ZHEEVR(1) LAPACK driver routine (version 3.2) ZHEEVR(1)NAME
ZHEEVR - computes selected eigenvalues and, optionally, eigenvectors of
a complex Hermitian matrix A
SYNOPSIS
SUBROUTINE ZHEEVR( 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
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
PURPOSE
ZHEEVR 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.
ZHEEVR first reduces the matrix A to tridiagonal form T with a call to
ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute eigen‐
spectrum using Relatively Robust Representations. ZSTEMR computes ei‐
genvalues by the dqds algorithm, while orthogonal eigenvectors are com‐
puted from various "good" L D L^T representations (also known as Rela‐
tively Robust Representations). Gram-Schmidt orthogonalization is
avoided as far as possible. More specifically, the various steps of the
algorithm are as follows.
For each unreduced block (submatrix) of T,
(a) Compute T - sigma I = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and
d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation com‐
pute
the corresponding eigenvector by forming a rank revealing
twisted
factorization. Go back to (c) for any clusters that remain. The
desired accuracy of the output can be specified by the input parameter
ABSTOL.
For more details, see DSTEMR's documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representa‐
tions
to compute orthogonal eigenvectors of symmetric tridiagonal matri‐
ces,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications,
Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.
Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested on
machines which conform to the ieee-754 floating point standard. ZHEEVR
calls DSTEBZ and ZSTEIN on non-ieee machines and
when partial spectrum requests are made.
Normal execution of ZSTEMR 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.
ARGUMENTS
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*16 array, dimension (LDA, N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular 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 triangle (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) DOUBLE PRECISION
VU (input) DOUBLE PRECISION 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) DOUBLE PRECISION
The absolute error tolerance for the eigenvalues. An approxi‐
mate 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 pre‐
cision. 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 impor‐
tant, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will
guarantee that eigenvalues are computed to high relative accu‐
racy 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. Demmel, "Computing
Accurate Eigensystems of Scaled Diagonally Dominant Matrices",
LAPACK Working Note #7, for a discussion of which matrices
define their eigenvalues 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) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
tain 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) 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 >= max(1,N).
ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices indi‐
cating 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*16 array, dimension (MAX(1,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 opti‐
mal efficiency, LWORK >= (NB+1)*N, where NB is the max of the
blocksize for ZHETRD and for ZUNMTR as returned by ILAENV. If
LWORK = -1, then a workspace query is assumed; the routine only
calculates the optimal sizes 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.
RWORK (workspace/output) DOUBLE PRECISION array, dimension
(MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the optimal (and mini‐
mal) 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 sizes of the
WORK, RWORK and IWORK arrays, returns these values as the first
entries of the WORK, RWORK and IWORK arrays, and no error mes‐
sage related to LWORK or LRWORK or LIWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
mal) 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 sizes of the
WORK, RWORK and IWORK arrays, returns these values as the first
entries of the WORK, RWORK and IWORK arrays, and no error mes‐
sage related to LWORK or LRWORK or 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
FURTHER DETAILS
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
Jason Riedy, Computer Science Division, University of
California at Berkeley, USA
LAPACK driver routine (version 3November 2008 ZHEEVR(1)
