ZSTEGR - compute selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix T
SYNOPSIS
SUBROUTINE ZSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, RANGE
INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK(
* )
COMPLEX*16 Z( LDZ, * )
PURPOSE
ZSTEGR computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix T.
Eigenvalues and
(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 sym
metric tridiagonal eigenvalue/eigenvector problem", by
Inderjit Dhillon, Computer Science Division Technical
Report No. UCB/CSD-97-971, UC Berkeley, May 1997.
Note 1 : Currently ZSTEGR is only set up to find ALL the n
Note 2 : Currently the routine ZSTEIN is called when an
appropriate sigma_i cannot be chosen in step (c) above.
ZSTEIN invokes modified Gram-Schmidt when eigenvalues are
close.
Note 3 : ZSTEGR works only on machines which follow
ieee-754 floating-point standard in their handling of
infinities and NaNs. Normal execution of ZSTEGR may cre
ate NaNs and infinities and hence may abort due to a
floating point exception in environments which do not con
form to the ieee standard.
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.
N (input) INTEGER
The order of the matrix. N >= 0.
D (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the n diagonal elements of the tridiago
nal matrix T. On exit, D is overwritten.
E (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the (n-1) subdiagonal elements of the
tridiagonal matrix T in elements 1 to N-1 of E;
E(N) need not be set. On exit, E is overwritten.
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 refer
enced if RANGE = 'A' or 'V'.
ABSTOL (input) DOUBLE PRECISION
The absolute error tolerance for the eigenval
ues/eigenvectors. IF JOBZ = 'V', the eigenvalues
bounded by ABSTOL, and the dot products between
different eigenvectors are bounded by ABSTOL. If
ABSTOL is less than N*EPS*|T|, then N*EPS*|T| will
be used in its place, where EPS is the machine
precision and |T| is the 1-norm of the tridiagonal
matrix. The eigenvalues are computed to an accu
racy of EPS*|T| irrespective of ABSTOL. If high
relative accuracy is important, set ABSTOL to
DLAMCH( 'Safe minimum' ). See Barlow and Demmel
"Computing Accurate Eigensystems of Scaled Diago
nally 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 eigen
values 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 contain the orthonormal eigenvectors
of the matrix T 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 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) DOUBLE PRECISION array,
dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
(and minimal) LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
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.
IWORK (workspace/output) INTEGER array, dimension
(LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal
LIWORK.
LIWORK (input) INTEGER
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 ille
gal value
> 0: if INFO = 1, internal error in DLARRE, if
INFO = 2, internal error in ZLARRV.
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
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