Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?spevd

Uses divide and conquer algorithm to compute all eigenvalues and (optionally) all eigenvectors of a real symmetric matrix held in packed storage.

Syntax

lapack_int
LAPACKE_sspevd
(
int
matrix_layout
,
char
jobz
,
char
uplo
,
lapack_int
n
,
float
*
ap
,
float
*
w
,
float
*
z
,
lapack_int
ldz
);
lapack_int
LAPACKE_dspevd
(
int
matrix_layout
,
char
jobz
,
char
uplo
,
lapack_int
n
,
double
*
ap
,
double
*
w
,
double
*
z
,
lapack_int
ldz
);
Include Files
  • mkl.h
Description
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix
A
 (held in packed storage). 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.
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'
,
ap
 stores the packed upper triangular part of
A
.
If
uplo
 =
'L'
,
ap
 stores the packed lower triangular part of
A
.
n
The order of the matrix
A
 (
n
≥
 0
).
ap
ap
 contains the packed upper or lower triangle of symmetric matrix
A
, as specified by
uplo
.
The dimension of
ap
 must be max(1,
n
*(
n
+1)/2)
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
,
z
Arrays:
w
, size at least max(1,
n
).
If
info
 = 0
, contains the eigenvalues of the matrix
A
 in ascending order. See also
info
.
z
(size max(1,
ldz
*
n
))
.
If
jobz
 =
'V'
, then this array is overwritten by the orthogonal matrix
Z
 which contains the eigenvectors of
A
. If
jobz
 =
'N'
, then
z
 is not referenced.
ap
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
 =
i
, then the algorithm failed to converge;
i
 indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.
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 hpevd.
See also syevd for matrices held in full storage, and sbevd for banded matrices.

Product and Performance Information

1

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