# Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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### solving linear system whose symmetrized matrix is positive definite

Are there iterative methods for the solution of nonsymmetric linear systems $Ax=b$ that can take (theoretical or practical) advantage from knowing that $A+A^T$ is positive definite? These matrices are ...
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### Preconditioning vs. regularization

I used to be more of a numerical linear algebra and computational science person, but recently, I've crossed into stats and machine learning. For this discussion, let's focus on matrices that are not ...
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### Are direct solvers affected by the condition number of a matrix?

If I were to solve a relatively small problem, that is, a problem that can be handled by a direct method like LU, then does the condition number of the linear operator affect the accuracy of the ...
197 views

### 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. ...
83 views

### Simplest solver for linear equation systems

Normally, this boards sees a lot of traffic about the most efficient and most powerful solvers for huge linear equation systems. But this time, I have the opposite problem: I need to implement a ...
158 views

### Linear solver recommendation(s) for small problems

I am interested in solving many linear systems $Ax = b$, where $A$ is symmetric positive definite and small (i.e. less than 25,000 rows) --- $b$ will be changing. We can assume that $A$ arises from ...
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### Is there an overview of the runtime speed up of LP/MIP solvers throughout the years?

whenever I read papers on OR that use an LP/MIP approach, they include the time solver used, as well as the version and the year. I would like to know how much faster the same experiment would be ...
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### How is Krylov-accelerated Multigrid (using MG as a preconditioner) motivated?

Multigrid (MG) may be used to solve a linear system $Ax=b$ by constructing an initial guess $x_0$ and repeating the following for $i=0,1..$ until convergence: Compute the residual $r_i = b-Ax_i$ ...
64 views

### Solve for large array of PD matrices

I have N matrices that are positive definite, and I have to solve for a M vectors. As M is large in my case, doing all solves simultaneously using np.linalg.solve ...
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### 2-norm of solution update suddenly becomes zero after a few iterations

I am trying to solve the Poisson equation in 2D for heterostructure devices. I have linearized the equation and discretized it using FDM. I am using BiCGStab to iteratively solve for the solution as ...
263 views

### When is it easy to invert a sparse matrix?

(Crossposted on cstheory.SE) When is it easy to invert a sparse matrix? Specifically, I'm wondering about the cases in which matrix inversion has similar cost to sparse matrix multiplication, hence ...
51 views

### Algebraic multigrid as solver and as preconditioner

My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
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### Solving PDEs in parallel

I have read different approaches on how to solve pdes in parallel which are discretized using finite element method. For example: Non-overlapping domain decomposition approach as mentioned in https://...
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### Automatic selection of the SLE solver and preconditioner during simulation

To simulate the physical process necessary to solve the arising systems of linear algebraic equations. The SLE matrix has a highly sparse form. There are a couple dozen non-zero elements in the string,...
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### Iterative single variable solutions in large linear systems

I have a system where $A$ is a large $n\times n$ marix with fast MVMs. It may have many nonzero entries (albeit in a structured way so as to allow fast MVMs), and is not necessarily diagonally ...
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### How can the choice of coarsening factor affect Multigrid's convergence?

The linear system $Ax=b$ is coming from the discretization of an elliptic PDE. Multigrid method is used in order to solve it. Suppose $c_0$ is the coarsening factor on level 0 and $c_m$ the coarsening ...
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### Convergence of Conjugate Gradient Algorithm

I am trying to solve a linear elasticity model using finite element discretization in a rectangle domain [0,1]x[0,1]. For the solution of the the linear system $Ku=F$ I am using the CG algorithm. ...
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### Open-source iterative solvers robust to noise?

I need to solve $Ax=b$ in about 1 million dimensions. Furthermore, A is only accessible through matrix vector products, and these are noisy/inexact. Is there any solver with Python interface I can try ...
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### Solving MX=N where M is structured as a Gaussian 4th-moment tensor

I'm looking to solve numerically the following equation for $(d,d)$ variable $X$, in Einstein summation notation $$M_{ijkl}X_{kl}=N_{ij}$$ Where $M$ is a $(d,d,d,d)$ 4th-moment tensor of random ...
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### Is there any function to calculate condition number of sparse matrix in Eigen libraray?

The function JacobiSVD and BDCSVD can calcuate condtion number of a dense matrix via singular values. However I need to know condition number of a sparese matrix due to slow computation speed using ...
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### Solving an m x m symmetric linear system involving a matrix multiplication versus an (n+m) x (n+m) system

Suppose that $R$ and $D$ are an $n \times m$ and $m \times m$ matrices. Assume that $m \ll n$ and that $D$ is positive definite. We would like to solve the system $(R^T R + D) x = R^T b$. This ...
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### Is there any reason to scale a matrix before (sparse) Cholesky decomposition?

I have a sparse symmetric positive-definite matrix $M$ and I expect the entries in some rows/columns to have very different orders of magnitude (up to a factor of $10^8$) than the entries in others. ...
201 views

### What happens when I use a conjugate gradient solver with a symmetric positive semi-definite matrix?

I have a symmetric positive semi-definite matrix, i.e., a laplacian and wonder what may happen when I use a CG solver, that is an algorithm for positive definite matrices. What happens when the ...
166 views

### More stable method of back substitution?

I've been tinkering a little in Fortran (2008) and wrote the following to solve $Rx=b$ for $R\in\mathbb{R}^{n\times n}$ upper-triangular, $x,b\in\mathbb{R}^n$. My code looks like this: ...
131 views

### Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
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### Solving triangular matrix equations on a GPU

Suppose I have these two $N\times N$ lower triangular banded matrices: \$A = \begin{bmatrix} a_0 & & \\ a_1 & a_0 & \\ a_2 & a_1 & a_0 \\ a_3 & a_2 & a_1 & a_0 \\ &...