Contents

# ?ggev3

Computes the generalized eigenvalues and the left and right generalized eigenvectors for a pair of matrices.

## Syntax

Include Files
• mkl.h
Description
For a pair of
n
-by-
n
real or complex nonsymmetric matrices (
A
,
B
),
?ggev3
computes the generalized eigenvalues, and optionally, the left and right generalized eigenvectors.
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.
For real flavors:
The right eigenvector
v
j
corresponding to the eigenvalue
λ
j
of (
A
,
B
) satisfies
A
*
v
j
=
λ
j
*
B
*
v
j
.
The left eigenvector
u
j
corresponding to the eigenvalue
λ
j
of (
A
,
B
) satisfies
u
j
H
*
A
=
λ
j
*
u
j
H
*
B
where
u
j
H
is the conjugate-transpose of
u
j
.
For complex flavors:
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 eigenvalues
λ
j
of (
A
,
B
) satisfies
u
j
H
*
A
=
λ
j
*
u
j
H
*
B
where
u
j
H
is the conjugate-transpose of
u
j
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobvl
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
jobvr
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
n
The order of the matrices
A
,
B
,
VL
, and
VR
.
n
0.
a
Array, size
(
lda
*
n
)
.
On entry, the matrix
A
in the pair (
A
,
B
).
lda
a
.
lda
max(1,
n
).
b
Array, size
(
ldb
*
n
)
.
On entry, the matrix
B
in the pair (
A
,
B
).
ldb
b
.
ldb
max(1,
n
).
ldvl
The leading dimension of the matrix
VL
.
ldvl
1, and if
jobvl
= 'V',
ldvl
n
.
ldvr
The leading dimension of the matrix
VR
.
ldvr
1, and if
jobvr
= 'V',
ldvr
n
.
Output Parameters
a
On exit,
a
is overwritten.
b
On exit,
b
is overwritten.
alphar
Array, size (
n
).
alphai
Array, size (
n
).
alpha
Array, size (
n
).
beta
Array, size (
n
).
For real flavors:
On exit,
(
alphar
[
j
] +
alphai
[
j
]*i)/
beta
[
j
],
j
=0,...,
n
- 1
, are the generalized eigenvalues. If
alphai
[
j
- 1]
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
]
negative.
Note: the quotients
alphar
[
j
- 1]
/
beta
[
j
- 1]
and
alphai
[
j
- 1]
/
beta
[
j
- 1]
can easily over- or underflow, and
beta
(
j
) might even be zero. Thus, you should avoid computing the ratio
alpha
/
beta
by simply dividing
alpha
by
beta
. However,
alphar
and
alphai
are always less than and usually comparable with norm(
A
) in magnitude, and
beta
is always less than and usually comparable with norm(
B
).
For complex flavors:
On exit,
alpha
[
j
]/
beta
[
j
],
j
=0,...,
n
- 1
, are the generalized eigenvalues.
Note: the quotients
alpha
[
j
- 1]
/
beta
[
j
- 1]
may easily over- or underflow, and
beta
(
j
) can even be zero. Thus, you should avoid computing the ratio
alpha
/
beta
by simply dividing
alpha
by
beta
. However,
alpha
is always less than and usually comparable with norm(
A
) in magnitude, and
beta
is always less than and usually comparable with norm(
B
).
vl
Array, size
(
ldvl
*
n
)
.
For real flavors:
If
jobvl
= 'V', the left eigenvectors
u
j
are stored one after another in the columns of
vl
, in the same order as their eigenvalues. If the
j
-th eigenvalue is real, then
u
j
= the
j
-th column of
vl
. If the
j
-th and (
j
+1)-st eigenvalues form a complex conjugate pair, then
the real part of
u
j
= the
j
-th column of
vl
and the imaginary part of
v
j
= the (
j
+ 1)-st column of
vl
.
Each eigenvector is scaled so the largest component has abs(real part)+abs(imag. part)=1.
Not referenced if
jobvl
= 'N'.
For complex flavors:
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(real part) + abs(imag. part) = 1.
Not referenced if
jobvl
= 'N'.
vr
Array, size
(
ldvr
*
n
)
.
For real flavors:
If
jobvr
= 'V', the right eigenvectors
v
j
are stored one after another in the columns of
vr
, in the same order as their eigenvalues. If the
j
-th eigenvalue is real, then
v
j
= the
j
-th column of
vr
. If the
j
-th and (
j
+ 1)-st eigenvalues form a complex conjugate pair, then
the real part of
v
j
= the
j
-th column of
vr
and the imaginary part of
v
j
= the (
j
+ 1)-st column of
vr
.
Each eigenvector is scaled so the largest component has abs(real part)+abs(imag. part)=1.
Not referenced if
jobvr
= 'N'.
For complex flavors:
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(real part) + abs(imag. part) = 1.
Not referenced if
jobvr
= 'N'.
Return Values
This function returns a value
info
.
= 0: successful exit
< 0: if
info
= -
i
, the
i
-th argument had an illegal value.
=1,...,
n
:
for real flavors:
The QZ iteration failed. No eigenvectors have been calculated, but
alphar
[
j
],
alphar
[
j
] and
beta
[
j
] should be correct for
j
=
info
,...,
n
- 1.
for complex flavors:
The QZ iteration failed. No eigenvectors have been calculated, but
alpha
[
j
] and
beta
[
j
] should be correct for
j
=
info
,...,
n
- 1.
>
n
:
=
n
+ 1: other than QZ iteration failed in
?hgeqz
,
=
n
+ 2: error return from
?tgevc
.

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804