Purpose

LA_GGESX computes for a pair of $n\times n$ real or complex matrices $( A, B )$ the (generalized) real or complex Schur form, the generalized eigenvalues in the form of scalar pairs $(\alpha, \beta)$, and, optionally, the left and/or right Schur vectors.
If $A$ and $B$ are real then the real-Schur form is computed, otherwise the complex-Schur form is computed. The real-Schur form is a pair of real matrices $(S, T)$ such that 1) $S$ has block upper triangular form, with $1 \times 1$ and $2 \times 2$ blocks along the main diagonal, 2) $T$ has upper triangular form with nonnegative elements on the main diagonal, and 3) $S = Q^T A Z$ and $T = Q^T B Z$, where $Q$ and $Z$ are orthogonal matrices. The $2 \times 2$ blocks of $S$ are ``standardized'' by making the corresponding elements of $T$ have the form

$$\left[ \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right]$$

The complex-Schur form is a pair of matrices $(S, T)$ such that 1) $S$ has upper triangular form, 2) $T$ has upper triangular form with nonnegative elements on the main diagonal, and 3) $S = Q^H A Z$ and $T = Q^H B Z$, where $Q$ and $Z$ are unitary matrices.
In both cases the columns of $Q$ and $Z$ are called, respectively, the left and right Schur vectors.
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 $\alpha = 0$)
LA_GGESX also computes two reciprocal condition numbers for the average of the selected eigenvalues and reciprocal condition numbers for the right and left deflating subspaces corresponding to the selected eigenvalues.