# Questions tagged [linear-system]

For questions about solving linear systems of equations. These typically take the form Ax=b, where A is a matrix , while x and b are vectors.

78 questions
Filter by
Sorted by
Tagged with
6 views

### Solving huge dense square simmetric linear system

I have a linear system of the type $A x = y$ where A is a dense, square, symmetric, positive definite matrix. I know the Cholesky decomposition should be the fastest way to solve the system in a least ...
42 views

### Pass forward intermediate results during iterative optimization

To investigate a counter-current flow heat exchanger while considering temperature dependent physical properties (such as specific heat $c_\textrm{p,i}$, heat conductivity $\lambda_\textrm{i}$, ...
89 views

### Elementary question on numerical linear algebra

I’ve been facing a problem of solving linear systems Ax=b arising from discretized PDEs (Stokes equations in particular). Nively, it seems that solving Ax=b should not take much more time than simply ...
393 views

### When do not use preconditioners for sparse linear system of equations?

I'm implementing a solver of Finite Element Method, and to solve the linear system of equations I'm using gmres from MKL of Intel. Exists the option with and without a preconditioning. In what case it ...
189 views

### 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 ...
557 views

### Solution of the linear system using Sherman-Morrison formula for 1000000x1000000 (7450.6GB) matrix using MATLAB

Let $n = 10^6.$ Let $A \in \mathbb{R}^{n\times n}$ be the lower triangular matrix having 1's on and below the main diagonal. We want to solve the following linear system: $$(A + uv^T)x = b$$ by the ...
76 views

### Solution of an underdetermined system stemming from a PDE with Neumann BC

Consider the Poisson's equation in 1D with homogeneous b.c.'s $\mathrm{d} \phi/\mathrm{d} x=0$ with the seven point Laplacian (1 -54 783 -1460 783 -54 1 / 576 on a uniform grid). The resulting system ...
201 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. ...
89 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 ...
644 views

### Solve a system of coupled differential equations in Python

I have a system of two coupled differential equations, one is a third-order and the second is second-order. I am looking for a way to solve it in Python. I would be extremely grateful for any advice ...
69 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.
79 views

### Solving a linear system with block-banded matrix

It is required to solve a linear system $Ax=b$, where the matrix $A$ is symmetric, all the variables and coefficients are real. The structure of $A$ is A = \begin{pmatrix} A_{11} &...
68 views

### Kronecker-factored least-squares?

Suppose $a_i,b_i$ are $d\times 1$ matrices, $y_i$ is scalar, and I need to find least-squares solution $w$ of the following system of $n$ equations: $$y_i=(a_i^T\otimes b_i^T)w$$ Is there a ...
159 views

### Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular

I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ...
196 views

150 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 ...
97 views

### Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?

I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
57 views

194 views

145 views

### How to obtain linear tridiagonal system from PDE

I'm trying to re-solve the governing equations in hydraulic fracturing modeling as instructed step by step in a paper. After (A-9), the author stated that by substituting A-6, A-8 and A-9 into ...
124 views

### Iterative solution of ill-conditioned matrix systems

I want to solve a matrix system of the form $Ax=b$ where $A$ is ill-conditioned. The matrix system comes from a structural simulation problem which was discretized using finite elements. I do not have ...
36 views

### What will PDE discretization matrix look like for time and space? [duplicate]

Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
79 views

### Analytic vs discrete understanding of PDE

The PDE I am working with: $$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\ \partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\ \beta(x)>0$$ Integrate the PDE: \int_\Omega \partial_t u=\...
146 views

### Is steady linear elasticity inherently ill-conditioned?

Compared to the transient PDE for linear elasticity, the steady equations appear to less well-conditioned. Are they inherently ill-conditioned without the transient term? The condition number for ...
277 views

### Condition number of matrix and effects of round off errors

In my numerical linear algebra class, I learned that for some matrices, it could have an element that is a very small number that is approximately 0 (and many orders of magnitude different from all ...