Contents

# p?gehd2

Reduces a general matrix to upper Hessenberg form by an orthogonal/unitary similarity transformation (unblocked algorithm).

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?gehd2
function
reduces a real/complex general distributed matrix sub(
A
) to upper Hessenberg form
H
by an orthogonal/unitary similarity transformation:
Q'
*sub(
A
)*
Q
=
H
, where
sub(
A
) =
A
(
ia
+
n
-1 :
ia
+
n
-1,
ja
+
n
-1 :
ja
+
n
-1)
.
Input Parameters
n
(global) The order of the distributed submatrix
A
.
(
n
0)
.
ilo
,
ihi
(global) It is assumed that the matrix sub(
A
) is already upper triangular in rows
ia
:
ia
+
ilo
-2
and
ia
+
ihi
:
ia
+
n
-1
and columns
ja
:
ja
+
jlo
-2
and
ja
+
jhi
:
ja
+
n
-1
.
See
Application Notes
for further information.
If
n
0
,
1 ≤
ilo
ihi
n
; otherwise set
ilo
= 1
,
ihi
=
n
.
a
(local).
Pointer into the local memory to an array of size
lld_a
*
LOC
c
(
ja
+
n
-1)
.
On entry, this array contains the local pieces of the
n
-by-
n
general distributed matrix sub(
A
) to be reduced.
ia
,
ja
(global) The row and column indices in the global matrix
A
indicating the first row and the first column of sub(
A
), respectively.
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
work
(local).
This is a workspace array of size
lwork
.
lwork
(local or global)
The size of the array
work
.
lwork
is local input and must be at least
lwork
nb
+ max(
npa
0,
nb
)
, where
nb
=
mb_a
=
nb_a
,
iroffa
= mod(
ia
-1,
nb
),
iarow
=
indxg2p
(
ia
,
nb
,
myrow
,
rsrc_a
,
nprow
),
npa
0 =
numroc
(
ihi+iroffa
,
nb
,
myrow
,
iarow
,
nprow
)
.
indxg2p
and
numroc
are ScaLAPACK tool functions;
myrow
,
mycol
,
nprow
, and
npcol
can be determined by calling the
function
blacs_gridinfo
.
If
lwork
= -1
, then
lwork
is global input and a workspace query is assumed; the
function
only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by
pxerbla
.
Output Parameters
a
(local). On exit, the upper triangle and the first subdiagonal of sub(
A
) are overwritten with the upper Hessenberg matrix
H
, and the elements below the first subdiagonal, with the array
tau
, represent the orthogonal/unitary matrix
Q
as a product of elementary reflectors.
(see
Application Notes
below)
.
tau
(local).
Array of size
LOCc
(
ja
+
n
-2)
The scalar factors of the elementary reflectors (see
Application Notes
below). Elements
ja
:
ja
+
ilo
-2
and
ja
+
ihi
:
ja
+
n
-2
of the global vector
tau
are set to zero.
tau
is tied to the distributed matrix
A
.
work
On exit,
work

returns the minimal and optimal
lwork
.
info
(local)
If
info
= 0
, the execution is successful.
if
info
< 0: If the
i
-th argument is an array and the
j
-th entry
, indexed
j
-1,
info
= - (
i
*100+
j
)
, if the
i
-th argument is a scalar and had an illegal value, then
info
= -
i
.
Application Notes
The matrix
Q
is represented as a product of (
ihi-ilo
) elementary reflectors
`Q = H(ilo)*H(ilo+1)*...*H(ihi-1).`
Each
H
(
i
)
has the form
`H(i) = I - tau*v*v',`
where
tau
is a real/complex scalar, and
v
is a real/complex vector with
v
(1:
i
)=0
,
v
(
i
+1)=1
and
v
(
ihi
+1:
n
)=0
;
v
(
i
+2:
ihi
)
is stored on exit in
A
(
ia
+
ilo
+
i
:
ia
+
ihi
-1
,
ia
+
ilo
+
i
-2)
, and
tau
in
tau
[
ja
+
ilo
+
i
-3]
.
The contents of
A
(
ia
:
ia
+
n
-1
,
ja
:
ja
+
n
-1)
are illustrated by the following example, with
n
= 7
,
ilo
= 2
and
ihi
= 6
: where
a
denotes an element of the original matrix sub(
A
),
h
denotes a modified element of the upper Hessenberg matrix
H
, and
vi
denotes an element of the vector defining
H
(
ja
+
ilo
+
i
-2)
.

#### Product and Performance Information

1

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