This example computes the $\alpha$ and $\beta$ arguments, which defines the generalized eigenvalues and the corresponding left and right eigenvectors, of the matrix pair $\left(A,B\right)$ given by To compute generalized eigenvalues, it is required to call five routines: dggbal to balance the matrix, dgeqrf to perform the $QR$ factorization of $B$, dormqr to apply $Q$ to $A$, dgghrd to reduce the matrix pair to the generalized Hessenberg form and dhgeqz to compute the eigenvalues via the $QZ$ algorithm.

The computation of generalized eigenvectors is done by calling dtgevc to compute the eigenvectors of the balanced matrix pair. The routine dggbak is called to backward transform the eigenvectors to the user-supplied matrix pair. If both left and right eigenvectors are required then dggbak must be called twice.