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Is there a name/standard algorithm to solve the following equation for $X$?

$AX+X^TB=C$

Matrices $A$,$B$,$C$ are dense, diagonalizable, nearly singular, about $1000\times 1000$ in size. I've looked through SLICOT routines, and nothing looked directly applicable.

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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/10.13001/1081-3810.1479 . Analogous considerations for a more general class of equations and a Fortran-90 implementation of the last step of the resulting solution algorithm (the back-substitution on the triangular version of the equation) are in my paper https://doi.org/10.1002/nla.2261 . I don't think you will find something in Slicot, because it has no immediate control theory applications.

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  • $\begingroup$ thanks for the reference....it seems there's also a straightforward way to convert it to regular Sylvester equation and reuse those solvers $\endgroup$
    – Yaroslav Bulatov
    Commented Nov 1, 2020 at 17:14
  • $\begingroup$ That route has dubious stability properties, though. $\endgroup$
    – Federico Poloni
    Commented Nov 1, 2020 at 17:30
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    $\begingroup$ Always fun to find questions on here where just the right person is able to answer :-) $\endgroup$
    – Wolfgang Bangerth
    Commented Nov 2, 2020 at 16:51
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    $\begingroup$ BTW, evaluated a couple of reduction methods here -- wolframcloud.com/obj/yaroslavvb/newton/t-sylvester-solve.nb . Reducing to spectral decomposition gives acceptable quality, but the speed could use improvement -- 2.9 seconds end-to-end where diagonalization takes 1.3 seconds and Shur takes 1.0 seconds $\endgroup$
    – Yaroslav Bulatov
    Commented Nov 4, 2020 at 18:36

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