Questions tagged [matrix-factorization]

Decomposition of a matrix into a product of matrices with special properties. Common matrix factorizations include LU, QR, SVD, and Cholesky.

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Why the two Gram-Schmidt algorithms produce different results for qr factorization?

For the qr factorization using classic Gram-Schmidt algorithm, I found the 2 different implementations below. The first one uses the for loop to compute the upper ...
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Numerical methods. MDF (ILU) implementation

I am trying to implement Minimum Discarded Fill (MDF) Ordering algorithm for incomplete matrix factorization. The algorithm description is here on page 60 Preconditioning Techniques for a Newton–...
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1answer
112 views

Block-matrix: optimal fill-in reduction for LU factorization

Consider a square $N \times N$ block-matrix $\mathbf{A}$, where each $n \times n$ block $\mathbf{A}_{ii}$ is either a dense block or a zero-block. So, $N$ denotes the number of blocks, $n$ denotes the ...
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71 views

Reference for QR algorithm for complex matrix

I am trying to find out if the known QR algorithm to find the eigenvalues of a real matrix, which can be found in the book Fundamentals of Matrix Computations, can also be used for complex matrices ...
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Why does the matlab command **chol(A)** slower than **chol(A,'lower')** for a large sparse SPD matrix?

For a SPD matrix A, there exists Cholesky factorization $A=LL^T$ or $A=R^TR$, where L, R are a lower and upper triangular matrix, respectively. Also in matlab, there has a command R = chol(A) which ...
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Does a symmetric positive definite matrix also have $\mathbf{A} = \mathbf{L}^T\mathbf{L}$ (where $\mathbf{L}$ is a lower triangular matrix)?

As we know, for a symmetric positive definite (SPD) matrix $\mathbf{A}$, there is a theorem about the Cholesky factorization $\mathbf{A}= \mathbf{L}\mathbf{L}^T$, where $\mathbf{L}$ is a lower ...
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50 views

Low rank update of QR of inverse

I am in a situation where as part of a sort of inverse power method scheme, I want to very often perform the following step: Apply a symmetric rank one update $uu^\top$ to my inverse matrix $A^{-1}$ ...
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1answer
110 views

Do most statistical packages and libraries in high-level programming languages rely on LAPACK for their matrix inversion operations?

Possible an open-ended question, but I am wondering if most statistical packages and libraries, for instance, Stata, R, Python's NumPy and MATLAB rely on LAPACK algorithms to perform matrix operations,...
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55 views

How to avoid unnecessary checks when inverting this LU decomposition

Background for the question I am currently working on a Matlab code in which the systems of linear equations $Ax_1 = b_1$, $Ax_2 = b_2$, ... have to be solved. As the matrix $A$ is constant during ...
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1answer
70 views

Weighted QR Implementation

Say I want a QR decomposition of matrix $A$, where orthogonality of $Q$ is with respect to a generic non-degenerate positive-definite bilinear form $\phi$ (in my case, $\phi$ is "defined" by a finite-...
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1answer
185 views

Cholesky for ill-conditioned/singular covariance matrices

Can someone suggest a way to get Cholesky factorization of a singular covariance matrix? I need it to match Cholesky on full-rank matrices, ie coordinate order should be preserved. My attempt below ...
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1answer
72 views

Incomplete LU decomposition of sparse matrix

I have a sparse matrix stored in CSR format. For this matrix, I would like to get the incomplete LU decomposition. I tried to find algorithms which can utilize the CSR format but I could not find ...
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1answer
84 views

How to use QZ decomposition for single matrix in Matlab?

Can I use QZ decomposition on a single square matrix in Matlab? Like, [Aa,Q,Z]=qz(A);
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1answer
64 views

Classical vs. modified Gram-Schmidt

It is often said that modified Gram-Schmidt is more robust with respect to rounding errors than classical Gram-Schmidt, but it is very hard to find a good explanation / example of why this is so. Can ...
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1answer
103 views

Givens rotation vs 2x2 Householder reflection

The usual story of Givens rotations vs Householder reflections is that Householder reflections are better if you want to map a long vector to $e_1$, while Givens is better if you want to map a 2-...
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1answer
53 views

Software for parallel incomplete LU factorisation

I am looking for a software package to compute incomplete LU factorisations in parallel. Further considerations are: The package must allow for arbitrary level-of-fill or threshold-based truncation. ...
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Regularized least squares with QR factorization

Consider the regularized least squares problem $$ \min_x || b - A x ||^2 + \lambda^2 ||x||^2 $$ which is equivalent to $$ \min_x \left|\left| \pmatrix{b \\ 0} - \pmatrix{A \\ \lambda I} x \right|\...
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141 views

Inverting a matrix from LU decomposition

The LAPACK routines xGETRI compute the inverse of a matrix $A = PLU$ in its LU decomposed form by first computing $U^{-1}$, and then solving the system: $$ (A^{-1} P) L = U^{-1} $$ My question is: ...
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1answer
90 views

Re-using LU factorization within iterative (?) setup for a sum of two matrices

So, I would love to make at least some use of my preexisting data, no matter how small, and just out of ideas. Maybe I am just a prisoner of a Kahneman-like theatre-ticket paradox, and don't know ...
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1answer
92 views

Pivoted Cholesky vs Modified Cholesky

I am solving nonlinear least squares problems with the normal equations approach, so on each iteration, I need to solve: $$ J^T J \delta = -J^T f $$ for the step $\delta$, where $J$ is a large (...
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32 views

ILUTP in sparse.linalg.spilu?

In Matlab, an ILU with threshold and pivoting (ILUTP) can be passed by default as: setup.type = 'ilutp'; [L, U] = ilu(A, setup); Looking for an equivalent in ...
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Numerical analysis, pivoting and incomplete LU decomposition

When doing LU decomposition, the algorithm will break down if any of the diagonal element $x_{ii}$ is zero. Therefore, we can use pivoting on the matrix such that $x_{ii}$ is no longer zero. That is ...
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1answer
222 views

Numerically find the nearest positive semi definite matrix to a symmetric matrix

I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
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Factorize laplacian in terms of first derivative matrix

I am trying to factorize the following Laplacian matrix in terms of $ D^TD$, D is the first derivative matrix. The tridiagonal form of the secon derivative matrix using Neumann boundary condition is ...
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what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works ...
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2answers
127 views

Factoring the sum of two matrices

Given \begin{equation} A_i=B+C_i \end{equation} where $A_i$,$B$ and $C_i$, $i=1,\dotsc,N$ are large square matrices, $B$ is symmetric, $C_i$ are zero matrices aside for a square block on the diagonal. ...
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Term for the typical “linear in the larger dimension, quadratic in the smaller” cost for linear algebra

Many dense linear algebra decompositions (QR, SVD...) on an $m\times n$ matrix have cost $$ O(\max(m,n)\min(m,n)^2) $$ when implemented in practice on a computer. Is there a colloquial name or a more ...
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Reweighted least squares factorization

This is a continuation of the question asked here. I want to solve numerous least squares systems of the form $$ D_i A x \approx D_i b $$ where $D_i$ are $m \times m$ diagonal matrices with positive ...
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1answer
585 views

Ways to solve $Ax=b$ for a sparse (banded) $A$ with updates

I want to solve the time-dependent Schrodinger Equation using the Crank-Nicolson scheme. I end up with the following matrix equation ...
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2answers
253 views

inertia count sparse matrix with dense low-rank perturbation

I would like to determine the number of negative eigenvalues (inertia count) of the $(N \times N)$ symmetric real matrix $K - \sigma M$, with $K$ a positive-definite sparse matrix and $M$ a positive-...
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1answer
207 views

Least Squares with Dense-Block Diagonal Structure

I need to solve a least squares problem that takes the following form: $$p = \arg \min_{x}\Vert J V x - y \Vert_2, $$ where $J \in \mathbb{R}^{N \times N}$ is a general dense matrix, and $V \in \...
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1answer
231 views

How we can use CUR decomposition in place of SVD decomposition?

I have understood how CUR and SVD works, but have not been able to understand the following. How can we use CUR in place of the SVD decomposition? Do the $C$ and $R$ matrices in the CUR follow the ...
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64 views

Stable Method of orthogonal projection onto a subspace with the help of Moore-Penrose inverse,

Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v $. Below is ...
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1answer
116 views

Matrix factorization empty rows and columns

I would like to do non negative mf and I wanna ask a question about my main matrix. The question is should I include rows and columns that have no non-zero entry in them. I think if there is not a ...
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1answer
150 views

Supernodal vs multifrontal factorizations

What are the differences between supernodal and multifrontal matrix factorizations? Can you provide a few references or high-level points about the approaches?
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Rapidly determining whether or not a dense matrix is of low rank

In a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, ...
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1answer
182 views

Matrix Decomposition of Conics

I was reading about ellipse-ellipse intersection and I came across this article: https://math.stackexchange.com/questions/679622/intersection-between-conic-and-line-in-homogeneous-space/867428#...
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1answer
365 views

Fastest way to solve a sparse unsymmetric system many times

I have to solve a system $Ax^{(n)} = b^{(n)}$ many times, $A$ being a sparse (pentadiagonal in most part of its structure), unsymmetric, constant matrix. Currently, I am performing the LU ...
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2answers
173 views

numerically stable routines to compute $M = B A^{-1} B$

Rather than gesv -> solve $AX = B$ gemm -> compute $M = BX$, somehow I feel there are better ways to compute $M$ with lapack/mkl?
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1answer
42 views

Householder QR for abstract vectors

Assume I have a sequence of vectors $v_k \in V$ in some abstract vector space $V$ not necessarily equal to $\mathbb{C}^n$. Can I still use the Householder QR decomposition in this case, even though ...
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Updating factorization of Laplacian (add/remove edges)

For a graph $G=(V,E)$, recall that the unweighted Laplacian is $L:=D^\top D$, where $D\in\{-1,0,1\}^{|E|\times|V|}$ is the graph "gradient" operator that subtracts adjacent vertex values onto edges. ...
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Rank-one Update to a Rank Revealing QR (RRQR) Factorization?

Suppose we are given an RRQR factorization for some matrix $A \in \mathbb{R}^{m \times n}$, $A\Pi = QR$ where $m > n$. Is there a cheap way to update $A' = A + uv^{\top}$ given this factorization?...
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1answer
88 views

$LU$ Factorization of a nonsingular matrix with a particular pattern

Consider $S\in\mathbb{R}^{n\times n}$ whose nonzero elements have the following pattern for $n = 8$: $$\begin{pmatrix} 1 & 0 & 0 & 0 & \mu_1 & 0 & 0 & 0\\ 0 & 1 &...
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2answers
287 views

Solving $A=B+AB$ without matrix inverse

I have a linear system of equations that can be expressed $$A=B+AB,$$ where $A$ and $B$ are real, symmetric matrices. I would like to solve for $A$ given $B$. At present, I solve for $A$ directly via $...
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The fast, and The Backward-Stable (left) $3\times 3$ matrix inverse

I need to compute a lot of $3\times3$ matrix inverses (for Newton iteration polar decomposition), with very small number of degenerate cases ($<0.1\%$). Explicit inverse (via matrix minors divided ...
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1answer
490 views

Clever ways to update LU factorization for ridge regression

Ridge regression can be posed as minimizing the following objective function (over $x$): $$\frac{1}{2} \lVert Ax - b \lVert_2^2 ~+ \frac{\lambda}{2} \lVert x \lVert_2^2 $$ Which has a closed form ...
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1answer
424 views

Code to update dense QR and Cholesky factorizations

I am looking for some production-ready code to update dense QR and/or Cholesky factorizations (by adding / removing rows and columns or making small-rank updates -- yes, I need all these cases). I ...
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1answer
363 views

Solve rank one update to LU using plain vanilla LU routine

I have a large number of systems of the form: $$A_ix=b,$$ where each $A_i,i>0$ is a rank one update of $A_{i-1}$ and the $A_i$ are dense matrices. I was wondering whether it is possible to use the ...
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102 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
260 views

Symmetric nonnegative matrix factorization

Suppose $A\in\mathbb{R}_+^{n\times n}$ is symmetric. I would like to factorize $A\approx UU^\top$ by solving $$ \begin{array}{rl} \min_U & \sum_{ij} \left(A_{ij}\ln\frac{A_{ij}}{[UU^\top]_{ij}}+[...