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

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