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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restriction and interpolation in multigrid method

I need detailed explanation of the formula below A2=I1*A1*I2 I suppose this formula computes matrix A2 on a coarse grid and here A1 is original matrix on fine ...
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2answers
110 views

Choose a subset of $m$ columns that maximize $|A^T A|$?

I have a set of $n$-dimensional vectors, and would like to choose $m$ of them to become the columns of an $n\times m$ matrix. I would like to choose the subset that maximizes $|A^T A|$, where $A^T$ is ...
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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 ...
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2answers
194 views

Imposing invertibility on a Matrix

I have a symmetric positive semidefinite covariance matrix $A$, which is approximately computed as the output of a quadratic regression. I then need to invert $A$, but often it is close to singular. I'...
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142 views

What is the algorithm for computing block reflectors in xDLARFB

The theory behind computing a single Householder reflector to zero out part of a column of a matrix is pretty well described in Matrix Computations by Golub and Van Loan. However, the blocked ...
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249 views

Is there an efficient way to form this block matrix with numpy or scipy?

Is there an efficient way to form this block matrix with numpy or scipy? $$ \left[ \begin{array}{cccc} \mathbf{B} & \mathbf{0} & \cdots & \mathbf{0}\\ \mathbf{AB} & \mathbf{B} & \...
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313 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
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516 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 ...
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627 views

find a set of linearly independent columns in a $m\times n$ matrix

my question is between mathematics, physics and informatics. Suppose i have an Hamiltonian (hermitian matrix) that i can diagonalize. The matrix that allows this transformation is a unitary matrix ...
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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 ...
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191 views

What Linear Equation Solver should be used for a problem with many dirichlet conditions?

I am solving a laplace equation on a finite-element mesh (tetrahedral, triagonal) and have many say 99% dirichlet conditions compared to the number of unknowns. Is there an efficient way to solve this ...
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195 views

What is the difference between MATSEQMAIJ and MATSEQAIJ in PETSc?

What is the difference between MATSEQMAIJ and MATSEQAIJ in PETSc? Also, where can I find more information on each of the MatTypes? I went to the MatType documentation, but it didn't have anything but ...
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116 views

Fastest way to calculate the $2$-norm (or an upper bound for the $2$-norm) of the inverse of a matrix $A\in \mathbb{C}^{N\times N}$

I have a matrix $A\in \mathbb{C}^{N\times N}$ and I need to calculate $||A^{-1}||_{2}$ efficiently. Can it be done without having to evaluate the inverse explicitly? In general, I am looking for ...
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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 ...
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1answer
152 views

Finding all binary vectors with given A-length

I am given a $n \times n$ matrix $A$ with real entries and define the inner product $$\langle x,y\rangle = x^T A y.$$ I am also given an integer $k$ and need to find all binary vectors $x$ such that $...
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1answer
724 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 ...
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270 views

Out-of-core matrix transpose of row compressed data

Summary: Are there good algorithms for out-of-core dense matrix transpose if each row of the matrix is separately compressed? Details: The matrix is about 1 TB uncompressed, and is roughly but not ...
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186 views

Computing any element of the null space of a singular matrix

Given a singular matrix $A$, what is the fastest method to find a single non-zero solution to $Ax=0$? Note that we are not looking for the whole kernel, we just want any non-zero vector in it. I ...
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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'...
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132 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}^...
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1answer
2k views

How to calculate ALL of the eigenvalues/eigenvectors of a large, sparse, asymmetric matrix?

I am trying to calculate all of the eigenvectors/eigenvalues of large (40000x40000), sparse, asymmetric matrix. I am using MATLAB and have 3 GB of working memory. The way I am calculating them is by ...
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160 views

Is there a way to inspect the graph of a sparse matrix with PETSc?

I am currently trying to code the CA-CG method within the PETSc framework. A mandatory step in this process is the implementation of the "matrix powers kernel" algorithm for a generic sparse matrix. ...
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262 views

What category is this problem?

My first question, please excuse me if its too basic. I have a matrix of evenly spaced geographical points; say 10 x 10, which I will call seed points. Each seed ...
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626 views

Sparse matrix ordering in Python

I would like to implement custom, domain-specific algorithms for sparse matrix orderings. I am looking for Python packages for ordering sparse matrices. It would be nice to have: The underlying ...
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1answer
777 views

Eigenvalue Decomposition of Hermitian Matrix in Scala

I'm working on helping my friend create code to perform the Overlap Dirac Operator and have come across one part that I'm not sure what to do. I need to compute the Eigenvalues and corresponding ...
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129 views

Configuration shift to change the rank of a Gram matrix

Suppose a matrix $D\in\mathbb{R}^{n\times n}$ of Euclidean distances between $n$ points is given. To obtain a Gram matrix (matrix of inner-products of points that give rise to distances in $D$), one ...
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481 views

Sparse matrix format and sparse-matrix sparse-matrix multiplication

I'm having some performance problems with my code dealing with the multiplication of big sparse matrices (stiffness and aerodynamic influence coefficient matrices). Mainly I have to multiply such ...
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Library that performs sparse matrix-vector and matrix-transpose-vector multiplication

My main interest is sparse matrix-vector and matrix-transpose-vector multiplication of the form y=y+AA'x. Is there any library that performs ...
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1answer
523 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 ...
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153 views

Simplifying some operations on Gram matrices

Suppose two Gram matrices are given $A, B\in\mathbb{R}^{n\times n}$, such that $$A=XX^T,~~~~~~~~~~~~~B=YY^T,$$ for some $X, Y\in\mathbb{R}^{n\times k}$, $k\ll n$. Also, suppose a Gram matrix based on ...
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Extracting a 2D matrix from a multidimensional array in C

In my c language program, I have to store multiple dense $m\times m$ matrices corresponding to gridpoints $x_i$ with $i=1,...,n$. I decided to create a three dimensional array $A\in R^{n\times m\...
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164 views

Do I really need to invert this matrix

I need to calculate a matrix $A$ (at least some elements of it, see below) as defined by the following equation $$ A=B(\mathbb{1}-B)^{-1} $$ where B is a square matrix of dimension $N$ and $\mathbb{...
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Beating typical BLAS libraries matrix multiplication performance

A dull matrix multiplication algorithm where we use the formula $$C_{ij}=\sum_{k}A_{ik}B_{kj}$$ By literally following this in 3 loops we'll get a very slow program, because we don't utilize ...
4
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1answer
3k 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. ...
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357 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 ...
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4answers
228 views

Determining the algorithmic complexity

A few of the iterative matrix algorithms (CG,GMRES etc.) I have authored are acting rather funny. They converge to the right answers but take abnormally long time to run. I am in the process of ...
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1answer
120 views

In constructing matrices to model physical phenomena, are real matrices superior to complex matrices, in terms of computational cost?

Just studying some toy examples of $2\times 2$ and $3 \times 3$ matrices, complex number multiplication already gets a bit messy. From a numerical analysis point of view, if one were to try and build ...
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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 $z$...
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1answer
105 views

Finding the matrix inverse given a solver for the matrix equation $Ax=b$

So I'm given a solver that can solve for $x$ in the matrix equation $\underset{=}{A} \underline{x} = \underline{b}$ where $b$ can be anything we specify. (NB: A is an NxN matrix). I now want to find ...
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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 ...
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1answer
695 views

Libraries for distributed-memory Cholesky factorization?

What libraries can compute dense Cholesky factorizations in distributed-memory environments (preferably using MPI). I'm using C/C++, and have access to a cluster of between 1 and 40 nodes, where each ...
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5k views

Is there an MPI All Gather operation for matrices?

I have a distributed matrix, in block column format. I know that I can reshape the matrix into one long vector and use an all_gatherv operation. I just wanted to avoid the trouble of having to ...
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1answer
165 views

Accurate way of getting the square root inverse of a positive definite symmetric matrix

What is the most accurate algorithm to get the square root inverse of a positive definite symmetric matrix? I am not looking as much for efficiency, though using quadruple precision computation is out ...
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211 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 ...
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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 ...
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1answer
228 views

What is the most efficient way to obtain the max eigenvalue of a specific symmetric matrix via Eigen C++

Suppose I have a symmetric matrix $A_{1000\times 1000}$, which can be represented by: $A = J G J^T$ where $J$ in 1000x3 is full column rank dense matrix; $G$ in 3x3 is a nonsingular dense matrix. ...
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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 ...
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181 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 ...
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1answer
99 views

Does $\log(\det(A))$ equals sum of log of diagonal elements of D in LDLT decomposition?

For a large matrix $A$, I need to evaluate the $\log(\det(A))$. I already have it's LDLT decomposition. Is it possible to evaluate the $\log\det$ with the elements of the diagonal $D$ of the LDLT ...
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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 ...