CHEEVR(l) ) CHEEVR(l)
CHEEVR - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian
SUBROUTINE CHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, M, N
REAL ABSTOL, VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian
matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of
values or a range of indices for the desired eigenvalues.
Whenever possible, CHEEVR calls CSTEGR to compute the
eigenspectrum using Relatively Robust Representations. CSTEGR computes eigenvalues by the
dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T
representations (also known as Relatively Robust Representations). Gram-Schmidt orthogo-
nalization is avoided as far as possible. More specifically, the various steps of the
algorithm are as follows. For the i-th unreduced block of T,
(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
is a relatively robust representation,
(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
relative accuracy by the dqds algorithm,
(c) If there is a cluster of close eigenvalues, "choose" sigma_i
close to the cluster, and go to step (a),
(d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the input parameter ABSTOL.
For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenval-
ue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report
No. UCB//CSD-97-971, UC Berkeley, May 1997.
Note 1 : CHEEVR calls CSTEGR when the full spectrum is requested on machines which conform
to the ieee-754 floating point standard. CHEEVR calls SSTEBZ and CSTEIN on non-ieee
when partial spectrum requests are made.
Normal execution of CSTEGR may create NaNs and infinities and hence may abort due to a
floating point exception in environments which do not handle NaNs and infinities in the
ieee standard default manner.
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.
See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High
Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.
If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ).
Doing so will guarantee that eigenvalues are computed to high relative accuracy
when possible in future releases. The current code does not make any guarantees
about high relative accuracy, but furutre releases will. See J. Barlow and J. Dem-
mel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices",
LAPACK Working Note #7, for a discussion of which matrices define their eigenval-
ues to high relative accuracy.
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)
The first M elements contain the selected eigenvalues in ascending order.
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 JOBZ = 'N',
then Z is not referenced. Note: the user must ensure that at least max(1,M) col-
umns 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 >=
ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices indicating the nonzero
elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 )
through ISUPPZ( 2*i ).
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). 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/output) REAL array, dimension (LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK.
The length of the array RWORK. LRWORK >= max(1,24*N).
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the RWORK array, returns this value as the first entry of the
RWORK array, and no error message related to LRWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK.
The dimension of the array IWORK. LIWORK >= max(1,10*N).
If LIWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the IWORK array, returns this value as the first entry of the
IWORK array, and no error message related to LIWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA
LAPACK version 3.0 15 June 2000 CHEEVR(l)