dstebz(3P) Sun Performance Library dstebz(3P)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 * 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);
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)
= '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 ei‐
genvalues for the entire matrix will be ordered from smallest
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. Eigenvalues 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 small‐
est 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*DLAMCH('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 eigenval‐
ues. (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 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. = 4: RANGE='I', and the Gershgorin
interval initially used was too small. No eigenvalues were
computed.
6 Mar 2009 dstebz(3P)
