SSYEVR - compute selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix T
SYNOPSIS
SUBROUTINE SSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK,
M, N
REAL ABSTOL, VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ,
* )
PURPOSE
SSYEVR computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix T. Eigenvalues and
eigenvectors can be selected by specifying either a range
of values or a range of indices for the desired eigenval
ues.
Whenever possible, SSYEVR calls SSTEGR to compute the
eigenspectrum using Relatively Robust Representations.
SSTEGR 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 orthogonalization is
avoided as far as possible. More specifically, the various
steps of the algorithm are as follows. For the i-th unre
duced 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
Inderjit Dhillon, Computer Science Division Technical
Report No. UCB//CSD-97-971, UC Berkeley, May 1997.
Note 1 : SSYEVR calls SSTEGR when the full spectrum is
requested on machines which conform to the ieee-754 float
ing point standard. SSYEVR calls SSTEBZ and SSTEIN on
non-ieee machines and
when partial spectrum requests are made.
Normal execution of SSTEGR 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) REAL array, dimension (LDA, N)
On entry, the symmetric matrix A. If UPLO = 'U',
the leading N-by-N upper triangular part of A con
tains 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) 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'.
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) 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 guar
antee that eigenvalues are computed to high rela
tive 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. Demmel, "Computing
Accurate Eigensystems of Scaled Diagonally Domi
nant Matrices", LAPACK Working Note #7, for a dis
cussion 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) REAL array, dimension (N)
The first M elements contain the selected eigen
values in ascending order.
Z (output) REAL 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) columns are supplied
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) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
max(1,26*N). For optimal efficiency, LWORK >=
(NB+6)*N, where NB is the max of the blocksize for
SSYTRD and SORMTR 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.
IWORK (workspace/output) INTEGER array, dimension
(LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal
LWORK.
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: Internal error
FURTHER DETAILS
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA
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