Contents

# ?hbtrd

Reduces a complex Hermitian band matrix to tridiagonal form.

## Syntax

Include Files
• mkl.h
Description
The routine reduces a complex Hermitian band matrix
A
to symmetric tridiagonal form
T
by a unitary similarity transformation:
A
=
Q*T*Q
H
. The unitary matrix
Q
is determined as a product of Givens rotations.
If required, the routine can also form the matrix
Q
explicitly.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
vect
Must be
'V'
,
'N'
, or
'U'
.
If
vect
=
'V'
, the routine returns the explicit matrix
Q
.
If
vect
=
'N'
, the routine does not return
Q
.
If
vect
=
'U'
X
by forming
Q
*
X
.
uplo
Must be
'U'
or
'L'
.
If
uplo
=
'U'
,
ab
stores the upper triangular part of
A
.
If
uplo
=
'L'
,
ab
stores the lower triangular part of
A
.
n
The order of the matrix
A
(
n
0
).
kd
The number of super- or sub-diagonals in
A
(
kd
0
).
ab
ab
(size at least max(1,
ldab
*
n
) for column major layout and at least max(1,
ldab
*(
kd
+ 1)) for row major layout)
is an array containing either upper or lower triangular part of the matrix
A
(as specified by
uplo
) in band storage format.
q
q
(size max(1,
ldq
*
n
))
is an array.
If
vect
=
'U'
, the
q
array must contain an
n
-by-
n
matrix
X
.
If
vect
=
'N'
or
'V'
, the
q
parameter need not be set.'
ldab
ab
; at least
kd
+1
for column major layout and
n
for row major layout
.
ldq
q
. Constraints:
ldq
max(1,
n
)
if
vect
=
'V'
or
'U'
;
ldq
1
if
vect
=
'N'
.
Output Parameters
ab
On exit, the diagonal elements of the array
ab
are overwritten by the diagonal elements of the tridiagonal matrix
T
. If
kd
> 0, the elements on the first superdiagonal (if
uplo
=
'U'
) or the first subdiagonal (if
uplo
=
'L'
) are ovewritten by the off-diagonal elements of
T
. The rest of
ab
is overwritten by values generated during the reduction.
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).
q
If
vect
=
'N'
,
q
is not referenced.
If
vect
=
'V'
,
q
contains the
n
-by-
n
matrix
Q
.
If
vect
=
'U'
,
q
contains the product
X
*
Q
.
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 computed matrix
Q
differs from an exactly unitary matrix by a matrix
E
such that
||
E
||
2
=
O
(
ε
)
.
The total number of floating-point operations is approximately
20
n
2
*
kd
if
vect
=
'N'
, with
10
n
3
*(
kd
-1)/
kd