Generalized Nonsymmetric Eigenproblems (GNEP)

Given a matrix pair (*A*, *B*), where *A* and *B* are square *n* x *n*
matrices, the **generalized nonsymmetric eigenvalue problem** is to find
the **eigenvalues**
and corresponding
**eigenvectors**
such that

Note that these problems are equivalent with and

More precisely, *x* and *y* are called **right eigenvectors**.
Vectors
or
satisfying

are called

Sometimes the following, equivalent notation is used to refer to the
generalized eigenproblem for the pair (*A*,*B*): The object ,
where
is an indeterminate, is called a **matrix pencil**, or
just **pencil**.
So one can also refer to the generalized eigenvalues
and eigenvectors of the pencil .

If the determinant of
is identically
zero for all values of ,
the eigenvalue problem is called **singular**; otherwise it is **regular**.
Singularity of (*A*,*B*) is signaled by some
(in the presence of roundoff,
and
may be very small).
In this case, the eigenvalue problem is very ill-conditioned,
and in fact some of the other nonzero values of
and
may be indeterminate (see section 4.11.1.4 for further
discussion) [93,105,29].
The current routines in LAPACK are intended only for regular matrix pencils.

The generalized nonsymmetric eigenvalue problem can be solved via the
**generalized Schur decomposition**
of the matrix pair (*A*, *B*), defined in the *real case* as

where

where

The columns of *Q* and *Z* are called **left and right generalized Schur
vectors**
and span pairs of **deflating subspaces** of *A* and *B*
[93].
Deflating subspaces are a generalization of invariant subspaces:
For each *k*
,
the first *k* columns of *Z* span a right
deflating subspace mapped by both *A* and *B* into a left deflating subspace
spanned by the first *k* columns of *Q*.

More formally, let
*Q* = (*Q*_{1}, *Q*_{2}) and
*Z* = (*Z*_{1}, *Z*_{2}) be a conformal
partitioning with respect to the cluster of *k* eigenvalues in the
(1,1)-block of (*S*, *T*), i.e. where *Q*_{1} and *Z*_{1} both have *k* columns,
and *S*_{11} and *T*_{11} below are both *k*-by-*k*,

Then subspaces and form a pair of (left and right) deflating subspaces associated with the cluster of (

As for the standard nonsymmetric eigenproblem, two pairs of drivers are provided, one pair focusing on the generalized Schur decomposition, and the other pair on the eigenvalues and eigenvectors as shown in Table 2.6:

- xGGES:
a simple driver that computes all or part of the
generalized Schur decomposition of (
*A*,*B*), with optional ordering of the eigenvalues; - xGGESX:
an expert driver that can additionally compute condition
numbers for the average of a selected subset of eigenvalues,
and for the corresponding pair of deflating subspaces;
- xGGEV:
a simple driver that computes all the generalized
eigenvalues of (
*A*,*B*), and optionally the left or right eigenvectors (or both); - xGGEVX:
an expert driver that can additionally balance the
matrix pair to improve the conditioning of the eigenvalues and
eigenvectors, and compute condition numbers for the
eigenvalues and/or left and right eigenvectors (or both).

The subroutines xGGES and xGGEV are improved versions of the drivers, xGEGS and xGEGV, respectively. xGEGS and xGEGV have been retained for compatibility with Release 2.0 of LAPACK, but we omit references to these routines in the remainder of this users' guide.