Generalized Symmetric Definite Eigenproblems (GSEP)

Drivers are provided to compute all the eigenvalues and (optionally) the eigenvectors of the following types of problems:

1. $A z = \lambda B z$
2. $A B z = \lambda z$
3. $B A z = \lambda z$

where $A$ and $B$ are symmetric or Hermitian and $B$ is positive definite. For all these problems the eigenvalues $\lambda$ are real. The matrices $Z$ of computed eigenvectors satisfy $Z^T A Z = \Lambda$ (problem types 1 and 3) or $Z^{-1} A Z^{-T} = I$ (problem type 2), where $\Lambda$ is a diagonal matrix with the eigenvalues on the diagonal. $Z$ also satisfies $Z^T B Z = I$ (problem types 1 and 2) or $Z^T B^{-1} Z = I$ (problem type 3).
There are three types of driver routines for generalized symmetric and Hermitian eigenproblems. Originally LAPACK had just the simple and expert drivers described below, and the third driver was added after an improved algorithm was discovered.

• a simple driver (name ending -GV) computes all the eigenvalues and (optionally) eigenvectors.
  
• an expert driver (name ending -GVX) computes all or a selected subset of the eigenvalues and (optionally) eigenvectors. If few enough eigenvalues or eigenvectors are desired, the expert driver is faster than the simple driver.
  
• a divide-and-conquer driver (name ending -GVD) solves the same problem as the simple driver. It is much faster than the simple driver for large matrices, but uses more workspace. The name divide-and-conquer refers to the underlying algorithm (see sections 2.4.4 and 3.4.3 in the LAPACK Users' Guide[1]).

Different driver routines are provided to take advantage of special structure or storage of the matrices $A$ and $B$, as shown in Table 2.6.