Contents

# ?hbevd

Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian band matrix using divide and conquer algorithm.

## Syntax

Include Files
• mkl.h
Description
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian band matrix
A
. In other words, it can compute the spectral factorization of
A
as:
A
=
Z*
Λ*
Z
H
.
Here
Λ
is a real diagonal matrix whose diagonal elements are the eigenvalues
λ
i
, and
Z
is the (complex) unitary matrix whose columns are the eigenvectors
z
i
. Thus,
A*z
i
=
λ
i
*z
i
for
i
= 1, 2, ...,
n
.
If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the
QL
or
QR
algorithm.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobz
Must be
'N'
or
'V'
.
If
jobz
=
'N'
, then only eigenvalues are computed.
If
jobz
=
'V'
, then eigenvalues and eigenvectors are computed.
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 Hermitian matrix
A
(as specified by
uplo
) in band storage format.
ldab
ab
; must be at least
kd
+1
for column major layout and
n
for row major layout
.
ldz
The leading dimension of the output array
z
.
Constraints:
if
jobz
=
'N'
, then
ldz
1
;
if
jobz
=
'V'
, then
ldz
max(1,
n
) .
Output Parameters
w
Array, size at least max(1,
n
).
If
info
= 0
, contains the eigenvalues of the matrix
A
info
.
z
Array, size
max(1,
ldz
*
n
if
job
=
'V'
and at least 1 if
job
=
'N'
.
If
jobz
=
'V'
, then this array is overwritten by the unitary matrix
Z
which contains the eigenvectors of
A
. The
i
-th column of
Z
contains the eigenvector which corresponds to the eigenvalue
w
[
i
- 1]
.
If
jobz
=
'N'
, then
z
is not referenced.
ab
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form.
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 eigenvalues and eigenvectors are exact for a matrix
A
+
E
such that
||
E
||
2
=
O
(
ε
)||
A
||
2
, where
ε
is the machine precision.
The real analogue of this routine is sbevd.
See also heevd for matrices held in full storage, and hpevd for matrices held in packed storage.

#### 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