# Tag Info

Accepted

### Why does this preconditioner effectively reduce the condition number of a random SPD matrix?

If the eigenvalues of $A$ are $\lambda_1, \lambda_2, \dots,\lambda_n$, the eigenvalues of $A + \mu I$ are $\lambda_1 + \mu, \lambda_2 + \mu, \dots, \lambda_n + \mu$. It is an easy computation to ...

### "Cookbook" about iterative linear solvers and preconditioners

Have a look at Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Barrett et al.). You can find it here. Here's why I'm recommending this over other references: the ...
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### How to directly compute the inverse of an ill-conditioned dense matrix

Though it is a relatively rare situation when you actually have to calculate an inverse of the matrix, not all techniques were created equally. I would use the term badly-conditioned instead of ill-...
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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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### role of initial guess for iterative linear solver

Practical experience shows that trying to get good initial iterates has little value. For example, in the context of solving partial differential equations, if you take the solution from one mesh, ...

### Why does this preconditioner effectively reduce the condition number of a random SPD matrix?

The accepted answer is right: you are not making a preconditioner. To elaborate. For a matrix $A$, a preconditioner is a matrix $B$ such that $B^{-1}A$ has a smaller condition number than $A$. The ...
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### Efficient implementation of preconditioners for iterative solvers

I don't particularly care for the notation $M^{-1}$ precisely because of the confusion you find yourself in. I (and others) simply call the preconditioner $P$. The point, however, is that for ...
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### Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?

The preconditioner for the FDM method that corresponds to the one you outline for the FEM (i.e., the Sylvester-Wathen approach) will still contain the Schur complement of the FDM matrix. The Schur ...

### Does this partial eigen-expansion have a name?

I don't know for sure whether there is an existing name for this method, but @jessechan's suggestion of "truncated eigenvector expansion" sounds perfectly fine to me (and most people would understand ...

### Does this partial eigen-expansion have a name?

This is something like a truncated SVD or eigenvector expansion of your solution. If you take $$x_m = \sum_{j=1}^m \frac{q_j\cdot b}{\lambda_j}q_j$$ with $m=n$, this is the exact solution to $Ax=b$...
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### Correct use of scipy's sparse.linalg.spilu

If $\mathbf L$ and $\mathbf U$ give an approximate factorization of $\mathbf A$, you wouldn't want to use $\mathbf P = \mathbf L\cdot \mathbf U$ as a preconditioner (that's approximately $\mathbf A$), ...
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### Optimality of block-Jacobi preconditioner

Consider matrix a $2\times 2$ matrix $A$: $$A=\left(\begin{array}{cc} 1 & 0\\ 2 & 1 \end{array}\right)$$ Singular values of $A$ are: $$\sigma_1 = \sqrt{2}+1,\quad \sigma_2=\sqrt{2}-1$$ ...
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### Iterative linear solver for "ugly" saddle point system

You should stick with GMRES, it is the only method that is essentially guaranteed to get a solution here. The real problem appears to be you need a better preconditioner. You could try sticking with ...
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### Is it necessary to invert precondition matrix for iterative solver?

OK - so your original equation is $$Ax = b$$ Say you've come up with a good preconditioner for $A$, call this $M$. Also, say you have pre-computed an LU-decomposition for this $M$, i.e. M = L_m ...

### Is it necessary to invert precondition matrix for iterative solver?

Based on your comments, it looks like your main difficulty is how having an LU factorisation for a given matrix $M$ makes it easy to find a vector $\mathbf{y}$ such that $M\mathbf{y}=\mathbf{b}$ for a ...
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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 ...

### Correct use of scipy's sparse.linalg.spilu

This question has an example of how to create the preconditioner M with a scipy sparse matrix A of shape ...