Contents

# p?hetrd

Reduces a Hermitian matrix to Hermitian tridiagonal form by a unitary similarity transformation.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?hetrd
function
reduces a complex Hermitian matrix sub(
A
) to Hermitian tridiagonal form
T
by a unitary similarity transformation:
Q'
*sub(
A
)*
Q
=
T
where sub(
A
) =
A
(
ia
:
ia
+
n
-1,
ja
:
ja
+
n
-1).
Input Parameters
uplo
(global)
Specifies whether the upper or lower triangular part of the Hermitian matrix sub(
A
) is stored:
If
uplo
=
'U'
, upper triangular
If
uplo
=
'L'
, lower triangular
n
(global) The order of the distributed matrix sub(
A
)
(
n
0)
.
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 Hermitian distributed matrix sub(
A
).
If
uplo
=
'U'
n
-by-
n
upper triangular part of sub(
A
) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced.
If
uplo
=
'L'
n
-by-
n
lower triangular part of sub(
A
) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced.
(see
Application Notes
below).
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
work
, must be at least:
lwork
max
(
NB
*(
np
+1), 3*
NB
)
where
NB
=
mb_a
=
nb_a
,
np
=
numroc
(
n
,
NB
,
MYROW
,
iarow
,
NPROW
)
,
iarow
=
indxg2p
(
ia
,
NB
,
MYROW
,
rsrc_a
,
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
On exit,
If
uplo
=
'U'
, the diagonal and first superdiagonal of sub(
A
) are overwritten by the corresponding elements of the tridiagonal matrix
T
, and the elements above the first superdiagonal, with the array
tau
, represent the unitary matrix
Q
as a product of elementary reflectors;if
uplo
=
'L'
, the diagonal and first subdiagonal of sub(
A
) are overwritten by the corresponding elements of the tridiagonal matrix
T
, and the elements below the first subdiagonal, with the array
tau
, represent the unitary matrix
Q
as a product of elementary reflectors
(see
Application Notes
below)
.
d
(local)
Arrays of size
LOCc
(
ja
+
n
-1)
. The diagonal elements of the tridiagonal matrix
T
:
d
[
i
]
=
A
(
i
+1,
i
+1), 0
i
<
LOCc
(
ja
+
n
-1)
.
d
is tied to the distributed matrix
A
.
e
(local)
Arrays of size
LOCc
(
ja
+
n
-1)
if
uplo
=
'U'
;
LOCc
(
ja
+
n
-2)
- otherwise.
The off-diagonal elements of the tridiagonal matrix
T
:
e
[
i
]
=
A
(
i
+1,
i
+2), 0
i
<
LOCc
(
ja
+
n
-1)
if
uplo
=
'U'
,
e
[
i
]
=
A
(
i
+2,
i
+1)
if
uplo
=
'L'
.
e
is tied to the distributed matrix
A
.
tau
(local)
Array of size
LOCc
(
ja
+
n
-1)
. This array contains the scalar factors of the elementary reflectors.
tau
is tied to the distributed matrix
A
.
work
[0]
On exit
work
[0]
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
If
uplo
=
'U'
, the matrix
Q
is represented as a product of elementary reflectors
Q
=
H
(
n
-1)*...*
H
(2)*
H
(1).
Each
H
(
i
) has the form
H
(
i
) =
i
-
tau
*
v
*
v'
,
where
tau
is a complex scalar, and
v
is a complex vector with
v
(
i
+1:
n
) = 0 and
v
(
i
) = 1;
v
(1:
i
-1) is stored on exit in
A
(
ia
:
ia
+
i
-2,
ja
+
i
), and
tau
in
tau
[
ja
+
i
-2]
.
If
uplo
=
'L'
, the matrix
Q
is represented as a product of elementary reflectors
Q
=
H
(1)*
H
(2)*...*
H
(
n
-1).
Each
H
(
i
) has the form
H
(
i
) =
i
-
tau
*
v
*
v'
,
where
tau
is a complex scalar, and
v
is a complex vector with
v
(1:
i
) = 0 and
v
(
i
+1) = 1;
v
(
i
+2:
n
) is stored on exit in
A
(
ia
+
i
+1:
ia
+
n
-1,
ja
+
i
-1), and
tau
in
tau
[
ja
+
i
-2]
.
The contents of sub(
A
) on exit are illustrated by the following examples with
n
= 5
:
If
uplo
=
'U'
:
If
uplo
=
'L'
:
where
d
and
e
denote diagonal and off-diagonal elements of
T
, and
v
i
denotes an element of the vector defining
H
(
i
).

#### Product and Performance Information

1

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