DLARRD2(3) ScaLAPACK routine of NEC Numeric Library Collection DLARRD2(3)
NAME
DLARRD2 - computes the eigenvalues of a symmetric tridiagonal matrix T
to limited initial accuracy
SYNOPSIS
SUBROUTINE DLARRD2( RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D,
E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU,
IBLOCK, INDEXW, WORK, IWORK, DOL, DOU, INFO )
CHARACTER ORDER, RANGE
INTEGER DOL, DOU, IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION PIVMIN, RELTOL, VL, VU, WL, WU
INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ), W( *
), WERR( * ), WORK( * )
PURPOSE
DLARRD2 computes the eigenvalues of a symmetric tridiagonal matrix T to
limited initial accuracy. This is an auxiliary code to be called from
DLARRE2A.
DLARRD2 has been created using the LAPACK code DLARRD which itself
stems from DSTEBZ. The motivation for creating DLARRD2 is efficiency:
When computing eigenvalues in parallel and the input tridiagonal matrix
splits into blocks, DLARRD2 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.
ORDER (input) CHARACTER
= 'B': ("By Block") the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= 'E': ("Entire matrix")
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
N (input) INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION
If RANGE='V', the lower and upper bounds of the interval to be
searched for eigenvalues. Eigenvalues less than or equal to
VL, or greater than VU, will not be returned. 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; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
GERS (input) DOUBLE PRECISION array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval is
(GERS(2*i-1), GERS(2*i)).
RELTOL (input) DOUBLE PRECISION
The minimum relative width of an interval. When an interval is
narrower than RELTOL times the larger (in magnitude) endpoint,
then it is considered to be sufficiently small, i.e., con-
verged. Note: this should always be at least radix*machine
epsilon.
D (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
E2 (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) squared off-diagonal elements of the tridiagonal
matrix T.
PIVMIN (input) DOUBLE PRECISION
The minimum pivot allowed in the sturm sequence for T.
NSPLIT (input) INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
ISPLIT (input) INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix 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.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)
M (output) INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
W (output) DOUBLE PRECISION array, dimension (N)
On exit, the first M elements of W will contain the eigenvalue
approximations. DLARRD2 computes an interval I_j = (a_j, b_j]
that includes eigenvalue j. The eigenvalue approximation is
given as the interval midpoint W(j)= ( a_j + b_j)/2. The corre-
sponding error is bounded by WERR(j) = abs( a_j - b_j)/2
WERR (output) DOUBLE PRECISION array, dimension (N)
The error bound on the corresponding eigenvalue approximation
in W.
WL (output) DOUBLE PRECISION
WU (output) DOUBLE PRECISION
The interval (WL, WU] contains all the wanted eigenvalues.
If RANGE='V', then WL=VL and WU=VU.
If RANGE='A', then WL and WU are the global Gerschgorin bounds
on the spectrum.
If RANGE='I', then WL and WU are computed by DLAEBZ from the
index range specified.
IBLOCK (output) INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the matrix T
is considered to split into a block diagonal matrix. On exit,
if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
number of blocks) the eigenvalue W(i) belongs. (DLARRD2 may
use the remaining N-M elements as workspace.)
INDEXW (output) INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th
eigenvalue W(i) is the j-th eigenvalue in block k.
WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
IWORK (workspace) INTEGER array, dimension (3*N)
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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE='I' only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE='A', and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER "FUDGE" may
make things work.
= 4: RANGE='I', and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER "FUDGE",
recompile, and try again.
INTERNAL PARAMETERS
FUDGE DOUBLE PRECISION, default = 2 originally, increased to 10.
A "fudge factor" to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.
ScaLAPACK routine 31 October 2017 DLARRD2(3)