?syevd
?syevd
Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric matrix using divide and conquer algorithm.
Syntax
lapack_int
LAPACKE_ssyevd
(
int
matrix_layout
,
char
jobz
,
char
uplo
,
lapack_int
n
,
float
*
a
,
lapack_int
lda
,
float
*
w
);
lapack_int
LAPACKE_dsyevd
(
int
matrix_layout
,
char
jobz
,
char
uplo
,
lapack_int
n
,
double
*
a
,
lapack_int
lda
,
double
*
w
);
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 , and . Thus,
Λ
is a diagonal matrix whose diagonal elements are the eigenvalues λ
i
Z
is the orthogonal matrix whose columns are the eigenvectors z
i
A
*z
i
λ
i
z
i
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, then only eigenvalues are computed.jobz='N'If, then eigenvalues and eigenvectors are computed.jobz='V'
- uplo
- Must be'U'or'L'.If,uplo='U'astores the upper triangular part ofA.If,uplo='L'astores the lower triangular part ofA.
- n
- The order of the matrixA().n≥0
- a
- Array, size (lda, *).a(size max(1,is an array containing either upper or lower triangular part of the symmetric matrixlda*n))A, as specified byuplo.
- lda
- The leading dimension of the arraya.Must be at least max(1,n).
Output Parameters
- w
- Array, size at least max(1,n).If, contains the eigenvalues of the matrixinfo= 0Ain ascending order. See alsoinfo.
- a
- If, then on exit this array is overwritten by the orthogonal matrixjobz='V'Zwhich contains the eigenvectors ofA.
Return Values
This function returns a value
info
.If , the execution is successful.
info
=0If , and ,
then the algorithm failed to converge;
info
= i
jobz
= 'N'
i
indicates the number of off-diagonal elements of an intermediate tridiagonal form which did not converge to zero. If , and , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns through
info
= i
jobz
= 'V'
info
/(n
+1)mod(
.info
,n
+1)If , the
info
= -i
i
-th parameter had an illegal value.Application Notes
The computed eigenvalues and eigenvectors are exact for a matrix such that
A
+E
||
, where E
||2
= O
(ε
)*||A
||2
ε
is the machine precision.The complex analogue of this routine is heevd