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

CHEEVR(l)					)					CHEEVR(l)

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, * )

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 orthogo-
       nalization 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 parameter ABSTOL.

       For  more  details,  see  "A  new O(n^2) algorithm for the symmetric tridiagonal eigenval-
       ue/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  trian-
	       gular  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 tri-
	       angle (if UPLO='U') of A, including the diagonal, is destroyed.

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

       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. Dem-
	       mel, "Computing Accurate Eigensystems of  Scaled  Diagonally  Dominant  Matrices",
	       LAPACK  Working Note #7, for a discussion of which matrices define their eigenval-
	       ues 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)  col-
	       umns  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 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.

	       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.

	       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
	       > 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 				CHEEVR(l)


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