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PDSTEBZ(l)			       )			    PDSTEBZ(l)

NAME
       PDSTEBZ	- compute the eigenvalues of a symmetric tridiagonal matrix in
       parallel

SYNOPSIS
       SUBROUTINE PDSTEBZ( ICTXT, RANGE, ORDER,	N, VL, VU, IL, IU, ABSTOL,  D,
			   E,  M,  NSPLIT,  W,	IBLOCK,	 ISPLIT,  WORK,	LWORK,
			   IWORK, LIWORK, INFO )

	   CHARACTER	   ORDER, RANGE

	   INTEGER	   ICTXT, IL, INFO, IU,	LIWORK,	LWORK, M, N, NSPLIT

	   DOUBLE	   PRECISION ABSTOL, VL, VU

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

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

PURPOSE
       PDSTEBZ computes	the eigenvalues	of a symmetric tridiagonal  matrix  in
       parallel.  The user may ask for all eigenvalues,	all eigenvalues	in the
       interval	[VL, VU], or the eigenvalues indexed IL	through	IU.  A	static
       partitioning  of	work is	done at	the beginning of PDSTEBZ which results
       in all processes	finding	an (almost) equal number of eigenvalues.

       NOTE : It is assumed that the user is on	an IEEE	machine. If the	user
	      is not on	an IEEE	mchine,	set the	compile	time flag NO_IEEE
	      to 1 (in SLmake.inc). The	features of IEEE arithmetic that
	      are needed for the "fast"	Sturm Count are	: (a) infinity
	      arithmetic (b) the sign bit of a single precision	floating
	      point number is assumed be in the	32nd bit position
	      (c) the sign of negative zero.

       See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal  Matrix",
       Report CS41, Computer Science Dept., Stanford
       University, July	21, 1966.

ARGUMENTS
       ICTXT   (global input) INTEGER
	       The BLACS context handle.

       RANGE   (global input) CHARACTER
	       Specifies  which	 eigenvalues  are to be	found.	= 'A': ("All")
	       all eigenvalues will be found.
	       = 'V': ("Value")	all eigenvalues	in the interval	[VL, VU]  will
	       be found.  = 'I': ("Index") the IL-th through IU-th eigenvalues
	       (of the entire matrix) will be found.

       ORDER   (global input) CHARACTER
	       Specifies the order in which the	eigenvalues  and  their	 block
	       numbers	are  stored  in	W and IBLOCK.  = 'B': ("By Block") the
	       eigenvalues will	be grouped by split-off	block (see IBLOCK, IS-
	       PLIT) 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       (global input) INTEGER
	       The order of the	tridiagonal matrix T.  N >= 0.

       VL      (global input) DOUBLE PRECISION
	       If  RANGE='V',  the  lower bound	of the interval	to be searched
	       for eigenvalues.	 Eigenvalues less than	VL  will  not  be  re-
	       turned.	Not referenced if RANGE='A' or 'I'.

       VU      (global input) DOUBLE PRECISION
	       If  RANGE='V',  the  upper bound	of the interval	to be searched
	       for eigenvalues.	 Eigenvalues greater than VU will not  be  re-
	       turned.	 VU  must  be  greater	than  VL.   Not	 referenced if
	       RANGE='A' or 'I'.

       IL      (global input) INTEGER
	       If RANGE='I', the index	(from  smallest	 to  largest)  of  the
	       smallest	 eigenvalue  to	 be  returned.	IL must	be at least 1.
	       Not referenced if RANGE='A' or 'V'.

       IU      (global input) INTEGER
	       If RANGE='I', the index	(from  smallest	 to  largest)  of  the
	       largest	eigenvalue to be returned.  IU must be at least	IL and
	       no greater than N.  Not referenced if RANGE='A' or 'V'.

       ABSTOL  (global 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  the  underflow threshold
	       DLAMCH('U'), not	zero.  Note  :	If  eigenvectors  are  desired
	       later by	inverse	iteration ( PDSTEIN ), ABSTOL should be	set to
	       2*PDLAMCH('S').

       D       (global input) DOUBLE PRECISION array, dimension	(N)
	       The n diagonal elements of the tridiagonal matrix T.  To	 avoid
	       overflow,  the  matrix must be scaled so	that its largest entry
	       is no greater than overflow**(1/2) * underflow**(1/4) in	 abso-
	       lute  value,  and  for greatest accuracy, it should not be much
	       smaller than that.

       E       (global input) DOUBLE PRECISION array, dimension	(N-1)
	       The (n-1) off-diagonal elements of the  tridiagonal  matrix  T.
	       To  avoid  overflow,  the  matrix  must	be  scaled so that its
	       largest entry is	 no  greater  than  overflow**(1/2)  *	under-
	       flow**(1/4)  in	absolute  value, and for greatest accuracy, it
	       should not be much smaller than that.

       M       (global output) INTEGER
	       The actual number of eigenvalues	found. 0 <= M <= N.  (See also
	       the description of INFO=2)

       NSPLIT  (global output) INTEGER
	       The  number of diagonal blocks in the matrix T.	1 <= NSPLIT <=
	       N.

       W       (global output) DOUBLE PRECISION	array, dimension (N)
	       On exit,	the first M elements of	W contain the  eigenvalues  on
	       all processes.

       IBLOCK  (global 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
	       IBLOCK(i)  specifies  which  block  (from  1  to	 the number of
	       blocks) the eigenvalue W(i) belongs to.	NOTE:  in  the	(theo-
	       retically  impossible)  event  that bisection does not converge
	       for some	or all eigenvalues, INFO is set	to 1 and the ones  for
	       which it	did not	are identified by a negative block number.

       ISPLIT  (global 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    (local workspace) DOUBLE	PRECISION array,
	       dimension ( MAX(	5*N, 7 ) )

       LWORK   (local input) INTEGER
	       size of array WORK must be >= MAX( 5*N, 7 ) If LWORK = -1, then
	       LWORK is	global input and a workspace  query  is	 assumed;  the
	       routine	only  calculates  the minimum and optimal size for all
	       work arrays. Each of these values is returned in	the first  en-
	       try  of	the  corresponding work	array, and no error message is
	       issued by PXERBLA.

       IWORK   (local workspace) INTEGER array,	dimension ( MAX( 4*N, 14 ) )

       LIWORK  (local input) INTEGER
	       size of array IWORK must	be >= MAX( 4*N,	14, NPROCS ) If	LIWORK
	       =  -1, then LIWORK is global input and a	workspace query	is as-
	       sumed; the routine only calculates the minimum and optimal size
	       for  all	 work  arrays. Each of these values is returned	in the
	       first entry of the corresponding	work array, and	no error  mes-
	       sage is issued by PXERBLA.

       INFO    (global 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 : 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 arithmetic
	       which  is  less	accurate  than PDLAMCH says.  =	2 : There is a
	       mismatch	between	the number of eigenvalues output and the  num-
	       ber desired.  = 3 : RANGE='i', and the Gershgorin interval ini-
	       tially used was incorrect. No eigenvalues were computed.	 Prob-
	       able  cause: your machine has sloppy floating point arithmetic.
	       Cure: Increase the PARAMETER "FUDGE", recompile,	and try	again.

PARAMETERS
       RELFAC  DOUBLE PRECISION, default = 2.0
	       The relative tolerance.	An interval [a,b] lies	within	"rela-
	       tive  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 = 2.0
	       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 the accuracy	of the
	       solution.

ScaLAPACK version 1.7		13 August 2001			    PDSTEBZ(l)

NAME | SYNOPSIS | PURPOSE | ARGUMENTS | PARAMETERS

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