DSTEBZ(3F)DSTEBZ(3F)NAMEDSTEBZ - 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( * )
PURPOSEDSTEBZ 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 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
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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) 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.)
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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 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 "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.
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