## Developer Reference

• 0.10
• 10/21/2020
• Public Content
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

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