# Tag Info

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### What is this regularization technique?

All they are doing is taking the problem of minimizing $x^T x$ subject to $A x = b$ and then forming a Lagrangian, like so: $$L(x, \lambda) = x^T x + \lambda^T (A x - b)$$ From here, you want to find ...
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### When do not use preconditioners for sparse linear system of equations?

In my experience, you always need (or better use) some form of preconditioning. The type and complexity of the precondition would vary depending on the task though. From Y. Saad, Iterative Methods for ...
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### Without positive definiteness, does an iterative solver work?

No, positive definiteness (and symmetry) are only precondition to using the Conjugate Gradient method. But there are plenty of other iterative methods such as MinRes and GMRES that can be used for ...
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### Is it really necessary to solve a system of linear equations in the Finite Element Method?

I think your question is actually pretty fundamental and deserves a thoughtful answer. Paraphrasing a bit, your question is perhaps motivated by the observation that engineering design is often ...
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• 11.8k
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### How to verify solution to pre-conditioned linear systems solver?

You've started with a singular linear system of equations $Ax=b$. As a practical matter, it's unlikely that $b$ lies exactly in the range of $A$, so at best you can find a least squares solution that ...
• 18.9k
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### How "sparse" should a sparse matrix be to see benefits?

This kind of scaling is fairly common and sparse direct factorization methods are commonly used on matrices of up to a few hundred thousand rows and columns. By the time you get to $N=20$, youâ€™ll ...
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### How to implement flexible gmres in matlab?

First of all, MATLAB's gmres assumes that the preconditioner you use is linear. This is important! Actually it is the main difference between FGMRES and GMRES. ...
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### Solving $(I-Q)x={\bf 1}$ for sub-stochastic sparse $Q$ of dimension 5M $\times$ 5M

Yea, iterative solvers are often more effective for sparse problems, as long as the condition number isn't too large. IterativeSolvers.jl provides several common methods. The general recommendation ...

### What is this regularization technique?

Spektr's answer is already great but note that you can also argue about this in the following manner $2x + A^T \tau = 0$ and $Ax=b$ implies that $x = -\frac{1}{2} A^T\tau$ and plugging it in the ...
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### Solve $Ax=b$ repeatedly where $A$ is a sparse weighted Laplacian matrix with changing weights

Michael Saunders wrote a sparse LU package that can do rank-1 updates, LUSOL. You could try to use that, since you write that direct solvers are viable for your problem.
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### Solve $Ax=b$ repeatedly where $A$ is a sparse weighted Laplacian matrix with changing weights

It appears to me that you are dealing with a sequence of linear systems $A_j x_j = b$, where $A_{j+1}$ is a low rank modification of $A_{j}$. In your case, I would investigate if the Krylov subspace \$...
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### How to obtain linear tridiagonal system from PDE

Welcome to the site. You are actually virtually finished, but may not have realised it yet. A tridiagonal linear system is another name for a matrix problem which only has non zero entries on the ...
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### Why OpenFOAM uses its own data structures and linear solvers?

In the light of more recent information: OpenFOAM follows the C++11 standard without any exception at the time of writing. Therefore, you can use any C++ containers of this standard within OpenFOAM. ...
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### Solving an m x m symmetric linear system involving a matrix multiplication versus an (n+m) x (n+m) system

In my opinion, you need a really good excuse to take a symmetric positive definite system and then turn it into an indefinite system. The first thing I would try is the conjugate gradient method ...
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### Accurately Computing a Positive Vector in the Nullspace of a Matrix

Quick answer to summarize my comments. Keep in mind that a delicate point is the choice of the truncation threshold in the SVD (what is "numerically zero" and what is not). If you do not ...
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### 1D FEM for nonlinear diffusion coefficient

You would need to linearize the problem. I prefer to do it before discretization but it's possible to do also after discretization. (I'm a bit skeptical of linearization after discretization because I ...
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