NAME

dstebz - 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);

PURPOSE

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 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 eigenvalues 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 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. (SSTEBZ 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. (SSTEBZ 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.