zstemr(3P) Sun Performance Library zstemr(3P)NAMEzstemr - computes selected eigenvalues and, optionally, eigenvectors of
a real symmetric tridiagonal matrix T.
SYNOPSIS
SUBROUTINE ZSTEMR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W,
Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
LOGICAL TRYRAC
INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
DOUBLE PRECISION VL, VU
INTEGER ISUPPZ(*), IWORK(*)
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)
COMPLEX*16 Z(LDZ, *)
SUBROUTINE ZSTEMR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W,
Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
LOGICAL TRYRAC
INTEGER*8 IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
INTEGER*8 ISUPPZ(*), IWORK(*)
DOUBLE PRECISION VL, VU
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)
COMPLEX*16 Z(LDZ, *)
F95 INTERFACE
SUBROUTINE STEMR(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, M, W,
Z, [LDZ], NZC, ISUPPZ, TRYRAC, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE
LOGICAL TRYRAC
INTEGER :: IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
DOUBLE PRECISION :: VL, VU
DOUBLE PRECISION, DIMENSION(:) :: D, E, W, WORK
COMPLEX*16, DIMENSION(:,:) :: Z
SUBROUTINE STEMR_64(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, M, W,
Z, [LDZ], NZC, ISUPPZ, TRYRAC, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE
LOGICAL TRYRAC
INTEGER(8) :: IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
DOUBLE PRECISION :: VL, VU
DOUBLE PRECISION, DIMENSION(:) :: D, E, W, WORK
COMPLEX*16, DIMENSION(:,:) :: Z
C INTERFACE
#include <sunperf.h>
void zstemr(char jobz, char range, int n, double *d, double *e, double
vl, double vu, int il, int iu, int *m, double *w, doublecom‐
plex *z, int ldz, int nzc, int *isuppz, int *tryrac, int
*info);
void zstemr_64(char jobz, char range, long n, double *d, double *e,
double vl, double vu, long il, long iu, long *m, double *w,
doublecomplex *z, long ldz, long nzc, long *isuppz, long
*tryrac, long *info);
PURPOSE
ZSTEMR computes selected eigenvalues and, optionally, eigenvectors of a
real symmetric tridiagonal matrix T. Any such unreduced matrix has a
well defined set of pairwise different real eigenvalues, the corre‐
sponding real eigenvectors are pairwise orthogonal.
The spectrum may be computed either completely or partially by specify‐
ing either an interval (VL,VU] or a range of indices IL:IU for the
desired eigenvalues.
Depending on the number of desired eigenvalues, these are computed
either by bisection or the dqds algorithm. Numerically orthogonal
eigenvectors are computed by the use of various suitable L D L^T fac‐
torizations near clusters of close eigenvalues (referred to as RRRs,
Relatively Robust Representations). An informal sketch of the algorithm
follows.
For each unreduced block (submatrix) of T,
(a) Compute T - sigma I = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and
d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation com‐
pute
the corresponding eigenvector by forming a rank revealing
twisted
factorization. Go back to (c) for any clusters that remain.
For more details, see:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representa‐
tions
to compute orthogonal eigenvectors of symmetric tridiagonal matri‐
ces,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications,
Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.
Notes:
1. ZSTEMR works only on machines which follow IEEE-754 floating-point
standard in their handling of infinities and NaNs. This permits the
use of efficient inner loops avoiding a check for zero divisors.
2. LAPACK routines can be used to reduce a complex Hermitian matrix to
real symmetric tridiagonal form. (Any complex Hermitian tridiagonal
matrix has real values on its diagonal and potentially complex numbers
on its off-diagonals. By applying a similarity transform with an appro‐
priate diagonal matrix, the complex Hermitian matrix can be transformed
into a real symmetric matrix and complex arithmetic can be entirely
avoided.) While the eigenvectors of the real symmetric tridiagonal
matrix are real, the eigenvectors of original complex Hermitean matrix
have complex entries in general. Since LAPACK drivers overwrite the
matrix data with the eigenvectors, ZSTEMR accepts complex workspace to
facilitate interoperability with CUNMTR or CUPMTR.
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) INTEGER
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) INTEGER
See the description of VL.
IL (input) INTEGER
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) INTEGER
See the description of IL.
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) 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 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 can be computed with a workspace query by setting NZC =
-1, see below.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ
= 'V', LDZ >= max(1,N).
NZC (input) INTEGER The number of eigenvectors to be held
in the array Z.
If RANGE = 'A', then NZC >= max(1,N).
If RANGE = 'V', then NZC >= the number of eigenvalues in
(VL,VU].
If RANGE = 'I', then NZC >= IU-IL+1.
If NZC = -1, then a workspace query is assumed; the routine
calculates the number of columns of the array Z that are
needed to hold the eigenvectors. This value is returned as
the first entry of the Z array, and no error message related
to NZC is issued by XERBLA.
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
). This is relevant in the case when the matrix is split.
ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
TRYRAC (input/output) LOGICAL
If TRYRAC.EQ..TRUE., indicates that the code should check
whether the tridiagonal matrix defines its eigenvalues to
high relative accuracy. If so, the code uses relative-accu‐
racy preserving algorithms that might be (a bit) slower
depending on the matrix. If the matrix does not define its
eigenvalues to high relative accuracy, the code can uses pos‐
sibly faster algorithms. If TRYRAC.EQ..FALSE., the code is
not required to guarantee relatively accurate eigenvalues and
can use the fastest possible techniques. On exit, a .TRUE.
TRYRAC will be set to .FALSE. if the matrix does not define
its eigenvalues to high relative accuracy.
WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
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
JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = '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 mes‐
sage 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
the eigenvectors are desired, and LIWORK >= max(1,8*N) if
only the eigenvalues are to be computed. If LIWORK = -1,
then a workspace query is assumed; the routine only calcu‐
lates 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 SLARRE, if INFO = 2X,
internal error in CLARRV. Here, the digit X = ABS( IINFO ) <
10, where IINFO is the nonzero error code returned by SLARRE
or CLARRV, respectively.
FURTHER DETAILS
Based on contributions by
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
6 Mar 2009 zstemr(3P)
