Contents

# p?sytd2/p?hetd2

Reduces a symmetric/Hermitian matrix to real symmetric tridiagonal form by an orthogonal/unitary similarity transformation (local unblocked algorithm).

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?sytd2/p?hetd2
function
reduces a real symmetric/complex Hermitian matrix sub(
A
) to symmetric/Hermitian tridiagonal form
T
by an orthogonal/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 symmetric/Hermitian matrix sub(
A
) is stored:
=
'U'
: upper triangular
=
'L'
: lower triangular
n
(global)
The number of rows and columns to be operated on, that is, the order of the distributed matrix sub(
A
).
n
0
.
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
symmetric/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 the strictly lower triangular part of sub(
A
) is not referenced.
If
uplo
=
'L'
n
-by-
n
lower triangular part of sub(
A
) contains the lower triangular part of the matrix, and the strictly upper triangular part of sub(
A
) is not referenced.
ia
,
ja
(global)
The row and column indices in the global matrix
A
indicating the first row and the first column of the sub(
A
), respectively.
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
work
(local)
The array
work
is a temporary workspace array of size
lwork
.
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 orthogonal/unitary matrix
Q
as a product of elementary reflectors;
if
uplo
=
'L'
, the diagonal and first subdiagonal of
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 orthogonal/unitary matrix
Q
as a product of elementary reflectors.
See the
Application Notes
below.
d
(local)
Array of size
LOCc
(
ja
+
n
-1)
. The diagonal elements of the tridiagonal matrix
T
:
d
[
i
] =
A
(
i
+1,
i
+1), where
i
=0,1, ...,
LOCc
(
ja
+
n
-1) -1
;
d
is tied to the distributed matrix
A
.
e
(local)
Array 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)
if
uplo
=
'U'
,
e
[i] =
A
(
i
+2,
i
+1)
if
uplo
=
'L'
,
where
i
=0,1, ...,
LOCc
(
ja
+
n
-1) -1.
e
is tied to the distributed matrix
A
.
tau
(local)
Array of size
LOCc
(
ja
+
n
-1)
.
The scalar factors of the elementary reflectors.
tau
is tied to the distributed matrix
A
.
work

On exit,
work

returns the minimal and optimal value of
lwork
.
lwork
(local or global)
The size of the workspace array
work
.
lwork
is local input and must be at least
lwork
3
n
.
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
.
info
(local)
: successful exit
< 0
: if the
i
-th argument
, indexed
i
-1,
is an array and the
j
-th entry had an illegal value,
then
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 real/complex scalar, and
v
is a real/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 real/complex scalar, and
v
is a real/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
: where
d
and
e
denotes diagonal and off-diagonal elements of
T
, and
v
i
denotes an element of the vector defining
H
(i).
The distributed matrix sub(
A
) must verify some alignment properties, namely the following expression should be true:
(
mb_a
==
nb_a
&&
iroffa
==
icoffa
)
where
iroffa
= mod(
ia
- 1,
mb_a
)
and
icoffa
= mod(
ja
-1,
nb_a
)
.

#### Product and Performance Information

1

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