On the Symmetric and Unsymmetric Solution Set of Interval Systems
Götz E. Alefeld, Günter Mayer · SIAM Journal on Matrix Analysis and Applications · 1995
We consider the solution set S of real linear systems $Ax = b$ with the $n \times n$ coefficient matrix A varying between a lower bound $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} $ and an upper bound $\bar A$, and with b similarly varying between $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} $, $\bar b$. First we list some properties on the shape of S if all matrices A are nonsingular. Then we restrict A to be nonsingular and symmetric deriving a complete description for the boundary of the corresponding symmetric solution set $S_{{\text{sym}}} $ in the $2 \times 2$ case. Finally we derive a new criterion for the feasibility of the Cholesky method with which bounds for $S_{{\text{sym}}} $ can be found.