sstevr(3P) Sun Performance Library sstevr(3P)NAMEsstevr - compute selected eigenvalues and, optionally, eigenvectors of
a real symmetric tridiagonal matrix T
SYNOPSIS
SUBROUTINE SSTEVR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER ISUPPZ(*), IWORK(*)
REAL VL, VU, ABSTOL
REAL D(*), E(*), W(*), Z(LDZ,*), WORK(*)
SUBROUTINE SSTEVR_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
INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER*8 ISUPPZ(*), IWORK(*)
REAL VL, VU, ABSTOL
REAL D(*), E(*), W(*), Z(LDZ,*), WORK(*)
F95 INTERFACE
SUBROUTINE STEVR(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
INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: D, E, W, WORK
REAL, DIMENSION(:,:) :: Z
SUBROUTINE STEVR_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
INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: D, E, W, WORK
REAL, DIMENSION(:,:) :: Z
C INTERFACE
#include <sunperf.h>
void sstevr(char jobz, char range, int n, float *d, float *e, float vl,
float vu, int il, int iu, float abstol, int *m, float *w,
float *z, int ldz, int *isuppz, int *info);
void sstevr_64(char jobz, char range, long n, float *d, float *e, float
vl, float vu, long il, long iu, float abstol, long *m, float
*w, float *z, long ldz, long *isuppz, long *info);
PURPOSEsstevr 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.
Whenever possible, SSTEVR 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 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
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 : SSTEVR calls SSTEGR when the full spectrum is requested on
machines which conform to the ieee-754 floating point standard. SSTEVR
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)
= '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
A. On exit, D may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.
E (input/output)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A in elements 1 to N-1 of E; E(N) need not be set. On
exit, E may be multiplied by a constant factor chosen to
avoid over/underflow in computing the eigenvalues.
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)
See the description of VL.
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)
See the description of IL.
ABSTOL (input)
The absolute error tolerance for the eigenvalues. An approx‐
imate eigenvalue is accepted as converged when it is deter‐
mined 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 guarantee that eigenvalues
are computed to high relative accuracy when possible in
future releases. The current code does not make any guaran‐
tees about high relative accuracy, but future releases will.
See J. Barlow and J. Demmel, "Computing Accurate Eigensystems
of Scaled Diagonally Dominant Matrices", LAPACK Working Note
#7, for a discussion of which matrices define their eigenval‐
ues 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 A corre‐
sponding to the selected eigenvalues, with the i-th column of
Z holding the eigenvector associated with W(i). 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) 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)
On exit, if INFO = 0, WORK(1) returns the optimal (and mini‐
mal) LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= 20*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 (and mini‐
mal) LIWORK.
LIWORK (input)
The dimension of the array IWORK. LIWORK >= 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: Internal error
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 sstevr(3P)
