For my research I need to solve a generalised eigenvalue problem $Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{R}$ and $x^T$ is also left eigenvector without solving the full problem. Does this problem has a name? I would appreciate some references. I am interested in iterative algorithms for its solution, such as conjugate gradient. However, people typically put additional constraints that $A$ and $B$ are symmetric and $B$ is positive definite. In the present case it is known that $A$ and $B$ are not symmetric.