Contents

# ?hgeqz

Implements the QZ method for finding the generalized eigenvalues of the matrix pair (H,T).

## Syntax

Include Files
• mkl.h
Description
The routine computes the eigenvalues of a real/complex matrix pair (
H
,
T
), where
H
is an upper Hessenberg matrix and
T
is upper triangular, using the double-shift version (for real flavors) or single-shift version (for complex flavors) of the
QZ
method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a real/complex matrix pair (
A
,
B
):
A
=
Q
1
*
H
*
Z
1
H
,
B
=
Q
1
*
T
*
Z
1
H
,
as computed by
?gghrd
.
For real flavors
:
If
job
=
'S'
, then the Hessenberg-triangular pair (
H
,
T
) is reduced to generalized Schur form,
H
=
Q
*
S
*
Z
T
,
T
=
Q
*
P
*
Z
T
,
where
Q
and
Z
are orthogonal matrices,
P
is an upper triangular matrix, and
S
is a quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks. The 1-by-1 blocks correspond to real eigenvalues of the matrix pair (
H
,
T
) and the 2-by-2 blocks correspond to complex conjugate pairs of eigenvalues.
Additionally, the 2-by-2 upper triangular diagonal blocks of
P
corresponding to 2-by-2 blocks of
S
are reduced to positive diagonal form, that is, if
S
j
+ 1,
j
is non-zero, then
P
j
+ 1,
j
=
P
j
,
j
+ 1
= 0
,
P
j
,
j
> 0
, and
P
j
+ 1,
j
+ 1
> 0
.
For complex flavors
:
If
job
=
'S'
, then the Hessenberg-triangular pair (
H
,
T
) is reduced to generalized Schur form,
H
=
Q
*
S
*
Z
H
,
T
=
Q
*
P
*
Z
H
,
where
Q
and
Z
are unitary matrices, and
S
and
P
are upper triangular.
For all function flavors
:
Optionally, the orthogonal/unitary matrix
Q
from the generalized Schur factorization may be post-multiplied by an input matrix
Q
1
, and the orthogonal/unitary matrix
Z
may be post-multiplied by an input matrix
Z
1
.
If
Q
1
and
Z
1
are the orthogonal/unitary matrices from
?gghrd
that reduced the matrix pair (
A
,
B
) to generalized upper Hessenberg form, then the output matrices
Q
1
Q
and
Z
1
Z
are the orthogonal/unitary factors from the generalized Schur factorization of (
A
,
B
):
A
= (
Q
1
Q
)*
S
*(
Z
1
Z
)
H
,
B
= (
Q
1
Q
)*
P
*(
Z
1
Z
)
H
.
To avoid overflow, eigenvalues of the matrix pair (
H
,
T
) (equivalently, of (
A
,
B
)) are computed as a pair of values (
alpha
,
beta
). For
chgeqz
/
zhgeqz
,
alpha
and
beta
are complex, and for
shgeqz
/
dhgeqz
,
alpha
is complex and
beta
real. If
beta
is nonzero,
λ
=
alpha
/
beta
is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
A
*
x
=
λ
*
B
*
x
and if
alpha
is nonzero,
μ
=
beta
/
alpha
is an eigenvalue of the alternate form of the GNEP
μ
*
A
*
y
=
B
*
y
.
Real eigenvalues (for real flavors) or the values of
alpha
and
beta
for the i-th eigenvalue (for complex flavors) can be read directly from the generalized Schur form:
alpha
=
S
i
,
i
,
beta
=
P
i
,
i
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
job
Specifies the operations to be performed. Must be
'E'
or
'S'
.
If
job
=
'E'
, then compute eigenvalues only;
If
job
=
'S'
, then compute eigenvalues and the Schur form.
compq
Must be
'N'
,
'I'
, or
'V'
.
If
compq
=
'N'
, left Schur vectors (
q
) are not computed;
If
compq
=
'I'
,
q
is initialized to the unit matrix and the matrix of left Schur vectors of (
H
,
T
) is returned;
If
compq
=
'V'
,
q
must contain an orthogonal/unitary matrix
Q
1
on entry and the product
Q
1
*Q
is returned.
compz
Must be
'N'
,
'I'
, or
'V'
.
If
compz
=
'N'
, right Schur vectors (
z
) are not computed;
If
compz
=
'I'
,
z
is initialized to the unit matrix and the matrix of right Schur vectors of (
H
,
T
) is returned;
If
compz
=
'V'
,
z
must contain an orthogonal/unitary matrix
Z
1
on entry and the product
Z
1
*Z
is returned.
n
The order of the matrices
H
,
T
,
Q
, and
Z
(
n
0
).
ilo
,
ihi
ilo
and
ihi
mark the rows and columns of
H
which are in Hessenberg form. It is assumed that
H
is already upper triangular in rows and columns 1:
ilo
-1 and
ihi
+1:
n
.
Constraint:
If
n
> 0
, then
1
ilo
ihi
n
;
if
n
= 0
, then
ilo
= 1
and
ihi
= 0
.
h
,
t
,
q
,
z
Arrays:
On entry,
h
(size max(1,
ldh
*
n
))
contains the
n
-by-
n
upper Hessenberg matrix
H
.
On entry,
t
(size max(1,
ldt
*
n
))
contains the
n
-by-
n
upper triangular matrix
T
.
q
(size max(1,
ldq
*
n
))
:
On entry, if
compq
=
'V'
, this array contains the orthogonal/unitary matrix
Q
1
used in the reduction of (
A
,
B
) to generalized Hessenberg form.
If
compq
=
'N'
, then
q
is not referenced.
z
(size max(1,
ldz
*
n
))
:
On entry, if
compz
=
'V'
, this array contains the orthogonal/unitary matrix
Z
1
used in the reduction of (
A
,
B
) to generalized Hessenberg form.
If
compz
=
'N'
, then
z
is not referenced.
ldh
h
; at least max(1,
n
).
ldt
t
; at least max(1,
n
).
ldq
q
;
If
compq
=
'N'
, then
ldq
1
.
If
compq
=
'I'
or
'V'
, then
ldq
max(1,
n
)
.
ldz
z
;
If
compq
=
'N'
, then
ldz
1
.
If
compq
=
'I'
or
'V'
, then
ldz
max(1,
n
)
.
Output Parameters
h
For real flavors
:
If
job
=
'S'
, then on exit
h
contains the upper quasi-triangular matrix
S
from the generalized Schur factorization.
If
job
=
'E'
, then on exit the diagonal blocks of
h
match those of
S
, but the rest of
h
is unspecified.
For complex flavors
:
If
job
=
'S'
, then, on exit,
h
contains the upper triangular matrix
S
from the generalized Schur factorization.
If
job
=
'E'
, then on exit the diagonal of
h
matches that of
S
, but the rest of
h
is unspecified.
t
If
job
=
'S'
, then, on exit,
t
contains the upper triangular matrix
P
from the generalized Schur factorization.
For real flavors
:
2-by-2 diagonal blocks of
P
corresponding to 2-by-2 blocks of
S
are reduced to positive diagonal form, that is, if
h
(j+1,j) is non-zero, then
t
(j+1,j)=
t
(j,j+1)=0
and
t
(j,j) and
t
(j+1,j+1) will be positive.
If
job
=
'E'
, then on exit the diagonal blocks of
t
match those of
P
, but the rest of
t
is unspecified.
For complex flavors
:
if
job
=
'E'
, then on exit the diagonal of
t
matches that of
P
, but the rest of
t
is unspecified.
alphar
,
alphai
Arrays, size at least max(1,
n
). The real and imaginary parts, respectively, of each scalar
alpha
defining an eigenvalue of GNEP.
If
alphai
[
j
- 1]
is zero, then the
j
-th eigenvalue is real; if positive, then the
j
-th and (
j
+1)-th eigenvalues are a complex conjugate pair, with
alphai
[
j
]
= -
alphai
[
j
- 1]
.
alpha
Array, size at least max(1,
n
).
The complex scalars
alpha
that define the eigenvalues of GNEP.
alphai
[
i
- 1]
=
S
i
,
i
in the generalized Schur factorization.
beta
Array, size at least max(1,
n
).
For real flavors
:
The scalars
beta
that define the eigenvalues of GNEP.
Together, the quantities
alpha
= (
alphar
[