CHEEVX(l) ) CHEEVX(l)
CHEEVX - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian
SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, LWORK, RWORK, IWORK, IFAIL, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
REAL ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
CHEEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian
matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of
values or a range of indices for the desired eigenvalues.
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input) CHARACTER*1
= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I':
the IL-th through IU-th eigenvalues will be found.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX array, dimension (LDA, N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper trian-
gular part of A contains the upper triangular part of the matrix A. If UPLO =
'L', the leading N-by-N lower triangular part of A contains the lower triangular
part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper tri-
angle (if UPLO='U') of A, including the diagonal, is destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
VL (input) REAL
VU (input) REAL If RANGE='V', the lower and upper bounds of the interval to
be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
IL (input) INTEGER
IU (input) INTEGER If RANGE='I', the indices (in ascending order) of the
smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL
= 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
ABSTOL (input) REAL
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is
accepted as converged when it is determined to lie in an interval [a,b] of width
less than or equal to
ABSTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABSTOL is less than or equal to zero, then
EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal
matrix obtained by reducing A to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is set to twice the
underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0,
indicating that some eigenvectors did not converge, try setting ABSTOL to
See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High
Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.
M (output) INTEGER
The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and
if RANGE = 'I', M = IU-IL+1.
W (output) REAL array, dimension (N)
On normal exit, the first M elements contain the selected eigenvalues in ascending
Z (output) COMPLEX array, dimension (LDZ, max(1,M))
If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal
eigenvectors of the matrix A corresponding to the selected eigenvalues, with the
i-th column of Z holding the eigenvector associated with W(i). If an eigenvector
fails to converge, then that column of Z contains the latest approximation to the
eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ =
'N', then Z is not referenced. Note: the user must ensure that at least max(1,M)
columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not
known in advance and an upper bound must be used.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >=
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,2*N-1). For optimal efficiency,
LWORK >= (NB+1)*N, where NB is the max of the blocksize for CHETRD and for CUNMTR
as returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
RWORK (workspace) REAL array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO
> 0, then IFAIL contains the indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, then i eigenvectors failed to converge. Their indices are
stored in array IFAIL.
LAPACK version 3.0 15 June 2000 CHEEVX(l)