# Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

268 questions
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### Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
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### Solving a singular system of linear equations with smallest solution?

I have a problem where I am solving a system of linear equations but sometimes the system results in a singular matrix which cannot be easily solved. In this case I would like that those rows for ...
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### Preconditioning matrix with known spectrum

Assume I know all eigenvalues of a matrix $A$ fall into a certain set $\Omega \subset \mathbb{C}$. Is there any way I can exploit this knowledge to design a preconditioner for $A$? Some further ...
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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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### CHOLMOD implementation

I am working on a domain decomposition code in C that uses CHOLMOD to approximate grid values for a PDE in each sub-domain. The issue I have is that the methods use Matrix Market format, which is not ...
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### Open Source Linear Algebra Library

I am making a code in C that requires the equivalent of Matlab's '\' command for a linear system of the form AX=B where A is an NxN matrix and X, B are Nx1 vectors- i.e a code that performs X=A\B that ...
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### Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$u^{k+1} = u^k - \alpha_{k+1}(Au^k - f),$$ The second degree ...
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### Does algebraic multigrid reuse its coarse grids?

Admittedly, I'm new to the subject, so this is probably a really simple question. Let's assume I want to solve the (large sparse) linear system Ax = b multiple times with algebraic multigrid and b ...
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### Parallel linear algebra without OpenMP

I have searched through the archives without success. Apparently, the question is simple: What linear algebra library can I use that is parallel (shared memory) but without OpenMP? As far as I've ...
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### NONLINEAR ENERGY MINIMIZATION EXAMPLE

I am learning about FEM methods and nonlinear optimization. I would like to try my nonlinear trust region solver on some simple nonlinear problem. What would be good example to implement for ...
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### Once and for all: Which FEM plattform should I use for a very large multiphysics simulation?

I'm fooling around with the decision on how to build a multiphysics simulation for too long now (also several questions in this forum). First, I thought it would be possible/necessary to write most of ...
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### Numerical method for solving a system with positive definite blocks

I have a system with below coefficient matrix $$C = \begin{pmatrix} A & B^T \\ B & D \end{pmatrix},$$ where, $A$ and $D$ are square and positive definite. Furthermore, if $B$ be square, ...
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### Solving linear system $Ax=b$ with Hessenberg matrix using lapack

I need to solve a linear system of the form $$Ax = b$$ where $A$ is upper Hessenberg matrix with the lower bandwidth equal to 1, $b$ is the RHS vector and $x$ is the solution vector. I have a C++ ...
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### Robust smoothers for geometric multigrid

I'm searching for robust smoothers for geometric multigrids. By robust I mean: Effective for high order approximations (say spectral element, spectral Discontinuous Galerkin), Parallel (suitable for ...
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### “Cookbook” about iterative linear solvers and preconditioners

I'm using a lot of linear solvers and preconditioners, but mostly, they are magical black boxes to me. Since I'll also have to implement some of them in future, I would like to learn a bit more, ...
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### Single Precision a x plus y (SAXPY) terminology

I've been reading books which refers to vector update operations of the form: y := y + ax, where y and x are vector variables and a is a scalar as SAXPY. I understand ax plus y part, but why "single ...
I have a standard set of linear equations $Ax=b$ where the Hessian matrix $A$ has the special block structure as shown: $A= \begin{pmatrix} T & U\\ U^T & V \end{pmatrix}$, $x= \begin{... 1answer 156 views ### solving tridiagonal system with multiple right hand sides I need to solve a tridiagonal system (positive definite, diagonally dominant)$Ax = b$in a time stepping loop.$A \in \mathbb{R}^{N \times N}$remains constant but$b$changes during each time ... 1answer 247 views ### Linear Systems with Multiple Right Hand sides I am interested in solving a sequence of linear systems of the form: $$A x_i = b_i$$ That is, all the systems use the same matrix$A$but they have different right hand sides. The matrix$A$is sparse ... 1answer 238 views ### Methods for solving rectangular, full-rank systems of equations — which is best? Suppose I have a large, sparse,$m \times n$matrix$A$, with$m \gt n$and$\text{rank}(A) = n$. I wish to solve$Ax=b$. Suppose I know that$A$has the following characteristics:$A$is somewhat ... 3answers 274 views ### role of initial guess for iterative linear solver Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ... 1answer 1k views ### Solving linear systems with ill-conditioned matrices As per suggestions of the people from MathOverflow, I'm reposting my question here: I'm currently trying to solve a linear system$Ax = B$, where the matrix$A$is ill conditioned (i.e. nearly ... 3answers 258 views ### How to assemble Global matrix (for coupled) problem? I'm trying to assemble global matrices for the following system. $$\begin{bmatrix} K& Q\\ Q^T&S \end{bmatrix} \begin{bmatrix} u_h\\ p_h \end{bmatrix} = \begin{bmatrix} f_u\\ f_p \end{... 0answers 81 views ### Solve for D in R^{T}DSDR = Id Given that R is a rectangular matrix, D is a diagonal, square matrix and S being a square matrix along with the fact that both D and S are invertible. S in this specific case can be ... 1answer 157 views ### Resources for solving mixed left and right matrix equations I'm looking to solve a matrix equation and not sure where to start looking for resources. The equation is$$AX + XB = C\,,$$where$A\in\mathbb{R}^{n\times n}$,$B\in\mathbb{R}^{m\times m}$,$C\in\...
For a system $\mathbf{x=Da}$, there exist a lot of algorithms to estimate sparse vector $\mathbf{a}$. I wish to know the big-O mathematical complexity of 1) orthogonal matching pursuit (OMP) both ...