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RedHat 9 (Linux i386) - man page for dstebz (redhat section l)

DSTEBZ(l)					)					DSTEBZ(l)

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	  ORDER, RANGE

	   INTEGER	  IL, INFO, IU, M, N, NSPLIT

	   DOUBLE	  PRECISION ABSTOL, VL, VU

	   INTEGER	  IBLOCK( * ), ISPLIT( * ), IWORK( * )

	   DOUBLE	  PRECISION D( * ), E( * ), W( * ), WORK( * )

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,  Com-
       puter 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 consid-
	       ered 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  ei-
	       genvalues
	       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.

LAPACK version 3.0			   15 June 2000 				DSTEBZ(l)


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