# 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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### 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 ...
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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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