CXML
        	    
LAPACK version 3.0
dstebz(3)

PURPOSE
  DSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix T

SYNTAX
  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( * )

DESCRIPTION
  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) 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 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.)

  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.