15 votes
Accepted

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 ...
spektr's user avatar
  • 4,238
7 votes

What is this regularization technique?

I have never seen it used for underdetermined systems, but for overdetermined systems (tall thin $A$) the version obtained replacing $A$ with $A^T$ in your block matrix a common reformulation, known ...
Federico Poloni's user avatar
5 votes
Accepted

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 ...
Neil Lindquist's user avatar
5 votes

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 ...
lightxbulb's user avatar
  • 2,112
4 votes
Accepted

Implementing matrix term version of Gauss-seidel

The $L$ and $U$ of Gauss-Seidel are different from the $L$ and $U$ that come from the LU factorization. For Gauss-Seidel, $L$ and $U$ are what you get if you zero out the upper or lower part, ...
Neil Lindquist's user avatar
2 votes

Implementing matrix term version of Gauss-seidel

In the Gauss iterative method (Gauss-seidel) we decompose the linear system $Ax=b$ into $x^{k+1} = (L+U)^{-1}(b-Ux^{k})$ as exposed in the reference indicated (Scientific Computing An Introductory ...
Carllos Limma's user avatar
1 vote

Numerically stable way to implement Cramer's rule analog

A partial answer to your point 2, from a comment I wrote to a now-deleted answer: there is nothing wrong about $\det A_j$ being very small: determinants are notoriously poorly scaled. For instance the ...
Federico Poloni's user avatar

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