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Questions tagged [linear-algebra]

Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.

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433 views

Fast Eigenvalue and SVD Solver for Structured Matrices

I am looking for a fast Eigenvalue and SVD solver for small dense structured matrices (Hankel and Toeplitz). I have searched for efficient implementations in libraries like MKL but I am not able to ...
9
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0answers
148 views

Inverse problem in linear ODE

I have a linear ordinary differential equation (ODE) with a system matrix with constant coefficients: $$\dot{y}(t) = \mathcal{A}\; y(t), \quad y(0) = y_0$$ with $y(t) \in \mathbb{R}^{n \times 1}$ and $...
8
votes
0answers
178 views

Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
8
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0answers
208 views

Wanted: sequences of linear systems for recycling Krylov solver analysis

In the solution of sequences of linear systems $$A_ix_i=b_i\quad\text{for}\quad i=1,2,\dots$$ with Krylov subspace methods, data can be recycled from already solved linear systems in order to speed up ...
7
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0answers
207 views

Eigenvalue with largest imaginary part

Iterative eigensolvers such as ARPACK, give the option to find a subset of the eigenvalues which have the largest imaginary part. My question is how do these algorithms work. As I understand it, ...
6
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0answers
148 views

Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
6
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0answers
117 views

What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...
6
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0answers
107 views

How does an unpivoted QR fail to reveal rank?

An unpivoted QR factorization produces a triangular factor $R$. A rank-revealing QR factorization is typically done with column pivoting. My question is, how does an unpivoted QR factorization fail to ...
6
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0answers
160 views

Find the solution of linear equation using Wiedemann/ Krylov method

Let given $M =$ 1 0 1 0 1 1 1 1 1 and $b =$ 1 0 1 How to find the solution $x_3$ where $x=${$...
5
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0answers
254 views

How do I implement a working Lanczos biorthogonalization method for Krylov subspace?

I am trying to implement Lanczos biorthogonalization algorithm to construct a Krylov subspace basis. Given a matrix $A$, and vectors $v,w:(v,w)=1$, it produces three matrices $V,W,T$ such that $V^T W =...
5
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0answers
122 views

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?...
5
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270 views

Compute sparsity pattern of $A^2$

Suppose we have a sparse matrix $A$. Is there any way to compute just the sparsity pattern of $A^2 = A \cdot A$ (I do not actually need to know what exactly the nonzero value are) faster than to ...
5
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0answers
139 views

Galerkin FEM error when using even number of elements

Intro: I am developing a Galerkin FEM code in matlab - starting small with a simple 1D ODE. The equation I'm trying to solve is: $$a u_x = cos(x), x \in [0, 2\pi]$$ Which has a known exact solution $$...
5
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0answers
334 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula $$ (A+uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u} $$ results in small errors in relation to the standard matrix inverse operation after each application, ...
4
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0answers
61 views

Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
4
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0answers
99 views

How to stochastically estimate the trace of a matrix?

Specifically, the diagonal elements (can possibly both positive and negative) of the matrix can be computed efficiently but the total number is large ($\mathcal O(10^{18})$). My first thought about ...
4
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0answers
79 views

Computing Algebraic Riccati inequality

In my Robust model control, I have got a couple of quadratic Riccati inequalities which need to be solved numerically on MATLAB. My question, Is there function on MATLAB can solve quadratic Riccati ...
4
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0answers
145 views

Eigenvalues with absolute values close to 1

I have a generalized eigen-problem: $A\psi = \lambda B \psi$ with $A$ and $B$ are large (>1000) complex non-Hermitian matrices. I know that eigenvalues with largest and smallest absolute values can be ...
4
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0answers
98 views

Conjugate gradient: the 1-norm of the residual

I am trying to solve $Ax=b$ using the conjugate gradient method. However, it is important to me to obtain a bound not only on the usual residual $||b-Ax_k||_2$ but also on the quantity $||b-Ax_k||_1$. ...
4
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0answers
97 views

Tracking for two meshes

I'm dealing with physical simulation (position based dynamic). Now I'm trying to realize tracking for two meshes. In order to explain what does it mean, let's assume that we have two similar meshes: ...
3
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0answers
60 views

Block matrix and DSYRK

I want to compute the matrix $$ A = \sum_{i=1}^N v_i v_i^T $$ where each $v_i$ is a given vector of length $2500$, so that $A$ is $2500 \times 2500$, and my $N$ is about 2 million. Rather than call ...
3
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0answers
79 views

Why the MIRACLE of Lanczos/CG-like?

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
3
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0answers
65 views

Element Preconditioner

Im just working on a preconditioner for the linear equation system $Ax = b$ arising in FEM for elliptic PDE. $A$ is a s.p.d Matrix with real valued entries. I read something about the element by ...
3
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0answers
115 views

Computing Small Eigenvalues with Sparse Symmetric Indefinite Mass Matrix

I want the eigenvalues of the following generalized eigenvalue problem: $$ Av = \lambda M v $$ where $A\in\mathbb{R}^{n\times n}$ is sparse, symmetric, and positive semi-definite $M\in\mathbb{R}^{n\...
3
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0answers
88 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
3
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0answers
100 views

Iteratively solving a sparse, ill-conditioned system

I have a sparse (density = 0.2%), ill-conditioned system that I am trying to solve, with no luck. Background I have a sequence of sampled data, where two of every 8 samples have been zeroed due to a ...
3
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0answers
71 views

Left eigenvectors using ARPACK

I'm trying to find both the dominant $k$ left and right eigenvectors, that is, $$V_L\mathcal{A} = \Lambda V_L\\ \mathcal{A}V_R = V_R\Lambda\\ V_LV_R = I_{k\times k}$$ $V_L$ being the $k\times N$ ...
3
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0answers
67 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
votes
0answers
96 views

Eigenvalue decomposition of the sum: $AA^T$ + diag($u$)

Suppose $A\in\mathbb{R}^{n\times c}$,$u\in\mathbb{R}^n$,$n\gg c$. The time complexity of eigenvalue decomposing directly for matrix $AA^T+\text{diag}(u)$ is $O(n^3)$. And it is easy to avoid $O(n^3)$ ...
3
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0answers
233 views

Quasi Newton method for block diagonal Hessian

I am having an unconstrained optimization problem, where the Hessian is positive semidefinite and block diagonal. The function is strictly convex, hence, the curvature condition ($ s^{T}_{k}y_{k} > ...
3
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0answers
72 views

Does there exist a Fourier transform algorithm for perturbed data?

Assuming I have a length-$n$ real vector $x$ and have already computed its Fourier transform $\hat x$ (in time $O(n\log n)$), I would like to compute the Fourier transform of $y = x + \delta x$, where ...
3
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0answers
140 views

Find constrained vectors maximizing angles between them - methods?

This is related to a question I had asked earlier, with the distinction that earlier I did not have a non-linear objective functional to minimize. The problem is reproduced below with added ...
3
votes
0answers
361 views

full rank update to cholesky decomposition for multivariate normal distribution

This question is a specialization of full rank update to cholesky decomposition, to which I hope to get a more positive answer. When calculating the minus log of the multivariate normal distribution, ...
3
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0answers
165 views

How to accurately decompose positive semidefinite matrix and use the lower triangular part in linear equations

I have $n$ arbitrary $p\times 1$ vectors $x_i$, and $p\times k$ matrices $A_i$, and $n$ $p \times p$ positive semidefinite matrices $S_i$, where some (often most) of the $S_i$'s are same (for example ...
3
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0answers
102 views

Efficient principal pivots

It was suggested I should try posting this question here from Mathematics Background I'm working on a numerical linear algebra package in C#. I'm trying to implement a variety of "principal ...
3
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0answers
107 views

Up-/downdating methods for a series of normal equations

In an application I have to solve a series of positive definite linear systems of the form $A^TA x = A^Tb$ (i.e. normal equations). The next system is obtained from the previous one by adding and/or ...
2
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0answers
88 views

Discrete-time Algebraic Riccati Equation (DARE) solver in C++

I need to use a Discrete-time Algebraic Riccati Equation (DARE) solver for an embedded controller (with limited processing power) in a research project and sadly, I can't find any implementation of it ...
2
votes
0answers
81 views

A Bound for the inverse of the sum of identity and triangular matrix

I wonder if there are any theorems which can help me to calculate an upper bound for the spectral norm of: $$\left\| \left[ I + \sum_{i=1}^{\overline{n}\in\mathbb{N}} \big( C_i - I\big)\right]^{-1}\...
2
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0answers
24 views

Randomized Submatrix of a Sparse Matrix

I have a sparse square matrix $A$ with size $n \times n$ and number of nonzero entries $nnz$. The goal is making a sub-matrix $B$ with $s$ nonzeros which are randomly chosen from $A$. Duplicates are ...
2
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0answers
67 views

Simultaneous update to barycenters

Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex. Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
2
votes
0answers
101 views

Efficiently solve linear system with matrix quadratic form

Take the system $$A^TCAx=b$$ where $$A\in\mathbb{R}^{n\times m},\;C\in\mathbb{R}^{n\times n},\;x,b\in\mathbb{R}^{m}, \;m\leq n$$ and $$A^TA=I$$ and $$Cy=d$$ can be solved efficiently in general (...
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0answers
68 views

Acceleration of matrix geometric series

Suppose we want to find $x$ such that: $$x=b+Ax$$ where $A$ is a large sparse square matrix with eigenvalues in the unit circle. There are two representations of the solution: 1) $$x=(I-A)^{-1}b,$$...
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0answers
53 views

Parallel compact schemes using the Parallal Diagonal Dominant (PDD) algorithm

I would like to use the PDD algorithm developed by Sun to solve tridiagonal matrices in parallel for the following compact finite difference scheme: $ \begin{align} \dfrac{1}{4}f^{'}_{i-1} + f^{'}_i +...
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0answers
55 views

How can I numerically solve a saddle point problem with repeated constraints?

I am interested in numerically solving the following constrained minimization problem; Find the value of $x\in \mathbb{R}^n$ that minimizes $f$ where $f\colon \mathbb{R}^n\to \mathbb{R}$ is defined ...
2
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0answers
167 views

Connectivity and Clustering using Eigenvectors and the Fiedler Vector

Going off of the answer here: sorting adjacency matrix by the Fiedler vector So here, Jesse the answerer plotted the first 3 eigenvectors associated with non-zero eigenvalues of the Laplacian against ...
2
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0answers
144 views

Does applying the Newton-Raphson iteration for matrix reciprocal refine a matrix inverse from LU/GE?

This is a follow-up to this answer. Suppose you have a possibly very ill-conditioned matrix $A$, and you compute its inverse with LU/GE to get $X_{\text{lu}}\approx A^{-1}$. The Newton-Raphson ...
2
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0answers
46 views

Optimal ordering in Jacobi SVD algorithm

In Jacobi SVD algorithm as given here every pair of columns of the matrix is orthogonalized until convergence. I want to know that how does the order of selection of the pair of columns affect the ...
2
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0answers
359 views

Fixing a near singular covariance matrix

Given a near singular covariance matrix, the standard method of 'fixing' it seems to be to add a small damping coefficient $c>0$ to the diagonal, which serves to bump all the eigenvalues up by this ...
2
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0answers
51 views

SPECT reconstrction using MLEM

In Single-Photon Emission Computerized Tomography (SPECT) parallel beam reconstruction using Maximum-Likelihood Expectation–Maximization(MLEM), is it sufficient to scan the object around 180 degree? ...
2
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0answers
219 views

Computing eigenvectors from the QR algorithm

I've seen a few other posts on this topic but none have full answers. I'm trying to implement some eigen-decomposition algorithms. I've managed to get the Explicit QR algorithm and the Implicit (...