DLARRE2(3) ScaLAPACK routine of NEC Numeric Library Collection DLARRE2(3)
NAME
DLARRE2 - To find the desired eigenvalues of a given real symmetric
tridiagonal matrix T, DLARRE2 sets, via DLARRA, "small" off-diagonal
elements to zero
SYNOPSIS
SUBROUTINE DLARRE2( RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2,
SPLTOL, NSPLIT, ISPLIT, M, DOL, DOU, W, WERR, WGAP,
IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO )
CHARACTER RANGE
INTEGER DOL, DOU, IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), INDEXW( * )
DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ), W( *
),WERR( * ), WGAP( * ), WORK( * )
PURPOSE
To find the desired eigenvalues of a given real symmetric tridiagonal
matrix T, DLARRE2 sets, via DLARRA, "small" off-diagonal elements to
zero. For each block T_i, it finds
(a) a suitable shift at one end of the block's spectrum,
(b) the root RRR, T_i - sigma_i I = L_i D_i L_i^T, and
(c) eigenvalues of each L_i D_i L_i^T.
The representations and eigenvalues found are then returned to DSTEGR2
to compute the eigenvectors T.
DLARRE2 is more suitable for parallel computation than the original
LAPACK code for computing the root RRR and its eigenvalues.
When computing eigenvalues in parallel and the input tridiagonal matrix
splits into blocks, DLARRE2 can skip over blocks which contain none of
the eigenvalues from DOL to DOU for which the processor responsible. In
extreme cases (such as large matrices consisting of many blocks of
small size, e.g. 2x2, the gain can be substantial.
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) DOUBLE PRECISION
VU (input/output) DOUBLE PRECISION
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', DLARRE2 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
RTOL2 (input) DOUBLE PRECISION
Parameters for bisection.
An interval [LEFT,RIGHT] has converged if RIGHT-LEFT.LT.MAX(
RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
SPLTOL (input) DOUBLE PRECISION
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.
W (output) DOUBLE PRECISION 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 ( DLARRE2 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 might be reliable on
this processor when the eigenvalue computation is done in par-
allel.
WERR (output) DOUBLE PRECISION 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 might be reliable
on this processor when the eigenvalue computation is done in
parallel.
WGAP (output) DOUBLE PRECISION 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 might be reliable on this proces-
sor when 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) DOUBLE PRECISION array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval is
(GERS(2*i-1), GERS(2*i)).
PIVMIN (output) DOUBLE PRECISION
The minimum pivot in the sturm sequence for T.
WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
Workspace.
IWORK (workspace) INTEGER array, dimension (5*N)
Workspace.
INFO (output) INTEGER
= 0: successful exit
> 0: A problem occured in DLARRE2.
< 0: One of the called subroutines signaled an internal
probrem. Needs inspection of the corresponding
parameter INFO for further information.
=-1: Problem in DLARRD.
= 2: No base representation could be found in MAXTRY itera-
tions.
Increasing MAXTRY and recompilation might be a remedy.
=-3: Problem in DLARRB when computing the refined root
representation for DLASQ2.
=-4: Problem in DLARRB when preforming bisection on the
desired part of the spectrum.
=-5: Problem in DLASQ2.
=-6: Problem in DLASQ2.
ScaLAPACK routine 31 October 2017 DLARRE2(3)