Contents
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 * 1 RANGE, ORDER
INTEGER N, IL, IU, M, NSPLIT, INFO
INTEGER IBLOCK(*), ISPLIT(*), IWORK(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)
SUBROUTINE DSTEBZ_64(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E,
M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
CHARACTER * 1 RANGE, ORDER
INTEGER*8 N, IL, IU, M, NSPLIT, INFO
INTEGER*8 IBLOCK(*), ISPLIT(*), IWORK(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)
F95 INTERFACE
SUBROUTINE STEBZ(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M,
NSPLIT, W, IBLOCK, ISPLIT, [WORK], [IWORK], [INFO])
CHARACTER(LEN=1) :: RANGE, ORDER
INTEGER :: N, IL, IU, M, NSPLIT, INFO
INTEGER, DIMENSION(:) :: IBLOCK, ISPLIT, IWORK
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: D, E, W, WORK
SUBROUTINE STEBZ_64(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M,
NSPLIT, W, IBLOCK, ISPLIT, [WORK], [IWORK], [INFO])
CHARACTER(LEN=1) :: RANGE, ORDER
INTEGER(8) :: N, IL, IU, M, NSPLIT, INFO
INTEGER(8), DIMENSION(:) :: IBLOCK, ISPLIT, IWORK
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: D, E, W, WORK
C INTERFACE
#include <sunperf.h>
void dstebz(char range, char order, int n, double vl, double
vu, int il, int iu, double abstol, double *d,
double *e, int *m, int *nsplit, double *w, int
*iblock, int *isplit, int *info);
void dstebz_64(char range, char order, long n, double vl,
double vu, long il, long iu, double abstol, double
*d, double *e, long *m, long *nsplit, double *w,
long *iblock, long *isplit, long *info);
dstebz 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.
RANGE (input)
= '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)
= '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) The order of the tridiagonal matrix T. N >= 0.
VL (input)
If RANGE='V', the lower and upper bounds of the
interval to be searched for eigenvalues. Eigen-
values less than or equal to VL, or greater than
VU, will not be returned. VL < VU. Not
referenced if RANGE = 'A' or 'I'.
VU (input)
See the description of VL.
IL (input)
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'.
IU (input)
See the description of IL.
ABSTOL (input)
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) The n diagonal elements of the tridiagonal matrix
T.
E (input) The (n-1) off-diagonal elements of the tridiagonal
matrix T.
M (output)
The actual number of eigenvalues found. 0 <= M <=
N. (See also the description of INFO=2,3.)
NSPLIT (output)
The number of diagonal blocks in the matrix T. 1
<= NSPLIT <= N.
W (output)
On exit, the first M elements of W will contain
the eigenvalues. (DSTEBZ may use the remaining
N-M elements as workspace.)
IBLOCK (output)
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)
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)
dimension(4*N)
IWORK (workspace)
dimension(3*N)
INFO (output)
= 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. = 4: RANGE='I', and the
Gershgorin interval initially used was too small.
No eigenvalues were computed.