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.
92
questions
1
vote
2
answers
95
views
Implementing matrix term version of Gauss-seidel
I am trying to implement the below description from Ch. 11 of Heath's "Scientific Computing An Introductory Survey" of the Gauss-Seidel iterative method for solving a system of linear ...
- 233
2
votes
0
answers
76
views
Solving tridiagonal Toeplitz system using SIMD instructions
I need to solve many small Toeplitz systems that fit entirely in cache, meaning the computation is compute bound. Are there vectorizable algorithms for this?
I found a few older articles: (https://www....
- 21
2
votes
1
answer
170
views
Numerically stable way to implement Cramer's rule analog
Problem statement
Let $A$ be an $n\times n$ matrix and $b$ an $n$-dimensional vector. For $j\in \{1, \dots, n \}$, let $A_j$ be the matrix where we take $A$ and replace the $j^{\rm th}$ column with $b$...
- 31
4
votes
2
answers
232
views
Faster than forward substitution?
I have a matrix of the form:
$M:=\begin{pmatrix}
S_1 & & & \\\ Q_1 & S_2 & & \\\ & ... & ... & \\\ & & Q_n & S_n\end{pmatrix}$
where the blocks ...
- 157
3
votes
1
answer
247
views
Is it really necessary to solve a system of linear equations in the Finite Element Method?
When we solve some boundary value problem by Finite Element Method, the appropriate system of linear equations is built, $$Ax=b.$$
Usually we use the solution x just for plugging it into some ...
1
vote
0
answers
43
views
FEM Basis for functions of two variables in $\mathbb{R}^2$ (Applied to Linear Full Stokes Equations)
I'm trying to do FEM for a very basic version of the linear full Stokes equations in two dimensions. Say we are working in the grid $[0,1]\times[0,1]$ in the $xy-$plane.
To solve an FEM problem for a ...
- 111
6
votes
1
answer
214
views
Inverse power iteration and solving singular system
The algorithm for the inverse power iteration works as following :
\begin{align}
&v^{(0)} =\text{ some vector with }\|v^{(0)}\|=1\\
&\text{for }k = 1, 2, \ldots\\
&\qquad\text{Solve } (A - ...
- 169
0
votes
0
answers
42
views
Instability due to different lumping dimensions in second-order linear system
I have a problem that consists of a linear hyperbolic differential equation with some boundary and initial conditions which leads after discretization in space using finite elements to the following ...
- 472
0
votes
1
answer
66
views
Solving a linear system whose coefficient matrix is dense but symmetric
For solving a linear system,
$Ax = b$.
If $A$ is a dense but symmetric $n \times n$ matrix, how much memory is required?
$A$ is symmetric, which means only the upper (or lower) triangular part of $n \...
1
vote
0
answers
29
views
Non-Linear Distributed Delayed Kalman Filter
I have a system $\vec{x}_{i + 1} = \vec{x}_i + W_i$ where $W = N(\vec{\mu}, \Sigma)$. For some matrix $H_i$, let $y_i = H_i$ and let $z_i = y_i + R$. Where $R$ is some random variable. We are given $...
- 23
4
votes
0
answers
142
views
SINDy Vs standard methods for system identification
I have been trying to understand the recently proposed Sparse Identification of Nonlinear Dynamics SINDy. Despite several attempts, I seem to fail to understand the difference between SINDy and the ...
- 875
3
votes
1
answer
226
views
Correctness of direct numerical solution of ill-conditioned linear system
To what extent can you put trust in a numerical solution obtained by direct solver for an ill-conditioned linear system? In other words, how can you test the solution? Dropping it into the system says ...
3
votes
0
answers
124
views
Splitting system of equations into linear and nonlinear part and solving separately
I was working on a problem recently (calculating all flows in a network given input and output flows, basically what Hardy-Cross tries to achieve) which can be formulated as a well-determined system ...
- 131
0
votes
1
answer
68
views
Interpolation and Restriction operators in Multigrid
I saw in several places that interpolation operator ($P$) and restriction operator ($P^T$) are usually transposes of each other (up to multiplication by a constant).
As I understood it related to ...
- 109
0
votes
0
answers
276
views
Solving huge dense square symmetric linear system
I have a linear system of the type
$A x = y$
where A is a dense, square, symmetric, positive definite matrix, $x$ a vector of unknown parameters, and $y$ is a vector of observed quantity.
I know that ...
- 153
1
vote
1
answer
100
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 ...
- 21
4
votes
0
answers
51
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 $\lambda_\textrm{i}$, ...
- 223
3
votes
1
answer
526
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 ...
- 515
1
vote
0
answers
83
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 ...
0
votes
2
answers
109
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 ...
- 1,463
3
votes
1
answer
360
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 ...
- 281
0
votes
2
answers
2k
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 ...
- 33
1
vote
1
answer
152
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.
- 481
0
votes
0
answers
157
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
\begin{equation}
A = \begin{pmatrix} A_{11} &...
- 472
3
votes
0
answers
82
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 ...
- 2,273
2
votes
1
answer
376
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, ...
- 287
1
vote
1
answer
456
views
Solution of Coupled Differential equation for a 2d linear flow using RK4 method in python 3
I want to study the dynamics of a 2d linear flow, whose dynamical equation is- $\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \end{pmatrix}=\begin{pmatrix} 1 & 1\\ 4 & -2\\ \end{pmatrix}\begin{...
4
votes
2
answers
418
views
1D FEM for nonlinear diffusion coefficient
I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$
in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$.
...
- 309
0
votes
1
answer
155
views
Is it possible to predict solution oscillation before solving the system by looking at coefficient matrix?
Question
When it is about solving a system of equations, is it possible to predict that whether high-frequency noise (e.g. checker-boarding) is likely to appear in the converged solution by looking at ...
- 235
2
votes
2
answers
984
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 ...
user36184
5
votes
1
answer
198
views
Accurately Computing a Positive Vector in the Nullspace of a Matrix
I'm sure this question has been asked before yet after many hours of searching I am unable to find a definitive answer.
The problem at hand is solving the linear system: $$A \mathbf{x} = \mathbf{0}$$ ...
- 53
2
votes
1
answer
138
views
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 ...
- 123
4
votes
1
answer
456
views
Without positive definiteness, does an iterative solver work?
Question
Does lacking positive definiteness of the matrix of coefficients in a system of equations, make using iterative solvers impractical?
Description
Using the finite volume method, I have ...
- 235
1
vote
2
answers
451
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 ...
3
votes
3
answers
537
views
Using matrix exponential to solve linear system
Consider the system of linear equations:
$$
Ax=b
\tag{1}
\label{eq1}
$$
where
$A\in\mathbb F^{n\times n}$, diagonalizable dense matrix, over the field $\mathbb F$ of real or complex numbers,
$x\...
- 8,602
1
vote
0
answers
123
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 (...
- 207
1
vote
1
answer
83
views
How to properly compute weights for Weighted Least Squares (WLS)?
I want to apply the weighted least squares method in order to identify parameters of a dynamic process. The process is described by a second order differential equation of the form:
$$ \ddot{y}+a_1\...
- 221
5
votes
0
answers
130
views
Levinson Recursion for Non Square Toeplitz Matrices
Given a rectangular Toeplitz Matrix $ H $, how could one solve:
$$ y = H x $$
For instance, $ H $ can be Linear Convolution Matrix of the filter $ h $:
$$ H = \begin{bmatrix}
{h}_{1} & 0 & ...
- 332
3
votes
1
answer
2k
views
How to implement flexible gmres in matlab?
About the flexible GMRES (fgmres), we know that it is a variant of right preconditioned gmres. And the robust command gmres in matlab as follows:
...
- 961
1
vote
0
answers
121
views
How to compute the computational cost and storage of the Full Orthogonalization Method?
About the analysis of Full Orthogonalization Method (FOM) in Prof. Saad's book, wrote as follows:
Algorithm 6.4 (FOM):
\begin{array}{l}
r_0=b-Ax_0,\beta=\|r_0\|_2,v_1 = r_0/\beta\\
Define \quad H_m ...
- 961
3
votes
0
answers
218
views
How to reproduce the numerical examples in Prof. Saad's Book about Krylov subspace methods?
After reading Prof. Saad' Book, "Iterative methods for Sparse Linear Systems, 2nd version", I want to do the numerical examples about the Krylov subspace methods not only to reproduce the results in ...
- 961
2
votes
2
answers
473
views
How to understand the choice of Krylov subspace orthonormal basis?
This semester, I study the Krylov subspace iterative methods (about Ax=b) using the book H. A. Van der Vorst. Iterative Krylov Methods for Large Linear Systems,
volume 13. Cambridge University Press, ...
- 961
4
votes
3
answers
224
views
Solving for a vector in a linear system that is both left and right multiplied
I have a linear system where I am given 2 matrices, $A$ and $B$, and 2 vectors, $v$ and $c$, and I need to solve for the vector $x$. $A$ is $n\times n$, $B$ is $n \times n \times n$, and the vectors $...
- 2,069
3
votes
1
answer
79
views
Does the k-th approximate solution of a stationary iteration belong to the k-th Krylov subspace?
For an stationary iteration method solving $Ax=b$ as follows:
$$
Mx_k = Nx_{k-1}+b,
$$
I have known that when $M = I$, i.e., the Richardson iteration, the k-th solution $x_k = x_{k-1}+r_{k-1}$ is in ...
- 961
2
votes
1
answer
111
views
What is the standard, extrapolation, and modified version of Richardson iteration method?
I have been studying the iterative methods recently. For classical iterative methods solving $Ax=b$, I have seen that the most simplest iteration method is the so-called "Richardson iteration". But I ...
- 961
2
votes
1
answer
560
views
Why Krylov subspace iterative methods are faster than classical iteration?
This semester, I have been studying the most popular iterative methods, i.e., Krylov subspace iteration methods.
For a large sparse system linear
$$
Ax=b,
$$
where $A$ is nonsingular, I know that ...
- 961
1
vote
0
answers
43
views
What kind of problem or matrices are suitable for multigrid method?
For Poisson or Convection-diffusion equation as follows:
$$
-\Delta u=f,\qquad u|_\Omega = g.
$$
or
$$
-\Delta u +\vec{w}.\nabla u=f,\qquad u|_\Omega = g.
$$
using FDM or FEM discretization, we can ...
- 961
1
vote
1
answer
91
views
Why do not we choose the error solution norm as an iterative method's criterion?
For solving linear system
$$
Ax=b,
$$
using iterative mehods, we often use the terminate criterion as follows:
$$
\frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps.
$$where $x_0$ is the ...
- 961
2
votes
3
answers
307
views
Is there any other sparse matrix data in matlab built-in file?
I want to do some numerical examples solving large sparse linear system Ax=b. And I want to use some data from Maltab itself because this experiments are easily ...
- 961
3
votes
1
answer
500
views
What is the **contraction factor and convergence factor** of a iteration method?
For any iteration method from A=M-N, e.g.,
$$
Mx_{k+1}=Nx_{k}+b,\quad k=0,1,...
$$
we know that it converges iff $\rho(M^{-1}N)<1$. And when it converges, there exists a concept called asymtotic ...
- 961