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NAME
dstebz - compute the eigenvalues of a symmetric tridiagonal
matrix T
SYNOPSIS
SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
INFO )
CHARACTER ORDER, RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
#include <sunperf.h>
void dstebz(char range, char order, int n, double vl, double
vu, int il, int iu, double abstol, double *d, dou-
ble *e, int *m, int *nsplit, double *w, int
*iblock, int *isplit, int *info) ;
PURPOSE
DSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T. The user may ask for all eigenvalues, all eigen-
values in the half-open interval (VL, VU], or the IL-th
through IU-th eigenvalues.
To avoid overflow, the matrix must be scaled so that its
largest element 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.
See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiago-
nal Matrix", Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.
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 smal-
lest 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 refer-
enced if RANGE = 'A' or 'V'.
ABSTOL (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 twice the underflow threshold
2*DLAMCH('S'), not zero.
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.
M (output) INTEGER
The actual number of eigenvalues found. 0 <= M <=
N. (See also the description of INFO=2,3.)
NSPLIT (output) INTEGER
The number of diagonal blocks in the matrix T. 1
<= NSPLIT <= N.
W (output) DOUBLE PRECISION array, dimension (N)
On exit, the first M elements of W will contain
the eigenvalues. (DSTEBZ may use the remaining
N-M elements as workspace.)
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. (DSTEBZ may
use the remaining N-M elements as workspace.)
ISPLIT (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 (workspace) DOUBLE PRECISION array, dimension
(4*N)
IWORK (workspace) INTEGER array, dimension (3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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.
PARAMETERS
RELFAC DOUBLE PRECISION, default = 2.0e0 The relative
tolerance. An interval (a,b] lies within "rela-
tive 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 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 solu-
tion.
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