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Questions tagged [linear-system]

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2
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1answer
61 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 ...
3
votes
1answer
63 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 ...
1
vote
0answers
41 views

Thermal stress to displacement through finite volume method

I am attempting to solve for a displacement field where I know the thermal stresses on my discretized domain, which consists of hex cells. My first question is: is easier way to solve for the ...
1
vote
1answer
57 views

TDMA with 3rd order upwind scheme

I'm trying to implement a model I found in a paper, but there is something I do not understand. The authors say they use TDMA to solve their equations; however, they use a 3rd order upwind biased ...
4
votes
2answers
125 views

How “sparse” should a sparse matrix be to see benefits?

I have a matrix, whose size scales as $2^N$ (assume even $N$). In each row of the matrix, only about $2^{N/2}$ of the entries are filled ($N$ can be somewhere between 10 and 40, depending on what's ...
1
vote
2answers
87 views

What are the names of the variables in the linear system $Ax=b$

I've been discretizing PDEs and formulating $Ax=b$ systems, and yet I don't really know what the $A$ and $b$ are in words. I occasionally call the $A$ matrix the "Jacobian matrix," but for linear ...
2
votes
2answers
186 views

Automatic Differentiation - reverse accumulation of linear system solve

I am studying the reverse mode of automatic differentiation. The reverse mode of automatic differentiation allows the efficient computation of a the derivative of a single dependent variable $y$ with ...
4
votes
1answer
55 views

How to preserve or recover states meanings in canonical state-space realizations

Let assume that one would like to translate an input-output model, into a state-space one. It is well-known that the process of realization addresses this aspect. Let assume to have the following ...
0
votes
0answers
74 views

Forming the Cholesky decomposition of $A=D+X^TX$ without computing $X^TX$ [duplicate]

My understanding from this answer here, is that if I'm attempting to form the Cholesky decomposition of a positive-definite matrix $$A=X^TX$$ then it's best in terms of numerical accuracy to never ...
4
votes
2answers
126 views

Solve $Ax=b$ repeatedly where $A$ is a sparse weighted Laplacian matrix with changing weights

In the problem I am dealing with, I require to repeatedly solve $Ax=b$ where $A$ is a weighted Laplacian matrix of a sparse graph. The right-hand side remains constant. However each time I solve the ...
0
votes
1answer
137 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. ...
1
vote
0answers
142 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 ...
0
votes
1answer
175 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 ...
4
votes
0answers
113 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. Unfortunately, this is an ill-posed problem and possibly infinite versions of $p$ exist. ...
1
vote
0answers
44 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: ...
0
votes
2answers
96 views

Benefits of matrix multiply over inversion

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{...
1
vote
0answers
50 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 ...
2
votes
1answer
247 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 ...
0
votes
1answer
80 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 ...
0
votes
1answer
232 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}$...
2
votes
1answer
83 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}\...
0
votes
1answer
152 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 ...
5
votes
2answers
2k 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 ...
3
votes
1answer
207 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 ...
-1
votes
1answer
148 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,...
1
vote
0answers
82 views

Block preconditioners and condition number

I am working with block Jacobi like preconditioners which are very cheap for my problem. But I could not find much about the dynamics of basic preconditioners (block Jacobi, Gauss-Seidel, ILU etc). ...
3
votes
5answers
885 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 ...
7
votes
1answer
292 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\...
5
votes
2answers
189 views

Solving “Hadamard systems”

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 $$(A\circ B)x=b,$$ ...
3
votes
1answer
903 views

solving a linearly-constrained sparse linear least-squares problem

[ question reposted from https://math.stackexchange.com/questions/786612/solving-a-linearly-constrained-sparse-linear-least-squares-problem ] Given the system of equations $Ax=b$, subject to $Cx\le ...
5
votes
3answers
298 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
5
votes
3answers
1k views

What is the best solver for solving a large sparse indefinite system

What's the best solver that can solve a large sparse but indefinite matrix?
1
vote
2answers
1k views

Vortex Panel Method implementation

I'm trying to understand and implement panel methods for a two-dimensional airfoil. I haven't found yet a very detailed explanation on how to implement it, and there are some things I don't' still ...
10
votes
5answers
8k views

Best choice of solver for a large sparse symmetric (but not positive definite) system

I am presently working on solving very large symmetric (but not positive definite) systems, generated by some certain algorithms. These matrices have a nice block sparsity which can be used for ...
12
votes
5answers
2k views

Repeatedly solving $\mathbf{A} \mathbf{x} = \mathbf{b}$ with same $\mathbf{A}$, different $\mathbf{b}$

I am using MATLAB to solve a problem that involves solving $\mathbf{A} \mathbf{x}=\mathbf{b}$ at every timestep, where $\mathbf{b}$ changes with time. Right now, I am accomplishing this using MATLAB'...
5
votes
3answers
341 views

PRIMA gives an unstable result?

I am working with Modified Nodal Admittance representation of circuits. I am doing Model Order Reduction using PRIMA on MATLAB. I am considering these circuits as Descriptor State-Space systems. I ...
11
votes
1answer
526 views

Solving huge dense linear system?

Is there any hope in solving the following linear system efficiently with an iterative method? $A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^n, b \in \mathbb{R}^n \text{, with } n > 10^6$ $Ax=...
5
votes
2answers
729 views

Largest invertible dense matrix with standard solvers such as Lapack

I have a matrix which is complex symmetric. It is around 50,000 elements per side. It is a Method of Moments matrix. Is it feasible to use a standard direct solver such as Lapack to do a matrix ...