Provided by: scalapack-doc_1.5-11_all 
NAME
PDSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix in parallel
SYNOPSIS
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( * )
PURPOSE
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.
ARGUMENTS
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 IL-th through IU-th 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 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 (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 1-norm 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 (N-1)
The (n-1) off-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.
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 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.)
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 i-th 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.
PARAMETERS
RELFAC DOUBLE PRECISION, default = 2.0
The relative tolerance. An interval [a,b] lies within "relative tolerance" if b-
a < 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.