Contents

# ?gghrd

Reduces a pair of matrices to generalized upper Hessenberg form using orthogonal/unitary transformations.

## Syntax

Include Files
• mkl.h
Description
The routine reduces a pair of real/complex matrices (
A
,
B
) to generalized upper Hessenberg form using orthogonal/unitary transformations, where
A
is a general matrix and
B
is upper triangular. The form of the generalized eigenvalue problem is
A
*
x
=
λ
*
B
*
x
, and
B
is typically made upper triangular by computing its
QR
factorization and moving the orthogonal matrix
Q
to the left side of the equation.
This routine simultaneously reduces
A
to a Hessenberg matrix
H
:
Q
H
*
A
*
Z
=
H
and transforms
B
to another upper triangular matrix
T
:
Q
H
*
B
*
Z
=
T
in order to reduce the problem to its standard form
H
*
y
=
λ
*
T
*
y
, where
y
=
Z
H
*
x
.
The orthogonal/unitary matrices
Q
and
Z
are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices
Q
1
and
Z
1
, so that
Q
1
*
A
*
Z
1
H
= (
Q
1
*
Q
)*
H
*(
Z
1
*
Z
)
H
Q
1
*
B*
Z
1
H
= (
Q
1
*
Q
)*
T*
(
Z
1
*
Z
)
H
If
Q
1
is the orthogonal/unitary matrix from the
QR
factorization of
B
in the original equation
A
*
x
=
λ
*
B
*
x
, then the routine
?gghrd
reduces the original problem to generalized Hessenberg form.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
compq
Must be
'N'
,
'I'
, or
'V'
.
If
compq
=
'N'
, matrix
Q
is not computed.
If
compq
=
'I'
,
Q
is initialized to the unit matrix, and the orthogonal/unitary matrix
Q
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'
, matrix
Z
is not computed.
If
compz
=
'I'
,
Z
is initialized to the unit matrix, and the orthogonal/unitary matrix
Z
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
A
and
B
(
n
0).
ilo
,
ihi
ilo
and
ihi
mark the rows and columns of
A
which are to be reduced. It is assumed that
A
is already upper triangular in rows and columns 1:
ilo
-1 and
ihi
+1:
n
. Values of
ilo
and
ihi
are normally set by a previous call to ggbal; otherwise they should be set to 1 and
n
respectively.
Constraint:
If
n
> 0
, then 1
ilo
ihi
n
;
if
n
= 0
, then
ilo
= 1
and
ihi
= 0
.
a
,
b
,
q
,
z
Arrays:
a
(size max(1,
lda
*
n
))
contains the
n
-by-
n
general matrix
A
.
b
(size max(1,
ldb
*
n
))
contains the
n
-by-
n
upper triangular matrix
B
.
q
(size max(1,
ldq
*
n
))
If
compq
=
'N'
, then
q
is not referenced.
If
compq
=
'V'
, then
q
must contain the orthogonal/unitary matrix
Q
1
, typically from the
QR
factorization of
B
.
z
(size max(1,
ldz
*
n
))
If
compz
=
'N'
, then
z
is not referenced.
If
compz
=
'V'
, then
z
must contain the orthogonal/unitary matrix
Z
1
.
lda
a
; at least max(1,
n
).
ldb
b
; 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
compz
=
'N'
, then
ldz
1.
If
compz
=
'I'
or
'V'
, then
ldz
max(1,
n
).
Output Parameters
a
On exit, the upper triangle and the first subdiagonal of
A
are overwritten with the upper Hessenberg matrix
H
, and the rest is set to zero.
b
On exit, overwritten by the upper triangular matrix
T
=
Q
H
*
B*
Z
. The elements below the diagonal are set to zero.
q
If
compq
=
'I'
, then
q
contains the orthogonal/unitary matrix
Q
, ;
If
compq
=
'V'
, then
q
is overwritten by the product
Q
1
*
Q
.
z
If
compz
=
'I'
, then
z
contains the orthogonal/unitary matrix
Z
;
If
compz
=
'V'
, then
z
is overwritten by the product
Z
1
*
Z
.
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.

#### Product and Performance Information

1

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Notice revision #20110804