.TH SSTEBZ 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
SSTEBZ - the eigenvalues of a symmetric tridiagonal matrix T
.SH SYNOPSIS
.TP 19
SUBROUTINE SSTEBZ(
RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E,
M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
INFO )
.TP 19
.ti +4
CHARACTER
ORDER, RANGE
.TP 19
.ti +4
INTEGER
IL, INFO, IU, M, N, NSPLIT
.TP 19
.ti +4
REAL
ABSTOL, VL, VU
.TP 19
.ti +4
INTEGER
IBLOCK( * ), ISPLIT( * ), IWORK( * )
.TP 19
.ti +4
REAL
D( * ), E( * ), W( * ), WORK( * )
.SH 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.
.br
To avoid overflow, the matrix must be scaled so that its
.br
largest element is no greater than overflow**(1/2) *
.br
underflow**(1/4) in absolute value, and for greatest
.br
accuracy, it should not be much smaller than that.
.br
See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford
.br
University, July 21, 1966.
.br
.SH ARGUMENTS
.TP 8
RANGE (input) CHARACTER*1
= \(aqA\(aq: ("All") all eigenvalues will be found.
.br
= \(aqV\(aq: ("Value") all eigenvalues in the half-open interval
(VL, VU] will be found.
= \(aqI\(aq: ("Index") the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
.TP 8
ORDER (input) CHARACTER*1
= \(aqB\(aq: ("By Block") the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= \(aqE\(aq: ("Entire matrix")
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
.TP 8
N (input) INTEGER
The order of the tridiagonal matrix T. N >= 0.
.TP 8
VL (input) REAL
VU (input) REAL
If RANGE=\(aqV\(aq, 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 = \(aqA\(aq or \(aqI\(aq.
.TP 8
IL (input) INTEGER
IU (input) INTEGER
If RANGE=\(aqI\(aq, 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 = \(aqA\(aq or \(aqV\(aq.
.TP 8
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(\(aqS\(aq), not zero.
.TP 8
D (input) REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
.TP 8
E (input) REAL array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
.TP 8
M (output) INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
.TP 8
NSPLIT (output) INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
.TP 8
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.)
.TP 8
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.)
.TP 8
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.)
.TP 8
WORK (workspace) REAL array, dimension (4*N)
.TP 8
IWORK (workspace) INTEGER array, dimension (3*N)
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 0: some or all of the eigenvalues failed to converge or
.br
were not computed:
.br
=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=\(aqI\(aq only: Not all of the eigenvalues
.br
IL:IU were found.
.br
Effect: M < IU+1-IL
.br
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE=\(aqA\(aq, and pick
.br
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER "FUDGE" may
make things work.
= 4: RANGE=\(aqI\(aq, 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.
.SH PARAMETERS
.TP 8
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.)
.TP 8
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 solution.