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

CSTEGR(l)					)					CSTEGR(l)

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

SYNOPSIS
       SUBROUTINE CSTEGR( 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

	   REAL 	  ABSTOL, VL, VU

	   INTEGER	  ISUPPZ( * ), IWORK( * )

	   REAL 	  D( * ), E( * ), W( * ), WORK( * )

	   COMPLEX	  Z( LDZ, * )

PURPOSE
       CSTEGR  computes  selected  eigenvalues	and, optionally, eigenvectors of a real symmetric
       tridiagonal matrix T. Eigenvalues and

	  (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 : Currently CSTEGR is only set up to find ALL the n eigenvalues and eigenvectors of
       T in O(n^2) time
       Note 2 : Currently the routine CSTEIN is called when an appropriate sigma_i cannot be cho-
       sen in step (c) above. CSTEIN invokes modified Gram-Schmidt when eigenvalues are close.
       Note 3 : CSTEGR works only on machines which follow ieee-754  floating-point  standard  in
       their  handling	of  infinities	and NaNs.  Normal execution of CSTEGR may create NaNs and
       infinities and hence may abort due to a floating point exception in environments which  do
       not conform to the ieee standard.

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) REAL array, dimension (N)
	       On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is over-
	       written.

       E       (input/output) REAL array, dimension (N)
	       On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 1
	       to N-1 of E; E(N) need not be set.  On exit, E is overwritten.

       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/eigenvectors. IF JOBZ = 'V', the
	       eigenvalues and eigenvectors output have residual norms bounded by ABSTOL, and the
	       dot  products  between  different eigenvectors are bounded by ABSTOL. If ABSTOL is
	       less than N*EPS*|T|, then N*EPS*|T| will be used in its place, where  EPS  is  the
	       machine precision and |T| is the 1-norm of the tridiagonal matrix. The eigenvalues
	       are computed to an accuracy of EPS*|T| irrespective of ABSTOL.  If  high  relative
	       accuracy  is  important,  set  ABSTOL to DLAMCH( 'Safe minimum' ).  See Barlow and
	       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 T 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) REAL 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 >= max(1,18*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 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
	       >  0:   if  INFO  =  1,	internal  error in SLARRE, if INFO = 2, internal error in
	       CLARRV.

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 computational version 3.0	   15 June 2000 				CSTEGR(l)


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