zstegr(3P) Sun Performance Library zstegr(3P)NAMEzstegr - 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
SYNOPSIS
SUBROUTINE ZSTEGR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
DOUBLE COMPLEX Z(LDZ,*)
INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER ISUPPZ(*), IWORK(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)
SUBROUTINE ZSTEGR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
DOUBLE COMPLEX Z(LDZ,*)
INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER*8 ISUPPZ(*), IWORK(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)
F95 INTERFACE
SUBROUTINE STEGR(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, ABSTOL, M,
W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE
COMPLEX(8), DIMENSION(:,:) :: Z
INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: D, E, W, WORK
SUBROUTINE STEGR_64(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, ABSTOL,
M, W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE
COMPLEX(8), DIMENSION(:,:) :: Z
INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: D, E, W, WORK
C INTERFACE
#include <sunperf.h>
void zstegr(char jobz, char range, int n, double *d, double *e, double
vl, double vu, int il, int iu, double abstol, int *m, double
*w, doublecomplex *z, int ldz, int *isuppz, int *info);
void zstegr_64(char jobz, char range, long n, double *d, double *e,
double vl, double vu, long il, long iu, double abstol, long
*m, double *w, doublecomplex *z, long ldz, long *isuppz, long
*info);
PURPOSE
ZSTEGR computes selected eigenvalues and, optionally, eigenvectors of a
real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can
be selected by specifying either a range of values or a range of
indices for the desired eigenvalues. The eigenvalues are computed 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 unreduced 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 rel‐
ative 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, com‐
pute 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 ZSTEGR is only set up to find ALL the n eigenvalues
and eigenvectors of T in O(n^2) time
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 execu‐
tion of ZSTEGR 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)
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input)
= '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) The order of the matrix. N >= 0.
D (input/output)
On entry, the n diagonal elements of the tridiagonal matrix
T. On exit, D is overwritten.
E (input/output)
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)
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'.
VU (input)
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)
If RANGE='I', the indices (in ascending order) of the small‐
est 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'.
IU (input)
If RANGE='I', the indices (in ascending order) of the small‐
est 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)
The absolute error tolerance for the eigenvalues/eigenvec‐
tors. 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 com‐
puted 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)
The total number of eigenvalues found. 0 <= M <= N. If
RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output)
If JOBZ = 'V', then if INFO = 0, the first M columns of Z
contain the orthonormal eigenvectors of the matrix T corre‐
sponding 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)
The leading dimension of the array Z. LDZ >= 1, and if JOBZ
= 'V', LDZ >= max(1,N).
ISUPPZ (output)
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)
On exit, if INFO = 0, WORK(1) returns the optimal (and mini‐
mal) LWORK.
LWORK (input)
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)
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input)
The dimension of the array IWORK. LIWORK >= max(1,10*N)
If LIWORK = -1, then a workspace query is assumed; the rou‐
tine 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)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = 1, internal error in ZLARRE, 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
6 Mar 2009 zstegr(3P)
