Contents

# ?syevd

Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric 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 real symmetric matrix
A
. In other words, it can compute the spectral factorization of
A
as:
A
=
Z
*
λ
*
Z
T
.
Here
Λ
is a diagonal matrix whose diagonal elements are the eigenvalues
λ
i
, and
Z
is the orthogonal 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 real symmetric eigenvalue problems the default choice should be syevr function as its underlying algorithm is faster and uses less workspace.
?syevd
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
Array, size (
lda
, *).
a
(size max(1,
lda
*
n
))
is an array containing either upper or lower triangular part of the symmetric 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 orthogonal 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
indicates the number of off-diagonal elements of an intermediate tridiagonal form which 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 complex analogue of this routine is heevd

#### Product and Performance Information

1

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