Purpose

LA_GGEVX computes for a pair of $n\times n$ real or complex matrices $(A,\,B)$ the generalized eigenvalues in the form of scalar pairs ( $\alpha,\,\beta$) and, optionally, the left and/or right generalized eigenvectors.
A generalized eigenvalue of the pair $( A, B )$ is, roughly speaking, a scalar of the form $\lambda = \alpha/\beta$ such that the matrix $A - \lambda B$ is singular. It is usually represented as the pair $(\alpha, \beta)$, as there is a reasonable interpretation of the case $\beta = 0$ (even if $\alpha = 0$).
A right generalized eigenvector corresponding to a generalized eigenvalue $\lambda$ is a vector $v$ such that $(A - \lambda\, B)\, v = 0$. A left generalized eigenvector is a vector $u$ such that $u^H \, (A - \lambda\, B) = 0$, where $u^H$ is the conjugate-transpose of $u$.
The computation is based on the (generalized) real or complex Schur form of $(A,\,B)$. (See LA_GGES for details of this form.)
Optionally, LA_GGEVX also computes a balancing transformation (to improve the conditioning of the eigenvalues and eigenvectors), reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors. The balancing transformation consists of a permutation of rows and columns and/or a scaling of rows and columns.