DSTEGR(l) ) DSTEGR(l)NAME
DSTEGR - compute selected eigenvalues and, optionally, eigenvectors of
a real symmetric tridiagonal matrix T
SYNOPSIS
SUBROUTINE DSTEGR( 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( * ), Z( LDZ,
* )
PURPOSE
DSTEGR 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 param‐
eter ABSTOL.
For more details, see "A new O(n^2) algorithm for the symmetric tridi‐
agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer
Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May
1997.
Note 1 : Currently DSTEGR is only set up to find ALL the n eigenvalues
and eigenvectors of T in O(n^2) time
Note 2 : Currently the routine DSTEIN is called when an appropriate
sigma_i cannot be chosen in step (c) above. DSTEIN invokes modified
Gram-Schmidt when eigenvalues are close.
Note 3 : DSTEGR works only on machines which follow ieee-754 floating-
point standard in their handling of infinities and NaNs. Normal execu‐
tion of DSTEGR may create NaNs and infinities and hence may abort due
to a floating point exception in environments which do not conform 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 tridiagonal 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
referenced if RANGE = 'A' or 'V'.
ABSTOL (input) DOUBLE PRECISION
The absolute error tolerance for the eigenvalues/eigenvectors.
IF JOBZ = 'V', the eigenvalues and eigenvectors output have
residual norms 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 accuracy
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 discus‐
sion of which matrices define their eigenvalues to high rela‐
tive 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) DOUBLE PRECISION 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 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 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) 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 >= max(1,18*N)
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 illegal value
> 0: if INFO = 1, internal error in DLARRE, if INFO = 2,
internal error in DLARRV.
FURTHER DETAILS
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
LAPACK computational version 3.0 15 June 2000DSTEGR(l)
