# Questions tagged [linear-system]

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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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### 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 ...
197 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. ...
83 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 ...
174 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 ...
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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 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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### 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 $u$. ...
10k 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 ...
103 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Why does the initial guess for linear system usually choose by zero vector?

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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### 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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### 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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### 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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### 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 ...