# 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 ...
• 11.8k
Accepted

### 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-...
• 8,701
Accepted

### 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 ...
• 8,701

### 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 ...
• 1,340
Accepted

### 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 ...
• 56.1k
Accepted

### Optimality of block-Jacobi preconditioner

Consider a $2\times 2$ matrix $A$: $$A=\left(\begin{array}{cc} 1 & 0\\ 2 & 1 \end{array}\right)$$ Singular values of $A$ are: $$\sigma_1 = \sqrt{2}+1,\quad \sigma_2=\sqrt{2}-1$$ resulting ...
• 8,701
Accepted

### 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 ...
• 56.1k
Accepted

### 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 $\mathbf A$), ...
• 4,946
Accepted

### 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 ...
• 2,089
Accepted

• 11.8k
Accepted

### what preconditioner for incompressible hyperelasticity in 3d (similar to stokes equation?)?

There is no difference between the linear and nonlinear Stokes problems as far as preconditioning is concerned. That is because at the end of the day, you always have to linearize the nonlinear ...
• 56.1k