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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Convexity of Sum of $k$-smallest Eigenvalue

If I have a real positive definite matrix $A\in\mathbb{R}^{n\times n}$, and denote its eigenvalues as $\lambda_1\leq \lambda_2 \leq ... \leq \lambda_n $. Define the function as $f(A)=\sum_{i=1}^{k} \...
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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++: ...
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3answers
727 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'...
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1answer
137 views

Matrix condition number and reordering

Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
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2answers
3k 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. ...
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2answers
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Matlab preconditioned conjugate gradient on big matrix

I have a sparse $5\,656\,236 * 5\,656\,236$ matrix $A$ with $166\,526\,888$ non-zero elements. The matrix comes from using the finite element method on a linear elasticity problem and is positive semi-...
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2answers
591 views

Unimodular Matrix calculation

I know for a given matrix $M$, there exists a matrix $U$ over the integers with determinant $+1$ or $-1$ such that $UM=E$. I know $E$, but $M$ is not a square matrix. Is there any easy way to get $...
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?
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2answers
2k 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 ...
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1answer
4k views

How to change the dimensions of an Eigen Matrix in a loop?

I have a while loop, in which I use a Matrix A, vectors B and x with varying dimensions: <...
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1answer
315 views

LU Decomposition with memory-mapped matrices

I have a ~4.12 Tb structured relatively-sparse matrix dataset (about 8% of the matrix entries are non-zero) that i want to apply an LU decomposition, however, given the size of it, loading it in ...
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2answers
5k views

how can a 2-d fft be constructed to an equivalent matrix?

When I use the cvx matlab toolbox, I met a puzzled problem. The function of fft (or dct, wavelet, etc.) cannot be recognized by the type of 'cvx'. For the 1-d fft, it can be constructed to an ...
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Computing Permanents of $64 \times 64$ Matrices

I need to compute the Matrix Permanents of several $64 \times 64$, zero-one matrices. I have tried using the built in functions in both Sage and Maple, but both programs return out of memory errors. I ...
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1answer
107 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 ...
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1answer
681 views

Why is the speed of the parts of the LU-decomposition so different?

I know that an easy way to solve the matrix problem $$A\cdot x=b$$ is the LU decomposition $$\begin{split} L,\,U&=\text{lu}(A)\\ y&=\text{solve}(L,\,b)\\ x&=\text{solve}(U,\,y) \end{split}$...
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2answers
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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 ...
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3answers
553 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, ...
3
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1answer
129 views

Calculate 3x3 matrix to give lowest difference for data set

I'm building an application where I need to compare found data with the actual data it should be. I have 5 sets of data, each with 3 variables a,b,c. Let matrix A be a 3x1 matrix with data a,b,c ...
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1answer
149 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 ...
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1answer
241 views

How does matrix-matrix product scale with multiple CPUs?

These days, one can have 64 cores in a single node. I wonder how well the dense matrix-matrix product (SGEMM and DGEMM) scales ...
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1answer
2k views

Sparse matrix - matrix multiplication

How can a sparse matrix - matrix product be calculated? I know the 'classic' / mathematical way of doing it, but it seems pretty inefficient. I thought about storing the first matrix in CSR form and ...
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2answers
616 views

How to make Elemental Gemm run quickly?

How to make Elemental Gemm run quickly? I have the following code: ...
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1answer
228 views

Manipulating a generalized eigenvector problem to plain eigenvector problem

Let $X\in\mathbb{R}^{n\times p}$ denote a matrix with $p$ linearly-independent columns, and let $L\in\mathbb{R}^{n\times n}$ denote a symmetric matrix. Furthermore, let $D\in\mathbb{R}^{n\times n}$ ...
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1answer
897 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)...
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2answers
135 views

Parallel assembly of matrix

I have a matrix which I want to assembly quickly, which is in block form: $$ A = \pmatrix{ A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33}} $$ ...
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1answer
111 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 ...
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1answer
197 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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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 ...
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1answer
385 views

Linear Algebra / Numerical Solution Of Matrix With Nullspace

I have a question relating to linear algebra. We have a fluid solver that solves the poisson equation for pressures. When there are areas of the domain that are entirely enclosed by Neumann ...
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3answers
132 views

Smoothly varying dense matrices arising from computational science

I have written an algorithm to solve a dense system with smoothly varying entries. This means I assume there is no large jump from any entry to its neighbors. I would love to use real-application-...
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2answers
122 views

Time-stable SO(n) matrix synthesis algorithm

Consider an equation $S(t)b(t) = a$, where $a, b(t) \in S^{n-1}$ are given and the vector $b(t)$ is continuous, i.e. its endpoint traces a continuous curve on the unit sphere. The task is to find ...
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1answer
890 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 ...
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1answer
576 views

Efficient ways to numerically evaluate matrix exponentials

What are some computationally efficient ways to solve matrix exponentials, i.e. functions of the form : $f(X)=e^{X}$, where $X$ is a square matrix? So far I have been able to diagonalise some ...
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1answer
80 views

Calculate Transformation Matrix between two sensors

My question is if I can calculate the transformation matrix between two sensors. Each sensor provides a $4\times 4$ matrix for every timestep recorded. The sensors are moving and have some noise in ...
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1answer
105 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 ...
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1answer
83 views

Efficient computation of leading eigenvector of a matrix product of the form $ADA^T$, where $D$ is diagonal

Let $A=[A_1|\ldots|A_m] \in \mathbb R^{n \times m}$ with $n \gg m \gg 1$ and $D=\text{diag}(d_1,\ldots,d_m)$ where $d_1,\ldots,d_m > 0$, and consider the $n\times n$ positive-definite matrix $X=\...
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1answer
183 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 ...
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1answer
309 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 ...
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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...)
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1answer
491 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 ...
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2answers
272 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 ...
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2answers
753 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\...
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2answers
1k views

Finding eigenvalues of a complex symmetric tridiagonal matrix

I am trying to find specific eigenvalues and -vectors of a large complex symmetric tridiagonal matrix (at least 10000x10000, and ideally larger). I know roughly which eigenvalues I am looking for, so ...
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1answer
223 views

Inverting many small matrices in parallel

I am trying to find a good way to handle the following problem: Let C be an N by 3 array (corresponding to points in $\mathbb{R}^3$). There is a method I am interested in testing which requires ...
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2answers
128 views

Computing sparse matrix products into a dense result

I need to assemble a matrix (in dense form, of moderate size, say dimension 1000) which is most easily expressed as the product of several (4) sparse matrices. These matrices are most easily expressed ...
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1answer
461 views

Marker and Cell Method (MAC) - STOKES FLOW - boundaries?

please can you help me with my problem with Stokes flow written using Marker and cell method (MAC)? I need only to solve the eq. of continuity + momentum eq. for a given condition (steady state). I ...
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1answer
153 views

multiplications of graph adjacency matrix

Suppose $A$ is a directed graph adjacency matrix. Is there any good interpration of the $(i,j)-$entry of the matrix $(A^{32}\cdot (A^T)^{32})$ ?
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1answer
93 views

High-dimensional representation of arbitrary input

Given a symmetric matrix $A\in\mathbb{R}^{n\times n}$ with positive entries and zero diagonal, is it always possible to construct a high-dimensional configuration in Euclidean space, such that these ...
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1answer
1k views

Application of an orthogonal matrix to a 3D configuration of point

Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as $$Y=XQ.$$ ...
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1answer
229 views

3D to 2D projections, a generalization

Given some data points in 3D, $X\in\mathbb{R}^{n\times 3}$, could one say that $$Y=XP,$$ for some $P\in\mathbb{R}^{3\times 2}$ actually corresponds to a particular viewpoint on a 3D data? Basically, ...

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