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NAME
sstebz - compute the eigenvalues of a symmetric tridiagonal
matrix T
SYNOPSIS
SUBROUTINE SSTEBZ( 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
REAL ABSTOL, VL, VU
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * )
#include <sunperf.h>
void sstebz(char range, char order, int n, float vl, float
vu, int il, int iu, float abstol, float *d, float
*e, int *m, int *nsplit, float *w, int *iblock,
int *isplit, int *info) ;
PURPOSE
SSTEBZ 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) REAL
VU (input) REAL 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) 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 twice the underflow threshold
2*SLAMCH('S'), not zero.
D (input) REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix
T.
E (input) REAL 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) REAL array, dimension (N)
On exit, the first M elements of W will contain
the eigenvalues. (SSTEBZ 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. (SSTEBZ 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) REAL 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 REAL, default = 2.0e0 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 A "fudge factor" to widen the
Gershgorin intervals. Ideally, a value of 1
should work, but on machines with sloppy arith-
metic, 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.
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