Generalized Nonsymmetric Eigenvalue Problems:
LAPACK Computational Routines
This
topic
describes LAPACK routines for solving generalized nonsymmetric
eigenvalue problems, reordering the generalized Schur factorization of a pair
of matrices, as well as performing a number of related computational tasks.
A
generalized nonsymmetric eigenvalue problem
is as follows: given a
pair of nonsymmetric (or non-Hermitian)
n
-by-n
matrices
A
and
B
, find the
generalized eigenvaluesλ
and the
corresponding
generalized eigenvectorsx
and
y
that satisfy
the equations
Ax
=
λ
Bx
x
)
and
y
H
A
=
λ
y
H
B
y
).
Table
"Computational Routines for Solving Generalized
Nonsymmetric Eigenvalue Problems"
lists LAPACK routines used to solve the
generalized nonsymmetric eigenvalue problems and the generalized Sylvester
equation.
Routine name
| Operation performed
|
|---|---|
Reduces a pair of matrices to generalized upper
Hessenberg form using orthogonal/unitary transformations.
| |
Balances a pair of general real or complex
matrices.
| |
Forms the right or left eigenvectors of a
generalized eigenvalue problem.
| |
Reduces a pair of matrices to generalized upper Hessenberg form.
| |
Implements the QZ method for finding the
generalized eigenvalues of the matrix pair (H,T).
| |
Computes some or all of the right and/or left
generalized eigenvectors of a pair of upper triangular matrices
| |
Reorders the generalized Schur decomposition of
a pair of matrices (A,B) so that one diagonal block of (A,B) moves to another
row index.
| |
Reorders the
generalized Schur decomposition
of a pair of matrices (A,B) so that a selected cluster of eigenvalues appears
in the leading diagonal blocks of (A,B).
| |
Solves the generalized Sylvester equation.
| |
Estimates reciprocal condition numbers for
specified eigenvalues and/or eigenvectors of a pair of matrices in generalized
real Schur canonical form.
|