SSTEBZ(l) ) SSTEBZ(l)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( * )
PURPOSE
SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T.
The user may ask for all eigenvalues, all eigenvalues 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 Tridiagonal 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 eigen‐
values for the entire matrix will be ordered from smallest 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
referenced 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 eigenval‐
ues. (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 illegal 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 Ger‐
shgorin interval initially used was too small. No eigenvalues
were computed. Probable cause: your machine has sloppy float‐
ing-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 "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 REAL, 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.
LAPACK version 3.0 15 June 2000 SSTEBZ(l)
