Questions tagged [linear-system]

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How to verify solution to pre-conditioned linear systems solver?

I am solving Ax=b. A has a very large condition number (> O(10^10)) I am using the conjugate gradients method with point jacobi pre-conditioning. I obtained a solution 'x' that "looks" reasonable. ...
289 views

Why does PETSc take unexpectedly long to set up its KSP solver with a custom preconditioner? [closed]

I am attempting to solve a large system, $\bf{Ax} = \bf{b}$ with the help of PETSc. Due to the size of the problem, I'm using a matrix-free approach, where $\bf{A}$ is just a shell. I'm also providing ...
194 views

Bifurcation of linear PDE

I have a linear elliptic PDE (unfortunately not allowed to be shown here) with a constant parameter $\epsilon$ giving the stable solutions qualitatively shown by the functions below. As we smoothly ...
145 views

Solve ill-posed linear system without transposing matrices?

I am attempting to use an iterative solver to solve $p$ in $$Jp = -r$$ where $J$ is an $m\times m$ matrix ($m$ is in the order of $10^5$ and never explicitly stored). $J$ is a dense matrix ...
56 views

Integrating the 2d vorticity equation on periodic boundaries

This question is a follow-up of https://stackoverflow.com/questions/44718160/solve-a-linear-system-for-fft-coefficients At some time (kt), the FT of the vorticity (omega) satisfies: ...
125 views

I have two variations of an iterative algorithm. All the steps of both algorithms are equivalent except one. In this step: Algorithm 1 needs to compute the matrix $ABA^T$ for matrices $A \in \mathbb{... 0answers 63 views FEM/FVM/FD for structural modeling and stability issues due to large structural constants? I've read that in modeling structures problems, the finite element method (FEM) is typically used. I am unfamiliar with FEM, but I am wondering, in particular, if using FEM, as opposed to finite ... 1answer 464 views Fastest way to solve a sparse unsymmetric system many times I have to solve a system$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 ... 1answer 83 views MATLAB: Matrix whose elements depend on its indicies I am trying to put the function $$f(\mu,\nu) = i^{\nu-\mu} \sum_{0}^{19} H_{\mu-\nu}(7j) + \delta_{\mu,\nu}\ ,$$$\mu, \nu =-3,-2,...2,3$into a 7x7 matrix, where$H$is the Hankel function of the ... 1answer 273 views The system matrix and the right hand side for diffusion equation with staggered grid In the following staggered grid setting, I want to solve diffusion equation as a linear system. $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}... 1answer 104 views Solving system of related equations without completely recomputing LU decomposition for each equation Let \Sigma be positive definite and the D_i positive diagonal. Let the X_i be unknown square matrices. Consider the system of equations:$$(I+\Sigma D_i)X_i=\Sigma\hspace{5mm}\text{for}\... 2answers 339 views Solve a very large linear system (question about a library linear algebra to do this) I need to solve a very large linear system (coming from finite element method). I'm currently using the Intel MKL library, but the system has been delayed more than 20 hours. The matrix of the system ... 2answers 4k views Solving a system of linear equations with only an approximate solution I have a system of linear equations that is derived partially from experimental data. Theoretically, the system should have a single, exact solution; however, experimental error causes it to not have ... 1answer 584 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 ... 1answer 240 views Solving large system of equations, is linear programming best option? [closed] I have a problem where I am trying to solve many systems of equations, that have very few variables per equation, but a lot of equations. For example potentially 10 variables max in a single equation,... 5answers 1k views Speed of solving linear system with block diagonal matrix I have a bunch of 3x3 linear systems of the form$Ax=b$. In general, would it be faster to solve each individual system, or to formulate it as a giant block diagonal system and solve that? I expect ... 1answer 432 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\...
249 views

Suppose we have two matrices $A$ and $B$ (we can assume they're symmetric; if absolutely necessary I think they may be positive definite). Then, is there any technique for solving $$(A\circ B)x=b,$$ ...