SLARRE2A(3) ScaLAPACK routine of NEC Numeric Library Collection SLARRE2A(3)
NAME
SLARRE2A - To find the desired eigenvalues of a given real symmetric
tridiagonal matrix T, SLARRE2 sets any "small" off-diagonal elements to
zero, and for each unreduced block T_i
SYNOPSIS
SUBROUTINE SLARRE2A(
RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2,
SPLTOL, NSPLIT, ISPLIT, M, DOL, DOU, NEEDIL,
NEEDIU, W, WERR, WGAP, IBLOCK, INDEXW, GERS, SDIAM,
PIVMIN, WORK, IWORK, MINRGP, INFO )
CHARACTER RANGE
INTEGER DOL, DOU, IL, INFO, IU, M, N, NSPLIT, NEEDIL,
NEEDIU
REAL MINRGP, PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), INDEXW( * )
REAL D( * ), E( * ), E2( * ), GERS( * ), SDIAM( * ), W(
* ),WERR( * ), WGAP( * ), WORK( * )
PURPOSE
To find the desired eigenvalues of a given real symmetric tridiagonal
matrix T, SLARRE2 sets any "small" off-diagonal elements to zero, and
for each unreduced block T_i, it finds
(a) a suitable shift at one end of the block's spectrum,
(b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
(c) eigenvalues of each L_i D_i L_i^T.
NOTE:
The algorithm obtains a crude picture of all the wanted eigenvalues (as
selected by RANGE). However, to reduce work and improve scalability,
only the eigenvalues DOL to DOU are refined. Furthermore, if the matrix
splits into blocks, RRRs for blocks that do not contain eigenvalues
from DOL to DOU are skipped. The DQDS algorithm (subroutine SLASQ2) is
not used, unlike in the sequential case. Instead, eigenvalues are com-
puted in parallel to some figures using bisection.
ARGUMENTS
RANGE (input) CHARACTER
= 'A': ("All") all eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ("Index") the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
N (input) INTEGER
The order of the matrix. N > 0.
VL (input/output) REAL
VU (input/output) REAL
If RANGE='V', the lower and upper bounds for the eigenvalues.
Eigenvalues less than or equal to VL, or greater than VU, will
not be returned. VL < VU.
If RANGE='I' or ='A', SLARRE2A computes bounds on the desired
part of the spectrum.
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.
D (input/output) REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix T.
On exit, the N diagonal elements of the diagonal matrices D_i.
E (input/output) REAL array, dimension (N)
On entry, the first (N-1) entries contain the subdiagonal ele-
ments of the tridiagonal matrix T; E(N) need not be set.
On exit, E contains the subdiagonal elements of the unit bidi-
agonal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <=
NSPLIT, contain the base points sigma_i on output.
E2 (input/output) REAL array, dimension (N)
On entry, the first (N-1) entries contain the SQUARES of the
subdiagonal elements of the tridiagonal matrix T; E2(N) need
not be set.
On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have
been set to zero
RTOL1 (input) REAL
RTOL2 (input) REAL
Parameters for bisection.
.br
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
SPLTOL (input) REAL
The threshold for splitting.
NSPLIT (output) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
ISPLIT (output) INTEGER array, dimension (N)
The splitting points, at which T breaks up into blocks.
The first block consists of rows/columns 1 to ISPLIT(1), the
second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and
the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
through ISPLIT(NSPLIT)=N.
M (output) INTEGER
The total number of eigenvalues (of all L_i D_i L_i^T) found.
DOL (input) INTEGER
DOU (input) INTEGER
If the user wants to work on only a selected part of the repre-
sentation tree, he can specify an index range DOL:DOU.
Otherwise, the setting DOL=1, DOU=N should be applied.
Note that DOL and DOU refer to the order in which the eigenval-
ues are stored in W.
NEEDIL (output) INTEGER
NEEDIU (output) INTEGER
The indices of the leftmost and rightmost eigenvalues of the
root node RRR which are needed to accurately compute the rele-
vant part of the representation tree.
W (output) REAL array, dimension (N)
The first M elements contain the eigenvalues. The eigenvalues
of each of the blocks, L_i D_i L_i^T, are sorted in ascending
order ( SLARRE2A may use the remaining N-M elements as
workspace).
Note that immediately after exiting this routine, only the
eigenvalues from position DOL:DOU in W are reliable on this
processor because the eigenvalue computation is done in paral-
lel.
WERR (output) REAL array, dimension (N)
The error bound on the corresponding eigenvalue in W.
Note that immediately after exiting this routine, only the
uncertainties from position DOL:DOU in WERR are reliable on
this processor because the eigenvalue computation is done in
parallel.
WGAP (output) REAL array, dimension (N)
The separation from the right neighbor eigenvalue in W.
The gap is only with respect to the eigenvalues of the same
block as each block has its own representation tree.
Exception: at the right end of a block we store the left gap
Note that immediately after exiting this routine, only the gaps
from position DOL:DOU in WGAP are reliable on this processor
because the eigenvalue computation is done in parallel.
IBLOCK (output) INTEGER array, dimension (N)
The indices of the blocks (submatrices) associated with the
corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i)
belongs to the first block from the top, =2 if W(i) belongs to
the second block, etc.
INDEXW (output) INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th
eigenvalue W(i) is the 10-th eigenvalue in block 2
GERS (output) REAL array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval is
(GERS(2*i-1), GERS(2*i)).
PIVMIN (output) REAL
The minimum pivot in the sturm sequence for T.
WORK (workspace) REAL array, dimension (6*N)
Workspace.
IWORK (workspace) INTEGER array, dimension (5*N)
Workspace.
MINRGP (input) REAL
The minimum relativ gap threshold to decide whether an eigen-
value or a cluster boundary is reached.
INFO (output) INTEGER
= 0: successful exit
> 0: A problem occured in SLARRE2A.
< 0: One of the called subroutines signaled an internal
probrem. Needs inspection of the corresponding
parameter INFO for further information.
=-1: Problem in SLARRD2.
= 2: No base representation could be found in MAXTRY itera-
tions.
Increasing MAXTRY and recompilation might be a remedy.
=-3: Problem in SLARRB2 when computing the refined root
representation
=-4: Problem in SLARRB2 when preforming bisection on the
desired part of the spectrum.
= -9 Problem: M < DOU-DOL+1, that is the code found fewer
eigenvalues than it was supposed to
ScaLAPACK routine 31 October 2017 SLARRE2A(3)