## Developer Reference

• 0.10
• 10/21/2020
• Public Content
Contents

# p?gehrd

Reduces a general matrix to upper Hessenberg form.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?gehrd
function
reduces a real/complex general distributed matrix sub(
A
) to upper Hessenberg form
H
by an orthogonal or unitary similarity transformation
Q'
*sub(
A
)*
Q
=
H
,
where sub(
A
) =
A
(
ia
:
ia
+
n
-1,
ja
:
ja
+
n
-1).
Input Parameters
n
(global). The order of the distributed matrix sub(
A
)
(
n
0)
.
ilo
,
ihi
(global).
It is assumed that sub(
A
) is already upper triangular in rows
ia
:
ia
+
ilo
-2
and
ia
+
ihi
:
ia
+
n
-1
and columns
ja
:
ja
+
ilo
-2
and
ja
+
ihi
:
ja
+
n
-1
. (See
Application Notes
below).
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
*
LOCc
(
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 the submatrix
A
, respectively.
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
work
(local)
Workspace array of size
lwork
.
lwork
(local or global) size of the array
work
.
lwork
is local input and must be at least
lwork
NB
*
NB
+
NB
*
max
(
ihip
+1,
ihlp
+
inlq
)
where
NB
=
mb_a
=
nb_a
,
iroffa
=
mod
(
ia
-1,
NB
)
,
icoffa
=
mod
(
ja
-1,
NB
)
,
ioff
=
mod
(
ia
+
ilo
-2,
NB
),
iarow
=
indxg2p
(
ia
,
NB
,
MYROW
,
rsrc_a
,
NPROW
),
ihip
=
numroc
(
ihi
+
iroffa
,
NB
,
MYROW
,
iarow
,
NPROW
)
,
ilrow
=
indxg2p
(
ia
+
ilo
-1,
NB
,
MYROW
,
rsrc_a
,
NPROW
)
,
ihlp
=
numroc
(
ihi
-
ilo
+
ioff
+1,
NB
,
MYROW
,
ilrow
,
NPROW
)
,
ilcol
=
indxg2p
(
ja
+
ilo
-1,
NB
,
MYCOL
,
csrc_a
,
NPCOL
)
,
inlq
=
numroc
(
n
-
ilo
+
ioff
+1,
NB
,
MYCOL
,
ilcol
,
NPCOL
)
,
mod(
x
,
y
)
is the integer remainder of
x
/
y
.
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
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 at least
max(
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

contains the minimum value of
lwork
required for optimum performance.
info
(global)
= 0
: the execution is successful.
< 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,
ja
+
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
:
on entry on exit 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

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