

 
PDSTEBZ  compute the eigenvalues of a symmetric tridiagonal matrix in parallel
 SUBROUTINE PDSTEBZ(
 ICTXT, RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W,
IBLOCK, ISPLIT, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER ORDER, RANGE INTEGER ICTXT, IL, INFO, IU, LIWORK, LWORK, M, N, NSPLIT
DOUBLE PRECISION ABSTOL, VL, VU INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
PDSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix in parallel.
The user may ask for all eigenvalues, all eigenvalues in the interval [VL,
VU], or the eigenvalues indexed IL through IU. A static partitioning of work
is done at the beginning of PDSTEBZ which results in all processes finding an
(almost) equal number of eigenvalues.
NOTE : It is assumed that the user is on an IEEE machine. If the user
is not on an IEEE mchine, set the compile time flag NO_IEEE
to 1 (in SLmake.inc). The features of IEEE arithmetic that
are needed for the "fast" Sturm Count are : (a) infinity
arithmetic (b) the sign bit of a single precision floating
point number is assumed be in the 32nd bit position
(c) the sign of negative zero.
See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.
 ICTXT (global input) INTEGER
 The BLACS context handle.
 RANGE (global input) CHARACTER
 Specifies which eigenvalues are to be found. = 'A': ("All") all
eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the interval [VL, VU] will be
found. = 'I': ("Index") the ILth through IUth eigenvalues (of
the entire matrix) will be found.
 ORDER (global input) CHARACTER
 Specifies the order in which the eigenvalues and their block numbers are
stored in W and IBLOCK. = 'B': ("By Block") the eigenvalues will
be grouped by splitoff 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 (global input) INTEGER
 The order of the tridiagonal matrix T. N >= 0.
 VL (global input) DOUBLE PRECISION
 If RANGE='V', the lower bound of the interval to be searched for
eigenvalues. Eigenvalues less than VL will not be returned. Not referenced
if RANGE='A' or 'I'.
 VU (global input) DOUBLE PRECISION
 If RANGE='V', the upper bound of the interval to be searched for
eigenvalues. Eigenvalues greater than VU will not be returned. VU must be
greater than VL. Not referenced if RANGE='A' or 'I'.
 IL (global input) INTEGER
 If RANGE='I', the index (from smallest to largest) of the smallest
eigenvalue to be returned. IL must be at least 1. Not referenced if
RANGE='A' or 'V'.
 IU (global input) INTEGER
 If RANGE='I', the index (from smallest to largest) of the largest
eigenvalue to be returned. IU must be at least IL and no greater than N.
Not referenced if RANGE='A' or 'V'.
 ABSTOL (global input) DOUBLE PRECISION
 The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is
considered to be located if it has been determined to lie in an interval
whose width is ABSTOL or less. If ABSTOL is less than or equal to zero,
then ULP*T will be used, where T means the 1norm of T. Eigenvalues
will be computed most accurately when ABSTOL is set to the underflow
threshold DLAMCH('U'), not zero. Note : If eigenvectors are desired later
by inverse iteration ( PDSTEIN ), ABSTOL should be set to
2*PDLAMCH('S').
 D (global input) DOUBLE PRECISION array, dimension (N)
 The n diagonal elements of the tridiagonal matrix T. To avoid overflow,
the matrix must be scaled so that its largest entry is no greater than
overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.
 E (global input) DOUBLE PRECISION array, dimension (N1)
 The (n1) offdiagonal elements of the tridiagonal matrix T. To avoid
overflow, the matrix must be scaled so that its largest entry is no
greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for
greatest accuracy, it should not be much smaller than that.
 M (global output) INTEGER
 The actual number of eigenvalues found. 0 <= M <= N. (See also the
description of INFO=2)
 NSPLIT (global output) INTEGER
 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
N.
 W (global output) DOUBLE PRECISION array, dimension (N)
 On exit, the first M elements of W contain the eigenvalues on all
processes.
 IBLOCK (global 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 IBLOCK(i)
specifies which block (from 1 to the number of blocks) the eigenvalue W(i)
belongs to. NOTE: in the (theoretically impossible) event that bisection
does not converge for some or all eigenvalues, INFO is set to 1 and the
ones for which it did not are identified by a negative block number.
 ISPLIT (global output) 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 NSPLITth
consists of rows/columns ISPLIT(NSPLIT1)+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.)
 WORK (local workspace) DOUBLE PRECISION array,
 dimension ( MAX( 5*N, 7 ) )
 LWORK (local input) INTEGER
 size of array WORK must be >= MAX( 5*N, 7 ) If LWORK = 1, then LWORK
is global input and a workspace query is assumed; the routine only
calculates the minimum and optimal size for all work arrays. Each of these
values is returned in the first entry of the corresponding work array, and
no error message is issued by PXERBLA.
 IWORK (local workspace) INTEGER array, dimension ( MAX( 4*N, 14 ) )
 LIWORK (local input) INTEGER
 size of array IWORK must be >= MAX( 4*N, 14, NPROCS ) If LIWORK = 1,
then LIWORK is global input and a workspace query is assumed; the routine
only calculates the minimum and optimal size for all work arrays. Each of
these values is returned in the first entry of the corresponding work
array, and no error message is issued by PXERBLA.
 INFO (global output) INTEGER
 = 0 : successful exit
< 0 : if INFO = i, the ith argument had an illegal value
> 0 : some or all of the eigenvalues failed to converge or
were not computed:
= 1 : 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 arithmetic which is less accurate than PDLAMCH says. =
2 : There is a mismatch between the number of eigenvalues output and the
number desired. = 3 : RANGE='i', and the Gershgorin interval initially
used was incorrect. No eigenvalues were computed. Probable cause: your
machine has sloppy floating point arithmetic. Cure: Increase the PARAMETER
"FUDGE", recompile, and try again.
 RELFAC DOUBLE PRECISION, default = 2.0
 The relative tolerance. An interval [a,b] lies within "relative
tolerance" if ba < RELFAC*ulp*max(a,b), where "ulp"
is the machine precision (distance from 1 to the next larger floating
point number.)
 FUDGE DOUBLE PRECISION, default = 2.0
 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
the accuracy of the solution.
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