Provided by: scalapack-doc_1.5-11_all 
NAME
PSSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix in parallel
SYNOPSIS
SUBROUTINE PSSTEBZ( 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
REAL ABSTOL, VL, VU
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * )
PURPOSE
PSSTEBZ 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 PSSTEBZ 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 double precision floating
point number is assumed be in the 32nd or 64th 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) REAL
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) REAL
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) REAL
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
SLAMCH('U'), not zero. Note : If eigenvectors are desired later by inverse iteration ( PSSTEIN
), ABSTOL should be set to 2*PSLAMCH('S').
D (global input) REAL 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) REAL 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) REAL 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) REAL 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
PSLAMCH 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 REAL, 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 REAL, 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.
LAPACK version 1.5 12 May 1997 PSSTEBZ(l)