Computes all eigenvalues and (optionally) all eigenvectors of a complex Hermitian matrix using divide and conquer algorithm.
FORTRAN 77:
call cheevd(jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info)
call zheevd(jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info)
Fortran 95:
call heevd(a, w [,job] [,uplo] [,info])
C:
lapack_int LAPACKE_cheevd( int matrix_order, char jobz, char uplo, lapack_int n, lapack_complex_float* a, lapack_int lda, float* w );
lapack_int LAPACKE_zheevd( int matrix_order, char jobz, char uplo, lapack_int n, lapack_complex_double* a, lapack_int lda, double* w );
The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.
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*ZH.
Here Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex) unitary matrix whose columns are the eigenvectors zi. Thus,
A*zi = λi*zi 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.
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
CHARACTER*1. Must be 'N' or 'V'.
If jobz = 'N', then only eigenvalues are computed.
If jobz = 'V', then eigenvalues and eigenvectors are computed.
CHARACTER*1. 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.
INTEGER. The order of the matrix A (n ≥ 0).
COMPLEX for cheevd
DOUBLE COMPLEX for zheevd
Array, DIMENSION (lda, *).
a(lda,*) 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).
INTEGER. The leading dimension of the array a. Must be at least max(1, n).
COMPLEX for cheevd
DOUBLE COMPLEX for zheevd.
Workspace array, DIMENSION max(1, lwork).
INTEGER.
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 ≥ n2+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.
REAL for cheevd
DOUBLE PRECISION for zheevd
Workspace array, DIMENSION at least lrwork.
INTEGER.
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*n2+ 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.
INTEGER. Workspace array, its dimension max(1, liwork).
INTEGER.
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.
REAL for cheevd
DOUBLE PRECISION for zheevd
Array, DIMENSION at least max(1, n).
If info = 0, contains the eigenvalues of the matrix A in ascending order. See also info.
If jobz = 'V', then on exit this array is overwritten by the unitary matrix Z which contains the eigenvectors of A.
On exit, if lwork > 0, then the real part of work(1) returns the required minimal size of lwork.
On exit, if lrwork > 0, then rwork(1) returns the required minimal size of lrwork.
On exit, if liwork > 0, then iwork(1) returns the required minimal size of liwork.
INTEGER.
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.
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.
Specific details for the routine heevd interface are the following:
Holds the matrix A of size (n, n).
Holds the vector of length (n).
Must be 'N' or 'V'. The default value is 'N'.
Must be 'U' or 'L'. The default value is 'U'.
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.
If you are in doubt how much workspace to supply, use a generous value of lwork (liwork or lrwork) for the first run or set lwork = -1 (liwork = -1, lrwork = -1).
If you choose the first option and set any of admissible lwork (liwork or lrwork) sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork, rwork) on exit. Use this value (work(1), iwork(1), rwork(1)) for subsequent runs.
If you set lwork = -1 (liwork = -1, lrwork = -1), the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork, rwork). This operation is called a workspace query.
Note that if you set lwork (liwork, lrwork) to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.
The real analogue of this routine is ?syevd. See also ?hpevd for matrices held in packed storage, and ?hbevd for banded matrices.
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