## Developer Reference

• 2021.1
• 12/04/2020
• Public Content
Contents

# ?ggev

Computes the generalized eigenvalues, and the left and/or right generalized eigenvectors for a pair of nonsymmetric matrices.

## Syntax

Include Files
• mkl.h
Description
The
?ggev
routine computes the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors for a pair of
n
-by-
n
real/complex nonsymmetric matrices (
A
,
B
).
A generalized eigenvalue for a pair of matrices (
A
,
B
) is a scalar
λ
or a ratio
alpha
/
beta
=
λ
, such that
A
-
λ
*
B
is singular. It is usually represented as the pair (
alpha
,
beta
), as there is a reasonable interpretation for
beta
=0
and even for both being zero.
The right generalized eigenvector
v
(j)
corresponding to the generalized eigenvalue
λ
(j)
of (
A
,
B
) satisfies
A
*
v
(j) =
λ
(j)*
B
*
v
(j)
.
The left generalized eigenvector
u
(j)
corresponding to the generalized eigenvalue
λ
(j)
of (
A
,
B
) satisfies
u
(j)
H
*
A
=
λ
(j)*
u
(j)
H
*
B
where
u
(j)
H
denotes the conjugate transpose of
u
(j)
.
The
?ggev
routine replaces the deprecated
?gegv
routine.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobvl
Must be
'N'
or
'V'
.
If
jobvl
=
'N'
, the left generalized eigenvectors are not computed;
If
jobvl
=
'V'
, the left generalized eigenvectors are computed.
jobvr
Must be
'N'
or
'V'
.
If
jobvr
=
'N'
, the right generalized eigenvectors are not computed;
If
jobvr
=
'V'
, the right generalized eigenvectors are computed.
n
The order of the matrices
A
,
B
,
vl
, and
vr
(
n
0
).
a
,
b
Arrays:
a
(size at least max(1,
lda
*
n
))
is an array containing the
n
-by-
n
matrix
A
(first of the pair of matrices).
b
(size at least max(1,
ldb
*
n
))
is an array containing the
n
-by-
n
matrix
B
(second of the pair of matrices).
lda
The leading dimension of the array
a
. Must be at least max(1,
n
).
ldb
The leading dimension of the array
b
. Must be at least max(1,
n
).
ldvl
,
ldvr
The leading dimensions of the output matrices
vl
and
vr
, respectively.
Constraints:
ldvl
1
. If
jobvl
=
'V'
,
ldvl
max(1,
n
)
.
ldvr
1
. If
jobvr
=
'V'
,
ldvr
max(1,
n
)
.
Output Parameters
a
,
b
On exit, these arrays have been overwritten.
alphar
,
alphai
Arrays, size at least max(1,
n
) each. Contain values that form generalized eigenvalues in real flavors.
See
beta
.
alpha
Array, size at least max(1,
n
). Contain values that form generalized eigenvalues in complex flavors. See
beta
.
beta
Array, size at least max(1,
n
).
For real flavors
:
On exit,
(
alphar
[
j
] +
alphai
[
j
]*i)/
beta
[
j
],
j
=0,...,
n
- 1
, are the generalized eigenvalues.
If
alphai
[
j
]
is zero, then the
j
-th eigenvalue is real; if positive, then the
j
-th and (
j
+1)-st eigenvalues are a complex conjugate pair, with
alphai
[
j
+1]
negative.
For complex flavors:
On exit,
alpha
[
j
]/
beta
[
j
],
j
=0,...,
n
- 1
, are the generalized eigenvalues.
Application Notes
below.
vl
,
vr
Arrays:
vl
(size at least max(1,
ldvl
*
n
))
. Contains the matrix of left generalized eigenvectors
VL
.
If
jobvl
=
'V'
, the left generalized eigenvectors
u
j
are stored one after another in the columns of
VL
, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has
abs(Re) + abs(Im) = 1
.
If
jobvl
=
'N'
,
vl
is not referenced.
For real flavors
:
If the j-th eigenvalue is real,
then the
k
-th component of the
j
-th left eigenvector
u
j
is stored in
vl
[(
k
- 1) + (
j
- 1)*
ldvl
]
for column major layout and in
vl
[(
k
- 1)*
ldvl
+ (
j
- 1)]
for row major layout.
.
If the
j
-th and (
j
+1)-st eigenvalues form a complex conjugate pair, then for
i
= sqrt(-1)
,
the
k
-th components of the
j
-th left eigenvector
u
j
are
vl
[(
k
- 1) + (
j
- 1)*
ldvl
]
+ i*
vl
[(
k
- 1) +
j
*
ldvl
]
for column major layout and
vl
[(
k
- 1)*
ldvl
+ (
j
- 1)]
+ i*
vl
[(
k
- 1)*
ldvl
+
j
]
for row major layout. Similarly, the
k
-th components of left eigenvector
j
+1
u
j+1
are
vl
[(
k
- 1) + (
j
- 1)*
ldvl
]
- i*
vl
[(
k
- 1) +
j
*
ldvl
]
for column major layout and
vl
[(
k
- 1)*
ldvl
+ (
j
- 1)]
- i*
vl
[(
k
- 1)*
ldvl
+
j
]
for row major layout.
.
For complex flavors
:
The
k
-th component of the
j
-th left eigenvector
u
j
is stored in
vl
[(
k
- 1) + (
j
- 1)*
ldvl
]
for column major layout and in
vl
[(
k
- 1)*
ldvl
+ (
j
- 1)]
for row major layout.
vr
(size at least max(1,
ldvr
*
n
))
. Contains the matrix of right generalized eigenvectors
VR
.
If
jobvr
=
'V'
, the right generalized eigenvectors
v
j
are stored one after another in the columns of
VR
, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(Re) + abs(Im) = 1.
If
jobvr
=
'N'
,
vr
is not referenced.
For real flavors
:
If the
j
-th eigenvalue is real, then
The
k
-th component of the
j
-th right eigenvector
v
j
is stored in
vr
[(
k
- 1) + (
j
- 1)*
ldvr
]
for column major layout and in
vr
[(
k
- 1)*
ldvr
+ (
j
- 1)]
for row major layout.
.
If the
j
-th and (
j
+1)-st eigenvalues form a complex conjugate pair, then
the
k
-th components of the
j
-th right eigenvector
v
j
can be computed as
vr
[(
k
- 1) + (
j
- 1)*
ldvr
]
+ i*
vr
[(
k
- 1) +
j
*
ldvr
]
for column major layout and
vr
[(
k
- 1)*
ldvr
+ (
j
- 1)]
+ i*
vr
[(
k
- 1)*
ldvr
+
j
]
for row major layout. Similarly, the
k
-th components of the right eigenvector
j
+1
v
{
j
+1}
can be computed as
vr
[(
k
- 1) + (
j
- 1)*
ldvr
]
- i*
vr
[(
k
- 1) +
j
*
ldvr
]
for column major layout and
vr
[(
k
- 1)*
ldvr
+ (
j
- 1)]
- i*
vr
[(
k
- 1)*
ldvr
+
j
]
for row major layout.
.
For complex flavors
:
The
k
-th component of the
j
-th right eigenvector
v
j
is stored in
vr
[(
k
- 1) + (
j
- 1)*
ldvr
]
for column major layout and in
vr
[(
k
- 1)*
ldvr
+ (
j
- 1)]
for row major layout.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
If
info
=
i
, and
i
n
: the
QZ
iteration failed. No eigenvectors have been calculated, but
alphar
[
j
],
alphai
[
j
] (for real flavors), or
alpha
[
j
] (for complex flavors), and
beta
[
j
],
j=
info
,...,
n
- 1
should be correct.
i
>
n
: errors that usually indicate LAPACK problems:
i
=
n
+1
: other than
QZ
iteration failed in hgeqz;
i
=
n
+2
: error return from tgevc.
Application Notes
The quotients
alphar
[
j
]/
beta
[
j
]
and
alphai
[
j
]/
beta
[
j
]
may easily over- or underflow, and
beta
[
j
]
may even be zero. Thus, you should avoid simply computing the ratio. However,
alphar
and
alphai
(for real flavors) or
alpha
(for complex flavors) will be always less than and usually comparable with norm(
A
) in magnitude, and
beta
always less than and usually comparable with norm(
B
).

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.