Contents

# ?heevd

Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian 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 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.
Note that for most cases of complex Hermetian eigenvalue problems the default choice should be heevr function as its underlying algorithm is faster and uses less workspace.
?heevd
requires more workspace but is faster in some cases, especially for large matrices.
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'
,
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 Hermitian matrix
A
, as specified by
uplo
.
lda
The leading dimension of the array
a
. Must be at least max(1,
n
).
Output Parameters
w
Array, size at least max(1,
n
).
If
info
= 0
, contains the eigenvalues of the matrix
A
info
.
a
If
jobz
=
'V'
, then on exit this array is overwritten by the unitary matrix
Z
which contains the eigenvectors of
A
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
i
, and
jobz
= 'N'
, then the algorithm failed to converge;
i
off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
if
info
=
i
, and
jobz
= 'V'
, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns
info
/(
n
+1)
through
mod(
info
,
n
+1)
.
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 syevd. See also hpevd for matrices held in packed storage, and hbevd for banded matrices.

#### 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