# Tag Info

Accepted

### Solving $AX+X^TB=C$?

It is called a T-Sylvester equation, or *-Sylvester equation in the complex case. Solvability conditions and a pseudocode algorithm based on the Schur form are in https://doi.org/10.13001/1081-3810....
• 11.5k

### Compute $x = B^{-1}(2A+I)(C^{-1}+A)b$ without calculating matrix inverses

As was mentioned in the comment, calculating $x=M^{-1}y$ is equivalent to solving $Mx=y$. Here is the full solution: First, you can reformulate the equation to: $Bx=(2A+I)(C^{-1}+A)b$, and by ...
• 392
Accepted

### Matrix Balancing Algorithm

Took me quite a while to figure this out and as usual it becomes obvious after you find the culprit. After checking the problematic cases reported in David S. Watkins. A case where balancing is ...
• 393
Accepted

### Maximize a function of an orthogonal matrix

There are specialized methods for the minimization of a differentiable function $f(X)$ subject to the orthogonality constraint $X^{T}X=I$. See for example: Lai, Rongjie, and Stanley Osher. “A ...
• 18.8k

### Recurrence relation for matrices

This is an instance of the Riccati equation, which can be solved using the Matrix Sign function. Relevant section from Higham's "Functions of Matrices" book:
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• 760
Accepted

### Mass Matrix and how to handle it (ODEs) - References

Ignoring Newton's method here is the wrong approach! The fact that you're using Newton's method is what makes this cheap to add, and is what makes singular mass matrices possible. Essentially look at ...
• 12.3k
Accepted

• 6,109

### Solve linear system for only part of the solution vector

The only way that I'm aware of to take advantage of only needing a partial solution is in the triangular solves. But those are usually a small part of the total time in dense LU (unless you have many ...
Accepted

### Least Squares with Dense-Block Diagonal Structure

If $N$ is on the order of 100,000 and $m$ is on the order of $100$, Then $J$ requires about 80 gigabytes to store in double precision and $V$ requires a trivial amount of storage. The product $M=JV$ ...
• 18.8k
Accepted

### Convergence conditions of a stationary iteration method for linear systems

You can prove convergence by satisfying the spectral radius relationship you note, choosing $S$ and $T$ such that $\rho(S^{-1}T) < 1$. This comes about by first writing two equations based on your ...
• 4,258