Bug 840

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.

Algorithm
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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.

Challenges
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- 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