Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

p?hentrd

Reduces a complex Hermitian matrix
to Hermitian tridiagonal form.

Syntax

void pchentrd
(
const
char*
uplo
,
const
MKL_INT*
n
,
MKL_Complex8*
a
,
const
MKL_INT*
ia
,
const
MKL_INT*
ja
,
const
MKL_INT*
desca
,
float*
d
,
float*
e
,
MKL_Complex8*
tau
,
MKL_Complex8*
work
,
const
MKL_INT*
lwork
,
float*
rwork
,
const
MKL_INT*
lrwork
,
MKL_INT*
info
);
void pzhentrd
(
const
char*
uplo
,
const
MKL_INT*
n
,
MKL_Complex16*
a
,
const
MKL_INT*
ia
,
const
MKL_INT*
ja
,
const
MKL_INT*
desca
,
double*
d
,
double*
e
,
MKL_Complex16*
tau
,
MKL_Complex16*
work
,
const
MKL_INT*
lwork
,
double*
rwork
,
const
MKL_INT*
lrwork
,
MKL_INT*
info
);
Include Files
  • mkl_scalapack.h
Description
p?hentrd
 is a prototype version of
p?hetrd
 which uses tailored codes (either the serial,
?hetrd
, or the parallel code,
p?hettrd
) when adequate workspace is provided.
p?hentrd
 reduces a complex Hermitian matrix sub(
A
 ) to Hermitian tridiagonal form
T
 by an unitary similarity transformation:
Q
' * sub(
A
 ) *
Q
 =
T
, where sub(
A
 ) =
A
(
ia
:
ia
+
n
-1,
ja
:
ja
+
n
-1).
p?hentrd
 is faster than
p?hetrd
 on almost all matrices, particularly small ones (i.e.
n
 < 500 * sqrt(P) ), provided that enough workspace is available to use the tailored codes.
The tailored codes provide performance that is essentially independent of the input data layout.
The tailored codes place no restrictions on
ia
,
ja
, MB or NB. At present,
ia
,
ja
, MB and NB are restricted to those values allowed by
p?hetrd
 to keep the interface simple (see the Application Notes section for more information about the restrictions).
Input Parameters
uplo
(global)
Specifies whether the upper or lower triangular part of the Hermitian matrix sub(
A
 ) is stored:
= 'U': Upper triangular
= 'L': Lower triangular
n
(global)
The number of rows and columns to be operated on, i.e. the order of the distributed submatrix 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', the leading
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', the leading
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 array
a
 indicating the first row of sub(
A
 ).
ja
(global)
The column index in the global array
a
 indicating the first column of sub(
A
 ).
desca
(global and local)
Array of size
dlen_
.
The array descriptor for the distributed matrix
A
.
work
(local)
Array, size (
lwork
)
lwork
(local or global)
The size of the array
work
.
lwork
 is local input and must be at least
lwork
 >= MAX( NB * ( NP +1 ), 3 * NB ).
For optimal performance, greater workspace is needed:
lwork
 >= 2*(
ANB
+1 )*( 4*
NPS
+2 ) + (
NPS
 + 4 ) *
NPS
ANB
 =
pjlaenv
(
ICTXT
, 3, '
p?hettrd
', 'L', 0, 0, 0, 0 )
ICTXT
 =
desca
(
ctxt_
 )
SQNPC
 = INT(
sqrt
( REAL(
NPROW
 *
NPCOL
 ) ) )
NPS
 = MAX(
numroc
(
n
, 1, 0, 0,
SQNPC
 ), 2*
ANB
 )
numroc
 is a ScaLAPACK tool function.
pjlaenv
 is a ScaLAPACK environmental inquiry function.
NPROW
 and
NPCOL
 can be determined by calling the subroutine
blacs_gridinfo
.
rwork
(local)
Array, size (
lrwork
)
lrwork
(local or global)
The size of the array
rwork
.
lrwork
 is local input and must be at least
lrwork
 >= 1.
For optimal performance, greater workspace is needed, i.e.
lrwork
 >= MAX( 2 *
n
 )
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.
d
(local)
Array, size LOCc(
ja
+
n
-1)
The diagonal elements of the tridiagonal matrix
T
:
d
[
i
 - 1]
 =
A
(
i
,
i
).
d
 is tied to the distributed matrix
A
.
e
(local)
Array, size LOCc(
ja
+
n
-1) if
uplo
 = 'U', LOCc(
ja
+
n
-2) otherwise.
The off-diagonal elements of the tridiagonal matrix
T
:
e
[
i
 - 1]
 =
A
(
i
,
i
+1) if
uplo
 = 'U',
e
[
i
 - 1]
 =
A
(
i
+1,
i
) if
uplo
 = 'L'.
e
 is tied to the distributed matrix
A
.
tau
(local)
Array, size LOCc(
ja
+
n
-1).
This array contains the scalar factors
tau
 of the elementary reflectors.
tau
 is tied to the distributed matrix
A
.
work
On exit,
work
[0]
 returns the optimal
lwork
.
rwork
On exit,
rwork
[0]
 returns the optimal
lrwork
.
info
(global)
= 0: successful exit
< 0: If the
i
-th argument 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 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-1).
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-1).
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
vi
 denotes an element of the vector defining H(
i
).
Alignment requirements
The distributed submatrix sub(
A
 ) must verify some alignment properties, namely the following expression should be true:
(
mb_a
 =
nb_a
 and
IROFFA
 =
ICOFFA
 and
IROFFA
 = 0 ) with
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.