Positive semidefinite interval of matrix pencil and its applications for the generalized trust region subproblems

We are concerned with finding the set $I_{⪰}(A,B)$ of real values $\mu$ such that the matrix pencil $A+\mu B$ is positive semidefinite. If $A,B$ are not simultaneously diagonalizable via congruence (SDC), $I_{⪰}(A,B)$ either is empty or has only one value $\mu \mathrm{.}$ When $A,B$ are SDC, $I_{⪰}(A,B),$ if not empty, can be a singleton or an interval. Especially, if $I_{⪰}(A,B)$ is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval $I_{\succ }(A,B)\mathrm{.}$ If $A,B$ are both singular, then even $I_{⪰}(A,B)$ is an interval, its interior may not be $I_{\succ }(A,B),$ but $A,B$ are then decomposed to block diagonals of submatrices $A_1,B_1$ with $B_1$ nonsingular such that $I_{⪰}(A,B)=I_{⪰}(A_1,B_1)\mathrm{.}$ Applying $I_{⪰}(A,B),$ the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.