Contents

# ?hetrd

Reduces a complex Hermitian matrix to tridiagonal form.

## Syntax

Include Files
• mkl.h
Description
The routine reduces a complex Hermitian matrix
A
to symmetric tridiagonal form
T
by a unitary similarity transformation:
A
=
Q*T*Q
H
. The unitary matrix
Q
is not formed explicitly but is represented as a product of
n
-1 elementary reflectors. Routines are provided to work with
Q
in this representation.
(They are described later in this
topic
.)
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Must be
'U'
or
'L'
.
If
uplo
=
'U'
,
a
stores the upper triangular part of
A
.
If
uplo
=
'L'
,
a
stores the lower triangular part of
A
.
n
The order of the matrix
A
(
n
0
).
a
a
(size max(1,
lda
*
n
))
is an array containing either upper or lower triangular part of the matrix
A
, as specified by
uplo
. If
uplo
=
'U'
n
-by-
n
upper triangular part of
a
contains the upper triangular part of the matrix
A
, and the strictly lower triangular part of
A
is not referenced. If
uplo
=
'L'
n
-by-
n
lower triangular part of
a
contains the lower triangular part of the matrix
A
, and the strictly upper triangular part of
A
is not referenced.
lda
a
; at least max(1,
n
).
Output Parameters
a
On exit,
if
uplo
=
'U'
, the diagonal and first superdiagonal of
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 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 matrix
Q
as a product of elementary reflectors.
d
,
e
Arrays:
d
contains the diagonal elements of the matrix
T
.
The dimension of
d
must be at least max(1,
n
).
e
contains the off-diagonal elements of
T
.
The dimension of
e
must be at least max(1,
n
-1).
tau
Array, size at least max(1,
n
-1). Stores (
n
-1) scalars that define elementary reflectors in decomposition of the unitary matrix
Q
in a product of
n
-1 elementary reflectors.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
Application Notes
The computed matrix
T
is exactly similar to a matrix
A
+
E
, where
||
E
||
2
=
c
(
n
)*
ε
*||
A
||
2
,
c
(
n
)
is a modestly increasing function of
n
, and
ε
is the machine precision.
The approximate number of floating-point operations is
(16/3)
n
3
.
After calling this routine, you can call the following:
to form the computed matrix
Q
explicitly
to multiply a complex matrix by
Q
.
The real counterpart of this routine is ?sytrd.

#### 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