Visible to Intel only — GUID: GUID-9AD3B5B7-DC35-4DF7-A126-9A8730FE98CA
Visible to Intel only — GUID: GUID-9AD3B5B7-DC35-4DF7-A126-9A8730FE98CA
?heevd
Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian matrix using divide and conquer algorithm.
Syntax
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)
call heevd(a, w [,job] [,uplo] [,info])
Include Files
- mkl.fi, lapack.f90
Description
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.
Input Parameters
- jobz
-
CHARACTER*1. Must be 'N' or 'V'.
If jobz = 'N', then only eigenvalues are computed.
If jobz = 'V', then eigenvalues and eigenvectors are computed.
- uplo
-
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.
- n
-
INTEGER. The order of the matrix A (n≥ 0).
- a
-
COMPLEX for cheevd
DOUBLE COMPLEX for zheevd
Array, size (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).
- lda
-
INTEGER. 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
-
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.
- rwork
-
REAL for cheevd
DOUBLE PRECISION for zheevd
Workspace array, size at least lrwork.
- 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.
- iwork
-
INTEGER. Workspace array, its dimension max(1, liwork).
- 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.
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.
- info
-
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.
LAPACK 95 Interface Notes
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 LAPACK 95 Interface Conventions.
Specific details for the routine heevd interface are the following:
- a
-
Holds the matrix A of size (n, n).
- w
-
Holds the vector of length (n).
- jobz
-
Must be 'N' or 'V'. The default value is 'N'.
- uplo
-
Must be 'U' or 'L'. The default value is 'U'.
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.
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.