|
|
||
|
|
|
|
|
This webpage aims at collecting open and difficult problems covering diverse areas of
computation with polynomial matrices and polynomial control design methods. The list of
problems has ambitions to be a useful communication channel between researchers from
different groups and perhaps even different areas.
Conditioning and stability of numerical algorithms with polynomial matricesFormulating various control-theoretical problems using polynomial matrices turned out very convenient and natural as early as in the late 1970s. The actual numerical solution, however, did suffer serious troubles when diverse computational problems with polynomial matrices were attacked by extending the standard Gaussian elimination schemes. The performance of most numerical algorithms improved dramatically in the 1990s when polynomial matrices begun to be viewed as matrix polynomials, i.e., polynomials with matrix coefficients. This often leads to building and solving block Toeplitz linear systems using standard tools of numerical linear algebra like LU, QR and SVD decompositions. Nontheless, the polynomial methods still keep the reputation of numerically tricky methods that they gained in the 1980s. Apparently, the only way out is to support the experienced good numerical performance by rigorous analysis of conditioning and stability of the numerical algorithms. No research has been done in this area. |