SSTEGR2A(3) ScaLAPACK routine of NEC Numeric Library Collection SSTEGR2A(3)
NAME
SSTEGR2A - computes selected eigenvalues and initial representations
needed for eigenvector computations in SSTEGR2B
SYNOPSIS
SUBROUTINE SSTEGR2A(
JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ,
NZC, WORK, LWORK, IWORK, LIWORK, DOL, DOU, NEEDIL,
NEEDIU, INDERR, NSPLIT, PIVMIN, SCALE, WL, WU, INFO
)
CHARACTER JOBZ, RANGE
INTEGER DOL, DOU, IL, INDERR, INFO, IU, LDZ, LIWORK, LWORK,
M, N, NEEDIL, NEEDIU, NSPLIT, NZC
REAL PIVMIN, SCALE, VL, VU, WL, WU
INTEGER IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * )
REAL Z( LDZ, * )
PURPOSE
SSTEGR2A computes selected eigenvalues and initial representations
needed for eigenvector computations in SSTEGR2B. It is invoked in the
ScaLAPACK MRRR driver PSSYEVR and the corresponding Hermitian version
when both eigenvalues and eigenvectors are computed in parallel on mul-
tiple processors. For this case, SSTEGR2A implements the FIRST part of
the MRRR algorithm, parallel eigenvalue computation and finding the
root RRR. At the end of SSTEGR2A, other processors might have a part of
the spectrum that is needed to continue the computation locally. Once
this eigenvalue information has been received by the processor, the
computation can then proceed by calling the SECOND part of the parallel
MRRR algorithm, SSTEGR2B.
Please note:
1. The calling sequence has two additional INTEGER parameters,
(compared to LAPACK's SSTEGR), these are
DOL and DOU and should satisfy M>=DOU>=DOL>=1.
These parameters are only relevant for the case JOBZ = 'V'.
Globally invoked over all processors, SSTEGR2A computes
ALL the eigenVALUES specified by RANGE.
RANGE= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in (VL,VU] will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.
SSTEGR2A LOCALLY only computes the eigenvalues
corresponding to eigenvalues DOL through DOU in W. (That is,
instead of computing the eigenvectors belonging to W(1)
through W(M), only the eigenvectors belonging to eigenvalues
W(DOL) through W(DOU) are computed. In this case, only the
eigenvalues DOL:DOU are guaranteed to be fully accurate.
2. M is NOT the number of eigenvalues specified by RANGE, but it is
M = DOU - DOL + 1. Instead, M refers to the number of eigenvalues
computed on
this processor.
3. While no eigenvectors are computed in SSTEGR2A itself (this is
done later in SSTEGR2B), the interface
If JOBZ = 'V' then, depending on RANGE and DOL, DOU, SSTEGR2A
might need more workspace in Z then the original SSTEGR.
In particular, the arrays W and Z might not contain all the wanted
eigenpairs
locally, instead this information is distributed over other
processors.
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
input, but is used internally as workspace.
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.
Not referenced if RANGE = 'A' or 'V'.
M (output) INTEGER
Globally summed over all processors, M equals the total number
of eigenvalues found. 0 <= M <= N.
If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
The local output equals M = DOU - DOL + 1.
W (output) REAL array, dimension (N)
The first M elements contain approximations to the selected
eigenvalues in ascending order. Note that immediately after
exiting this routine, only the eigenvalues from position
DOL:DOU are to reliable on this processor because the eigen-
value computation is done in parallel.
The other entries outside DOL:DOU are very crude preliminary
approximations. Other processors hold reliable information on
these other parts of the W array.
This information is communicated in the ScaLAPACK driver.
Z (output) REAL array, dimension (LDZ, max(1,M) )
SSTEGR2A does not compute eigenvectors, this is done in
SSTEGR2B. The argument Z as well as all related other arguments
only appear to keep the interface consistent and to signal to
the user that this subroutine is meant to be used when eigen-
vectors are computed.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
'V', then 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.
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 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
message related to LWORK is issued.
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 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.
DOL (input) INTEGER
DOU (input) INTEGER
From all the eigenvalues W(1:M), only eigenvalues W(DOL:DOU)
are computed.
NEEDIL (output) INTEGER
NEEDIU (output) INTEGER
The indices of the leftmost and rightmost eigenvalues needed to
accurately compute the relevant part of the representation
tree. This information can be used to find out which processors
have the relevant eigenvalue information needed so that it can
be communicated.
INDERR (output) INTEGER
INDERR points to the place in the work space where the eigen-
value uncertainties (errors) are stored.
NSPLIT (output) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
PIVMIN (output) REAL
The minimum pivot in the sturm sequence for T.
SCALE (output) REAL
The scaling factor for the tridiagonal T.
WL (output) REAL
WU (output) REAL
The interval (WL, WU] contains all the wanted eigenvalues.
It is either given by the user or computed in SLARRE2A.
INFO (output) INTEGER
On exit, INFO
= 0: successful exit
other:if INFO = -i, the i-th argument had an illegal value
if INFO = 10X, internal error in SLARRE2A,
Here, the digit X = ABS( IINFO ) < 10, where IINFO is
the nonzero error code returned by SLARRE2A.
ScaLAPACK routine 31 October 2017 SSTEGR2A(3)