Contents

# p?latrd

Reduces the first
nb
rows and columns of a symmetric/Hermitian matrix
A
to real tridiagonal form by an orthogonal/unitary similarity transformation.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?latrd
function
reduces
nb
rows and columns of a real symmetric or complex Hermitian matrix sub(
A
)=
A
(
ia
:
ia
+
n
-1
,
ja
:
ja
+
n
-1)
to symmetric/complex tridiagonal form by an orthogonal/unitary similarity transformation
Q'
*sub(
A
)*
Q
, and returns the matrices
V
and
W
, which are needed to apply the transformation to the unreduced part of sub(
A
).
If
uplo
=
U
,
p?latrd
reduces the last
nb
rows and columns of a matrix, of which the upper triangle is supplied;
if
uplo
=
L
,
p?latrd
reduces the first
nb
rows and columns of a matrix, of which the lower triangle is supplied.
This is an auxiliary
function
called by
p?sytrd
/
p?hetrd
.
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
.
nb
(global)
The number of rows and columns to be reduced.
a
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 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 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.
ia
(global)
The row index in the global matrix
A
indicating the first row of sub(
A
).
ja
(global)
The column index in the global matrix
A
indicating the first column of sub(
A
).
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
iw
(global)
The row index in the global matrix
W
indicating the first row of sub(
W
).
jw
(global)
The column index in the global matrix
W
indicating the first column of sub(
W
).
descw
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
W
.
work
(local)
Workspace array of size
nb_a
.
Output Parameters
a
(local)
On exit, if
uplo
=
'U'
, the last
nb
columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub(
A
); the elements above the diagonal with the array
tau
represent the orthogonal/unitary matrix
Q
as a product of elementary reflectors;
if
uplo
=
'L'
, the first
nb
columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub(
A
); the elements below the diagonal with the array
tau
represent the orthogonal/unitary matrix
Q
as a product of elementary reflectors.
d
(local)
Array of size
LOCc
(
ja
+
n
-1)
.
The diagonal elements of the tridiagonal matrix
T
:
d
[
i
] =
A
(i+1,i+1),
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'
,
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)
. This array contains the scalar factors of the elementary reflectors.
tau
is tied to the distributed matrix
A
.
w
(local)
Pointer into the local memory to an array of size
lld_w
*
nb_w
. This array contains the local pieces of the
n
-by-
nb_w
matrix
w
required to update the unreduced part of sub(
A
).
Application Notes
If
uplo
=
'U'
, the matrix
Q
is represented as a product of elementary reflectors
`Q = H(n)*H(n-1)*...*H(n-nb+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
:
n
) = 0
and
v
(
i
-1) = 1
;
v
(1:
i
-1)
is stored on exit in
A
(
ia
:
ia
+
i
-1
,
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(nb)`
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 elements of the vectors
v
together form the
n
-by-
nb
matrix
V
which is needed, with
W
, to apply the transformation to the unreduced part of the matrix, using a symmetric/Hermitian rank-2
k
update of the form:
sub(
A
) := sub(
A
)-
vw
'-
wv
'
.
The contents of
a
on exit are illustrated by the following examples with
n
= 5
and
nb
= 2
: where
d
denotes a diagonal element of the reduced matrix,
a
denotes an element of the original matrix that is unchanged, 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.