  We are concerned with finding the set $I_{\succeq}(A,B)$ 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), $I_{\succeq}(A,B)$
either is empty or has only one value $\mu.$ When $A, B$ are SDC,
$I_{\succeq}(A,B),$ if not empty, can be a singleton or an interval.
Especially, if $I_{\succeq}(A,B)$ is an interval and at least one of the
matrices is nonsingular then its interior is the positive definite interval
$I_{\succ}(A,B).$ If $A, B$ are both singular, then even $I_{\succeq}(A,B)$ 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 $I_{\succeq}(A,B)=I_{\succeq}(A_1,B_1).$ Applying $I_{\succeq}(A,B),$
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.
