DSTEBZ(l) ) DSTEBZ(l)
DSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix T
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( * )
DSTEBZ 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
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, Com-
puter Science Dept., Stanford
University, July 21, 1966.
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
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
smallest 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
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) DOUBLE PRECISION
The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is consid-
ered 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
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 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 ei-
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
RELFAC DOUBLE PRECISION, 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 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 solution.
LAPACK version 3.0 15 June 2000 DSTEBZ(l)