Given two square matrices `A` and `B`,
the **generalized nonsymmetric eigenvalue problem** is to find
the **eigenvalues** and corresponding
**eigenvectors**
such that

*or*
find the eigenvalues and corresponding eigenvectors
such that

Note that these problems are equivalent with and if neither nor is zero. In order to deal with the case that or is zero, or nearly so, the LAPACK routines return two values, and , for each eigenvalue, such that and .

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

If the determinant of is zero for all values of ,
the eigenvalue problem is called **singular**, and 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 [21][80][71].

The generalized nonsymmetric eigenvalue problem can be solved via the
**generalized Schur factorization**
of the pair `A`,`B`, defined in the
real case as

where `Q` and `Z` are orthogonal matrices, `P` is upper triangular,
and `S` is an upper quasi-triangular matrix
with 1-by-1 and 2-by-2 diagonal blocks, the 2-by-2 blocks
corresponding to complex conjugate pairs of eigenvalues of `A`,`B`.
In the complex
case the Schur factorization is

where `Q` and `Z` are unitary and `S` and `P` are both upper triangular.

The columns of `Q` and `Z` are called **generalized Schur vectors**
and span pairs of **deflating subspaces** of `A` and `B` [72].
Deflating subspaces are a generalization of invariant subspaces:
For each `k` (1 < = `k` < = `n`),
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`.

Two simple drivers are provided for the nonsymmetric problem :

- xGEGS : computes
all or part of the generalized Schur factorization of the pencil ;
- xGEGV : computes
all the generalized eigenvalues, and (optionally) the
right or left eigenvectors (or both);

Tue Nov 29 14:03:33 EST 1994