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DSTEVR(3S)							    DSTEVR(3S)

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, *
			)

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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,

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DSTEVR(3S)							    DSTEVR(3S)

	    compute the corresponding eigenvector by forming a
	    rank-revealing twisted factorization.
     The desired accuracy of the output can be specified by the input
     parameter ABSTOL.

     For more details, see "A new O(n^2) algorithm for the symmetric
     tridiagonal 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'.

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DSTEVR(3S)							    DSTEVR(3S)

     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 approximate
	     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 reducing
	     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
	     Diagonally Dominant Matrices", LAPACK Working Note #7, for a
	     discussion of which matrices define their eigenvalues to high
	     relative 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 contain
	     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).

									Page 3

DSTEVR(3S)							    DSTEVR(3S)

     ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
	     The support of the eigenvectors in Z, i.e., the indices
	     indicating 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 minimal)
	     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

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.

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