 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
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1answer
594 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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2answers
3k 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 ...
10
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4answers
477 views

Are there any quad-double arithmetic sparse matrix package?

I am working on some ill-conditioned large sparse linear system of equations. I want to use double-double arithmetic or quad-double arithmetic to solve them. I know that there is a package named MPACK ...
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\...
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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1answer
2k views

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&...
6
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1answer
586 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 ...
5
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2answers
650 views

Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X=(X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...
5
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3answers
2k views

What is the best solver for solving a large sparse indefinite system

What's the best solver that can solve a large sparse but indefinite matrix?
5
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1answer
166 views

Algorithm to factorize matrix whose many rows are already of upper triangular form?

I have a matrix whose many rows are already in the upper triangular form. $$\begin{bmatrix} x_{11} & x_{12} & x_{13} & x_{14} & x_{5} \\ 0 & x_{22} & x_{23} & x_{...
5
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2answers
381 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
877 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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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
votes
3answers
694 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
votes
1answer
955 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)...
3
votes
1answer
140 views

Analytic formula for leading eigenvector of $uu^T + vv^T$?

Let $u$ and $v$ be nonzero column vectors of size $n$ and consider the $n \times n$ positive-definite matrix $A:=uu^T + vv^T$. In this post https://math.stackexchange.com/a/112201/168758, the ...
3
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2answers
2k 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
2answers
2k 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 ...
1
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3answers
577 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
65 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\...
11
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4answers
2k views

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 ...
6
votes
1answer
1k views

generalized eigenvalue problem

I need to solve a real generalized eigenvalue problem $Ax= \lambda Bx(*)$ A and B are calculated from equations below: $$A=\sum_{i,j=1}^{N}W_{ij}(K_{i}-K_{j})\beta\beta^{T}(K_{i}-K_{j})^{T}$$ $$B=\...
4
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1answer
54 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 ...
4
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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
votes
1answer
166 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
votes
1answer
131 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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2answers
410 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 ...