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

SSTEVR(l)					)					SSTEVR(l)

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

SYNOPSIS
       SUBROUTINE SSTEVR( 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( * ), Z( LDZ, * )

PURPOSE
       SSTEVR  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, SSTEVR calls SSTEGR to compute the
       eigenspectrum using Relatively Robust Representations.  SSTEGR 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 : SSTEVR calls SSTEGR when the full spectrum is requested on machines which conform
       to the ieee-754 floating point standard.  SSTEVR  calls	SSTEBZ	and  SSTEIN  on  non-ieee
       machines and
       when partial spectrum requests are made.

       Normal  execution  of  SSTEGR  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) REAL 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) REAL 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) 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 future 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) REAL 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).

       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 >= 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

LAPACK version 3.0			   15 June 2000 				SSTEVR(l)


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