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.
98
questions
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Solving linear system of equations with constraints on unknowns
I would like to solve a system of linear equations $y=Uh$ for an unknown vector $h$, where I have a few constraints on some of the elements of $h$. The matrix $U$ is composed of a vector $u$ (length $...
- 35
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0
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53
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Particular linear systems: sparse matrix + column
I am trying to understand a limitation in a routine in the interval arithmetic software Intlab. From matrices starting from a given size (in my particular problems),...
- 1,077
2
votes
1
answer
103
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2D cavity flow: Adding zero Dirichlet condition to the top boundary leads to NAN
I am following this guide to create a simple cfd solver for incompressible 2D fluid in a square cavity. The following Matlab code sample from the guide creates a Laplacian (coefficient) matrix with ...
- 21
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votes
3
answers
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What is this regularization technique?
I'm currently working on understanding some very old code. They are trying to solve an underdetermined system and the comments say that they want the minimum norm solution.
The system they solve is: $...
- 1,785
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0
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61
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Solving AU = F using linalg.cg results in 0 iterations
I am working on solving the following PDE: $$\left(\mu_{x}\frac{\partial^{2}u}{\partial x^{2}}+\mu_{y}\frac{\partial^{2}u}{\partial y^{2}}\right)=f(x,y) \tag 1$$
Which is then discretised:
$$- \mu_{x} ...
- 43
2
votes
1
answer
65
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Solving $(I-Q)x={\bf 1}$ for sub-stochastic sparse $Q$ of dimension 5M $\times$ 5M
I have a (right) sub-stochastic CSC sparse matrix $Q$ of dimension 5 million, with 200 million nonzero entries, which is a nonzero percentage of 0.0008%, so it is indeed extremely sparse. It is not ...
- 503
1
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2
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107
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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 ...
- 265
2
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0
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91
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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....
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2
votes
1
answer
181
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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$...
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3
votes
2
answers
256
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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 ...
- 259
3
votes
1
answer
300
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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
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0
answers
49
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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
300
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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 - ...
- 173
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0
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44
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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 ...
- 482
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1
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68
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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
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0
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31
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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 $...
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237
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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 ...
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3
votes
1
answer
230
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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
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0
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166
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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 ...
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1
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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 ...
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0
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319
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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 ...
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1
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1
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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 ...
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3
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0
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53
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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}$, ...
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1
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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
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0
answers
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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
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2
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130
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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
574
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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 ...
- 291
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votes
2
answers
2k
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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 ...
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1
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1
answer
268
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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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0
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171
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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} &...
- 482
3
votes
0
answers
87
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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$$
Is there a ...
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2
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1
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444
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Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular
I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ...
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1
answer
487
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Solution of Coupled Differential equation for a 2d linear flow using RK4 method in python 3
I want to study the dynamics of a 2d linear flow, whose dynamical equation is- $\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \end{pmatrix}=\begin{pmatrix} 1 & 1\\ 4 & -2\\ \end{pmatrix}\begin{...
4
votes
2
answers
522
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1D FEM for nonlinear diffusion coefficient
I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$
in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$.
...
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0
votes
1
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Is it possible to predict solution oscillation before solving the system by looking at coefficient matrix?
Question
When it is about solving a system of equations, is it possible to predict that whether high-frequency noise (e.g. checker-boarding) is likely to appear in the converged solution by looking at ...
- 235
2
votes
2
answers
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Solution of the linear system using Sherman-Morrison formula for 1000000x1000000 (7450.6GB) matrix using MATLAB
Let $n = 10^6.$ Let $A \in \mathbb{R}^{n\times n} $ be the lower triangular matrix having 1's on and below the main diagonal.
We want to solve the following linear system:
$$ (A + uv^T)x = b$$
by the ...
user36184
6
votes
1
answer
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Accurately Computing a Positive Vector in the Nullspace of a Matrix
I'm sure this question has been asked before yet after many hours of searching I am unable to find a definitive answer.
The problem at hand is solving the linear system: $$A \mathbf{x} = \mathbf{0}$$ ...
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2
votes
1
answer
167
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Solving an m x m symmetric linear system involving a matrix multiplication versus an (n+m) x (n+m) system
Suppose that $R$ and $D$ are an $n \times m$ and $m \times m$ matrices. Assume that $m \ll n$ and that $D$ is positive definite. We would like to solve the system $(R^T R + D) x = R^T b$. This ...
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4
votes
1
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Without positive definiteness, does an iterative solver work?
Question
Does lacking positive definiteness of the matrix of coefficients in a system of equations, make using iterative solvers impractical?
Description
Using the finite volume method, I have ...
- 235
1
vote
2
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Why OpenFOAM uses its own data structures and linear solvers?
I wonder why OpenFOAM code has its own data structures Lists, HashTables, ... etc. when there is the STL in C++?
Another ...
3
votes
3
answers
579
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Using matrix exponential to solve linear system
Consider the system of linear equations:
$$
Ax=b
\tag{1}
\label{eq1}
$$
where
$A\in\mathbb F^{n\times n}$, diagonalizable dense matrix, over the field $\mathbb F$ of real or complex numbers,
$x\...
- 8,712
1
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0
answers
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Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?
I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
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1
vote
1
answer
99
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How to properly compute weights for Weighted Least Squares (WLS)?
I want to apply the weighted least squares method in order to identify parameters of a dynamic process. The process is described by a second order differential equation of the form:
$$ \ddot{y}+a_1\...
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5
votes
0
answers
148
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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 & ...
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3
votes
1
answer
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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:
...
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1
vote
0
answers
140
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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 ...
- 981
3
votes
0
answers
241
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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 ...
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2
votes
2
answers
555
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, ...
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4
votes
3
answers
229
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 $...
- 2,099
3
votes
1
answer
83
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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 ...
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