Developer Reference for Intel® oneAPI Math Kernel Library for C
?heevd
Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian matrix using divide and conquer algorithm.
Syntax
lapack_int LAPACKE_cheevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , lapack_complex_float*a , lapack_intlda , float*w );
lapack_int LAPACKE_zheevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , lapack_complex_double*a , lapack_intlda , double*w );
Include Files
mkl.h
Description
cheevd zheevd heevd
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 zi . 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
-
COMPLEX for cheevd DOUBLE COMPLEX for zheevd Array, size ( lda , *).
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 .
The second dimension of a must be at least max(1, n ).
lda
The leading dimension of the array a . Must be at least max(1, n ).
- work
-
COMPLEX for cheevd DOUBLE COMPLEX for zheevd . Workspace array, size max(1, lwork) .
lwork
The dimension of the array work . Constraints: if n≤ 1 , then lwork≥ 1 ; if jobz = 'N' and n > 1, then lwork≥n+1 ; if jobz = 'V' and n > 1, then lwork≥n^{2}+2*n . If lwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work , rwork and iwork arrays, returns these values as the first entries of the work , rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla . See Application Notes for details.
- rwork
-
REAL for cheevd DOUBLE PRECISION for zheevd Workspace array, size at least lrwork .
lrwork
The dimension of the array rwork . Constraints: if n≤ 1 , then lrwork≥ 1 ; if job = 'N' and n > 1, then lrwork≥n ; if job = 'V' and n > 1 , then lrwork≥ 2*n^{2}+ 5*n + 1 . If lrwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work , rwork and iwork arrays, returns these values as the first entries of the work , rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla . See Application Notes for details.
iwork
Workspace array, its dimension max(1, liwork) .
liwork
The dimension of the array iwork . Constraints: if n ≤ 1, then liwork≥ 1 ; if jobz = 'N' and n > 1, then liwork≥ 1 ; if jobz = 'V' and n > 1, then liwork≥ 5*n+3 . If liwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work , rwork and iwork arrays, returns these values as the first entries of the work , rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla . See Application Notes for details.
Output Parameters
- w
-
REAL for cheevd DOUBLE PRECISION for zheevd Array, size at least max(1, n ). If info = 0 , contains the eigenvalues of the matrix A in ascending order. See also info .
- a
-
If jobz = 'V' , then on exit this array is overwritten by the unitary matrix Z which contains the eigenvectors of A .
- work(1)
-
On exit, if lwork > 0 , then the real part of work(1) returns the required minimal size of lwork .
- rwork(1)
-
On exit, if lrwork > 0 , then rwork(1) returns the required minimal size of lrwork .
- iwork(1)
-
On exit, if liwork > 0 , then iwork(1) returns the required minimal size of liwork .
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.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
The computed eigenvalues and eigenvectors are exact for a matrix \(A + E\) such that \(||E||_{2} = O(\varepsilon) ||A||_{2}\) , where \(\varepsilon\) 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.