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DSTEVR(l)			       )			     DSTEVR(l)

NAME
       DSTEVR  - compute selected eigenvalues and, optionally, eigenvectors of
       a real symmetric tridiagonal matrix T

SYNOPSIS
       SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,  W,
			  Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO )

	   CHARACTER	  JOBZ, RANGE

	   INTEGER	  IL, INFO, IU, LDZ, LIWORK, LWORK, M, N

	   DOUBLE	  PRECISION ABSTOL, VL, VU

	   INTEGER	  ISUPPZ( * ), IWORK( * )

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

PURPOSE
       DSTEVR computes selected eigenvalues and, optionally, eigenvectors of a
       real  symmetric	tridiagonal matrix T. Eigenvalues and eigenvectors can
       be selected by specifying either a  range  of  values  or  a  range  of
       indices for the desired eigenvalues.

       Whenever possible, DSTEVR calls SSTEGR to compute the
       eigenspectrum using Relatively Robust Representations.  DSTEGR computes
       eigenvalues by the dqds algorithm, while	 orthogonal  eigenvectors  are
       computed	 from  various	"good"	L D L^T representations (also known as
       Relatively Robust Representations). Gram-Schmidt	 orthogonalization  is
       avoided as far as possible. More specifically, the various steps of the
       algorithm are as follows. For the i-th unreduced block of T,
	  (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
	       is a relatively robust representation,
	  (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
	      relative accuracy by the dqds algorithm,
	  (c) If there is a cluster of close eigenvalues, "choose" sigma_i
	      close to the cluster, and go to step (a),
	  (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
	      compute the corresponding eigenvector by forming a
	      rank-revealing twisted factorization.
       The desired accuracy of the output can be specified by the input param‐
       eter ABSTOL.

       For  more details, see "A new O(n^2) algorithm for the symmetric tridi‐
       agonal eigenvalue/eigenvector problem", by Inderjit  Dhillon,  Computer
       Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
       1997.

       Note 1 : DSTEVR calls SSTEGR when the full  spectrum  is	 requested  on
       machines which conform to the ieee-754 floating point standard.	DSTEVR
       calls SSTEBZ and SSTEIN on non-ieee machines and
       when partial spectrum requests are made.

       Normal execution of DSTEGR may create NaNs and infinities and hence may
       abort  due  to  a floating point exception in environments which do not
       handle NaNs and infinities in the ieee standard default manner.

ARGUMENTS
       JOBZ    (input) CHARACTER*1
	       = 'N':  Compute eigenvalues only;
	       = 'V':  Compute eigenvalues and eigenvectors.

       RANGE   (input) CHARACTER*1
	       = 'A': all eigenvalues will be found.
	       = 'V': all eigenvalues in the half-open interval	 (VL,VU]  will
	       be  found.   = 'I': the IL-th through IU-th eigenvalues will be
	       found.

       N       (input) INTEGER
	       The order of the matrix.	 N >= 0.

       D       (input/output) DOUBLE PRECISION array, dimension (N)
	       On entry, the n diagonal elements of the tridiagonal matrix  A.
	       On  exit,  D  may  be multiplied by a constant factor chosen to
	       avoid over/underflow in computing the eigenvalues.

       E       (input/output) DOUBLE PRECISION array, dimension (N)
	       On entry, the (n-1) subdiagonal	elements  of  the  tridiagonal
	       matrix  A  in elements 1 to N-1 of E; E(N) need not be set.  On
	       exit, E may be multiplied by a constant factor chosen to	 avoid
	       over/underflow in computing the eigenvalues.

       VL      (input) DOUBLE PRECISION
	       VU	(input)	 DOUBLE	 PRECISION If RANGE='V', the lower and
	       upper bounds of the interval to be searched for eigenvalues. 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 error tolerance for the eigenvalues.  An  approxi‐
	       mate  eigenvalue is accepted as converged when it is determined
	       to lie in an interval [a,b] of width less than or equal to

	       ABSTOL + EPS *	max( |a|,|b| ) ,

	       where EPS is the machine precision.  If ABSTOL is less than  or
	       equal  to zero, then  EPS*|T|  will be used in its place, where
	       |T| is the 1-norm of the tridiagonal matrix obtained by	reduc‐
	       ing A to tridiagonal form.

	       See  "Computing	Small  Singular	 Values of Bidiagonal Matrices
	       with Guaranteed High Relative Accuracy," by Demmel  and	Kahan,
	       LAPACK Working Note #3.

	       If  high	 relative accuracy is important, set ABSTOL to DLAMCH(
	       'Safe minimum' ).  Doing so will guarantee that eigenvalues are
	       computed	 to  high  relative  accuracy  when possible in future
	       releases.  The current code does not make any guarantees	 about
	       high relative accuracy, but future releases will. See J. Barlow
	       and J. Demmel, "Computing Accurate Eigensystems of Scaled Diag‐
	       onally  Dominant	 Matrices", LAPACK Working Note #7, for a dis‐
	       cussion of which matrices define their eigenvalues to high rel‐
	       ative accuracy.

       M       (output) INTEGER
	       The  total number of eigenvalues found.	0 <= M <= N.  If RANGE
	       = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

       W       (output) DOUBLE PRECISION array, dimension (N)
	       The first  M  elements  contain	the  selected  eigenvalues  in
	       ascending order.

       Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
	       If  JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
	       tain the orthonormal eigenvectors of the matrix A corresponding
	       to  the selected eigenvalues, with the i-th column of Z holding
	       the eigenvector associated with	W(i).	Note:  the  user  must
	       ensure that at least max(1,M) columns are supplied in the array
	       Z; if RANGE = 'V', the exact value of M is not known in advance
	       and an upper bound must be used.

       LDZ     (input) INTEGER
	       The  leading dimension of the array Z.  LDZ >= 1, and if JOBZ =
	       'V', LDZ >= max(1,N).

       ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
	       The support of the eigenvectors in Z, i.e., the	indices	 indi‐
	       cating  the  nonzero  elements  in  Z.  The i-th eigenvector is
	       nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).

       WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal (and minimal)
	       LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK.	 LWORK >= 20*N.

	       If  LWORK  = -1, then a workspace query is assumed; the routine
	       only calculates the optimal size of  the	 WORK  array,  returns
	       this  value  as the first entry of the WORK array, and no error
	       message related to LWORK is issued by XERBLA.

       IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
	       On exit, if INFO = 0, IWORK(1) returns the optimal  (and	 mini‐
	       mal) LIWORK.

       LIWORK  (input) INTEGER
	       The dimension of the array IWORK.  LIWORK >= 10*N.

	       If  LIWORK = -1, then a workspace query is assumed; the routine
	       only calculates the optimal size of the	IWORK  array,  returns
	       this  value as the first entry of the IWORK array, and no error
	       message related to LIWORK is issued by XERBLA.

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value
	       > 0:  Internal error

FURTHER DETAILS
       Based on contributions by
	  Inderjit Dhillon, IBM Almaden, USA
	  Osni Marques, LBNL/NERSC, USA
	  Ken Stanley, Computer Science Division, University of
	    California at Berkeley, USA

LAPACK version 3.0		 15 June 2000			     DSTEVR(l)
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