This is an announcement for whom is interested in the same topic.
I'd like to write a module that provides solvers for Sylvester-like equations.
- Lyapunov equation (CT) AX + XA* = Q
- Lyapunov equation (DT) AXA* - X = Q
- Sylvester equation (CT) AX + XB = C (1)
- Sylvester equation (DT) AXB - X = C
More equations may be added after beta.
A classical solution is Bartels–Stewart algorithm. Take (1) for example. We compute the Schur forms of A and B, viz. UTU* and VSV*, transforming the equation into (2).
UTU*X + XVSV* = C (2)
Apply U* on the left and V on the right to transform (2) into the equation below.
T(U*XV) + (U*XV)S = U*CV
Parentheses are added to aid reading. A general SYCT (Sylvester CT) is reduced to triangular SYCT.
However, if A and B are to be used only once, a better way is *not* to compute the Schur form of the larger. We stop at cheaper Hessenberg form. Assume the larger is A without losing generality. Set the Hessenberg form of A is UHU*.
H(U*XV) + (U*XV)S = U*CV
Now we see the trade-off: we have to solve Hessenberg instead of triangular system.
- Blocking (different depths on-the-left and on-the-right)
- Do we need HessenbergView or PartialPivLU is enough?
- RealSchur or ComplexSchur for real matrices?
* Do we need QuasiTriangularView, or HessenbergView or even PartialPivLU is enough?
* Blocking is more tricky
Created attachment 482 [details]
Current incomplete work
I'm slowing down for my (medical) studies, but I'm still keeping an eye on this.
Hello, what is current status of this feature?
It is in the 'sylvester' branch in my clone of Eigen, but my lastest commit was in September 2016. :(
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