CSTEGR(3S)CSTEGR(3S)NAME
CSTEGR - compute selected eigenvalues and, optionally, eigenvectors of a
real symmetric tridiagonal matrix T
SYNOPSIS
SUBROUTINE CSTEGR( 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
REAL ABSTOL, VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * )
COMPLEX Z( LDZ, * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSE
CSTEGR 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
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parameter ABSTOL.
For more details, see "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
Computer Science Division Technical Report No. UCB/CSD-97-971, UC
Berkeley, May 1997.
Note 1 : Currently CSTEGR is only set up to find ALL the n eigenvalues
and eigenvectors of T in O(n^2) time
Note 2 : Currently the routine CSTEIN is called when an appropriate
sigma_i cannot be chosen in step (c) above. CSTEIN invokes modified
Gram-Schmidt when eigenvalues are close.
Note 3 : CSTEGR works only on machines which follow ieee-754 floating-
point standard in their handling of infinities and NaNs. Normal
execution of CSTEGR 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) REAL array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix T. On
exit, D is overwritten.
E (input/output) REAL 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) 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'.
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ABSTOL (input) REAL
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 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) 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 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) REAL 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.
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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 SLARRE, if INFO = 2,
internal error in CLARRV.
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
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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