# Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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### Stable iterative solver for complex symmetric linear systems

I am interested in the iterative solution (preferably Krylov-type solvers) of a problem Ax=b, with x, b ∈ ℂn x 1 and A ∈ ℂn x n. A is symmetric, invertible, and its real and imaginary parts are ...
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### Solving absolute value systems

Let $z, b \in \mathbb R^n$, $A \in M_n (\mathbb R)$ and $|z| := (|z_1|, \dots, |z_n|)$. I am searching for an efficient algorithm to solve the absolute value system: z - A |z| = b. \...
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### Solving multiple linear regression in parallel

I am working on a problem where I need to solve approximately 500 Million Linear Regressions (OLS). What would be the most efficient way to do this (e.g. using GPU or a some framework that can do this ...
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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 ...
279 views

### How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly

My post is structured in four parts: I give you some information about the context my principal questions refer to. I will tell you what I believe to know about the Thomas Algorithm. If I am wrong ...
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### Preconditioning the $[1 \quad-2 \quad 1]$ Finite Difference matrix

Let $A$ be the well known tridiagonal matrix coming from the 1D Finite difference discretization of the Laplacian, with stencil $\frac{[1 \quad-2 \quad 1]}{h^2}$. The system $Ax = b$ is very large, so ...
161 views

### Solving large sparse system

I am working on a problem with very large sparse matrices. I'd like to compute $A^{-1} B$, that is a crucial part of converting DAE to ODE (and there is no workaround). Here size of $A$ is 2E+5 x 2E+5 ...
158 views

### Efficient way to solve a set of linear equations $Ax=b$ when $A$ is sparse and some elements of $b$ are equal to zero

I have a set of linear equations, $Ax=b$. And about half of the elements in the right-hand side (vector $b$) are equal to zero. My system matrix $A$ is a sparse complex matrix. And $A$ is in the size ...
334 views

### Is there an iterative solver for dense matrices with possible zero diagonal entries?

Is there an iterative solver that can handle potentially zero entries on the central diagonal? I am implementing a polynomial fitting algorithm (up to $10^{th}$-order) and my matrix is a "...
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### Solver for large dense BVP system in python

I have a large system of boundary value problems of the form $$\frac{d^2 y }{dt^2} = C(t) y + b(t),$$ where the variable $y$ is a vector that has anywhere from 50 to around 500 components, $C$ is a ...
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### Solving a sparse linear system using transpose of lower triangular matrix without copying

I have a sparse lower-triangular matrix $L$, and a right-hand side $b$, and I'd like to solve the linear system $$L^T x = b$$ but without explicitly creating $L^T$. Ideally, I could write something ...
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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. ...
251 views

### Why OpenFOAM uses its own data structures and linear solvers?

I wonder why OpenFOAM code has its own data structures Lists, HashTables, ... etc. when there is the STL in C++? Another ...
310 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 ...
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### Solution of symmeric/non-symmetric linear system

I would like to understand what happens in the following: I have a really simple Poisson problem, in 1D, with $u_0 = u_N = 0$. I assembled the stiffness matrix and the right-hand side, and I applied ...
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: ...
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 \\ &...