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CHEEVR(3S)							    CHEEVR(3S)

NAME
     CHEEVR - compute selected eigenvalues and, optionally, eigenvectors of a
     complex Hermitian matrix T

SYNOPSIS
     SUBROUTINE CHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
			M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK,
			IWORK, LIWORK, INFO )

	 CHARACTER	JOBZ, RANGE, UPLO

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

	 REAL		ABSTOL, VL, VU

	 INTEGER	ISUPPZ( * ), IWORK( * )

	 REAL		RWORK( * ), W( * )

	 COMPLEX	A( LDA, * ), 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
     CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a
     complex Hermitian 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, CHEEVR calls CSTEGR to compute the
     eigenspectrum using Relatively Robust Representations.  CSTEGR 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

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CHEEVR(3S)							    CHEEVR(3S)

	    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
     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 : CHEEVR calls CSTEGR when the full spectrum is requested on
     machines which conform to the ieee-754 floating point standard.  CHEEVR
     calls SSTEBZ and CSTEIN on non-ieee machines and
     when partial spectrum requests are made.

     Normal execution of CSTEGR 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.

     UPLO    (input) CHARACTER*1
	     = 'U':  Upper triangle of A is stored;
	     = 'L':  Lower triangle of A is stored.

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

     A	     (input/output) COMPLEX array, dimension (LDA, N)
	     On entry, the Hermitian matrix A.	If UPLO = 'U', the leading N-
	     by-N upper triangular part of A contains the upper triangular
	     part of the matrix A.  If UPLO = 'L', the leading N-by-N lower
	     triangular part of A contains the lower triangular part of the
	     matrix A.	On exit, the lower triangle (if UPLO='L') or the upper
	     triangle (if UPLO='U') of A, including the diagonal, is
	     destroyed.

     LDA     (input) INTEGER
	     The leading dimension of the array A.  LDA >= max(1,N).

									Page 2

CHEEVR(3S)							    CHEEVR(3S)

     VL	     (input) REAL
	     VU	     (input) REAL 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) REAL
	     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 SLAMCH(
	     '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 furutre 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) REAL array, dimension (N)
	     The first M elements contain the selected eigenvalues in
	     ascending order.

     Z	     (output) COMPLEX 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).	If JOBZ = 'N', then Z is not
	     referenced.  Note: the user must ensure that at least max(1,M)
	     columns are supplied in the array Z; if RANGE = 'V', the exact

									Page 3

CHEEVR(3S)							    CHEEVR(3S)

	     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
	     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) COMPLEX array, dimension (LWORK)
	     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The length of the array WORK.  LWORK >= max(1,2*N).  For optimal
	     efficiency, LWORK >= (NB+1)*N, where NB is the max of the
	     blocksize for CHETRD and for CUNMTR as returned by ILAENV.

	     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.

     RWORK   (workspace/output) REAL array, dimension (LRWORK)
	     On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal)
	     LRWORK.

     LRWORK   (input) INTEGER
	     The length of the array RWORK.  LRWORK >= max(1,24*N).

	     If LRWORK = -1, then a workspace query is assumed; the routine
	     only calculates the optimal size of the RWORK array, returns this
	     value as the first entry of the RWORK array, and no error message
	     related to LRWORK 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 >= max(1,10*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

									Page 4

CHEEVR(3S)							    CHEEVR(3S)

	     > 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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