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.

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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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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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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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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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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 ...
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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 ...
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2 answers
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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 ...
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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 ...
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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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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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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} &...
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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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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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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{...
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2 answers
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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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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 ...
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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 ...
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5 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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1 answer
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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
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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 ...
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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 ...
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3 votes
3 answers
395 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\...
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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 answer
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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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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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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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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 ...
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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 answers
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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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3 answers
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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 $...
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3 votes
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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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2 votes
1 answer
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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 ...
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2 votes
1 answer
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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 ...
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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 ...
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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 ...
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3 answers
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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 ...
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3 votes
1 answer
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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 ...
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Book Recommendation: Analysis and design of mechanistic models - such as pharmacokinetics or hydrology models

I have been looking at an interesting book "Pharmacokinetic-Pharmacodynamic Modeling and Simulation" by Peter Bonate on pharmacokinetic models: the models of how medical drugs work their way through ...
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Numerical Method for Equation System of two depending Equation Systems

I am searching a solution method for the following equation system of equation systems: Let $A \in \mathbb{R}^{n \times n}$ be an invertible Matrix, $f, b_1, b_2 \in\mathbb{R}^n$ given vectors and $ ...
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1 vote
1 answer
137 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 ...
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1 vote
1 answer
149 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 ...
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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=\...
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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....
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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 ...
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2 votes
1 answer
301 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 ...
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1 vote
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92 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 ...
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1 vote
1 answer
189 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 ...
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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 ...
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1 vote
2 answers
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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 ...
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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 ...
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