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CSTEMR(1)	  LAPACK computational routine (version 3.2)	     CSTEMR(1)

NAME
       CSTEMR - computes selected eigenvalues and, optionally, eigenvectors of
       a real symmetric tridiagonal matrix T

SYNOPSIS
       SUBROUTINE CSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z,  LDZ,
			  NZC,	ISUPPZ,	 TRYRAC,  WORK,	 LWORK, IWORK, LIWORK,
			  INFO )

	   IMPLICIT	  NONE

	   CHARACTER	  JOBZ, RANGE

	   LOGICAL	  TRYRAC

	   INTEGER	  IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N

	   REAL		  VL, VU

	   INTEGER	  ISUPPZ( * ), IWORK( * )

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

	   COMPLEX	  Z( LDZ, * )

PURPOSE
       CSTEMR computes selected eigenvalues and, optionally, eigenvectors of a
       real  symmetric	tridiagonal  matrix T. Any such unreduced matrix has a
       well defined set of pairwise different  real  eigenvalues,  the	corre‐
       sponding real eigenvectors are pairwise orthogonal.
       The spectrum may be computed either completely or partially by specify‐
       ing either an interval (VL,VU] or a range  of  indices  IL:IU  for  the
       desired eigenvalues.
       Depending  on  the  number  of  desired eigenvalues, these are computed
       either by bisection  or	the  dqds  algorithm.  Numerically  orthogonal
       eigenvectors  are  computed by the use of various suitable L D L^T fac‐
       torizations near clusters of close eigenvalues (referred	 to  as	 RRRs,
       Relatively Robust Representations). An informal sketch of the algorithm
       follows.	 For each unreduced block (submatrix) of T,
	  (a) Compute T - sigma I  = L D L^T, so that L and D
	      define all the wanted eigenvalues to high relative accuracy.
	      This means that small relative changes in the entries of D and L
	      cause only small relative changes in the eigenvalues and
	      eigenvectors. The standard (unfactored) representation of the
	      tridiagonal matrix T does not have this property in general.
	  (b) Compute the eigenvalues to suitable accuracy.
	      If the eigenvectors are desired, the algorithm attains full
	      accuracy of the computed eigenvalues only right before
	      the corresponding vectors have to be computed, see steps c)  and
       d).
	  (c) For each cluster of close eigenvalues, select a new
	      shift close to the cluster, find a new factorization, and refine
	      the shifted eigenvalues to suitable accuracy.
	  (d) For each eigenvalue with a large enough relative separation com‐
       pute
	      the  corresponding  eigenvector  by  forming  a  rank  revealing
       twisted
	      factorization. Go back to (c) for any clusters that remain.  For
       more details, see:
       - Inderjit S. Dhillon and Beresford N. Parlett:	"Multiple  representa‐
       tions
	 to  compute  orthogonal  eigenvectors of symmetric tridiagonal matri‐
       ces,"
	 Linear Algebra and its Applications, 387(1), pp. 1-28,	 August	 2004.
       - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
	 Relative  Gaps,"  SIAM	 Journal  on Matrix Analysis and Applications,
       Vol. 25,
	 2004.	Also LAPACK Working Note 154.
       - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
	 tridiagonal eigenvalue/eigenvector problem",
	 Computer Science Division Technical Report No. UCB/CSD-97-971,
	 UC Berkeley, May 1997.
       Further Details
       floating-point standard in their handling of infinities and NaNs.  This
       permits	the  use  of  efficient	 inner loops avoiding a check for zero
       divisors.
       2. LAPACK routines can be used to reduce a complex Hermitean matrix  to
       real symmetric tridiagonal form.
       (Any complex Hermitean tridiagonal matrix has real values on its diago‐
       nal and potentially complex numbers on its off-diagonals. By applying a
       similarity transform with an appropriate diagonal matrix
       diag(1,e^{i							hy_1},
       ...		   ,		     e^{i		   hy_{n-1}}),
       the  complex  Hermitean matrix can be transformed into a real symmetric
       matrix and complex arithmetic can be entirely avoided.)
       While the eigenvectors of the real  symmetric  tridiagonal  matrix  are
       real,  the  eigenvectors of original complex Hermitean matrix have com‐
       plex entries in general.
       Since LAPACK drivers overwrite the matrix data with  the	 eigenvectors,
       CSTEMR  accepts	complex	 workspace to facilitate interoperability with
       CUNMTR or CUPMTR.

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 overwritten.

       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
	       input,  but  is	used  internally  as workspace.	 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.  Not referenced if RANGE = 'A' or
	       'V'.

       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', and if INFO = 0, then the first M columns of Z
	       contain the orthonormal eigenvectors of	the  matrix  T	corre‐
	       sponding 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	 value of M is not known in advance and can be
	       computed with a workspace query by setting NZC = -1, see below.

       LDZ     (input) INTEGER
	       The leading dimension of the array Z.  LDZ >= 1, and if JOBZ  =
	       'V', then LDZ >= max(1,N).

       NZC     (input) INTEGER
	       The number of eigenvectors to be held in the array Z.  If RANGE
	       = 'A', then NZC >= max(1,N).  If RANGE = 'V', then NZC  >=  the
	       number  of eigenvalues in (VL,VU].  If RANGE = 'I', then NZC >=
	       IU-IL+1.	 If NZC = -1, then a workspace query is	 assumed;  the
	       routine	calculates  the	 number of columns of the array Z that
	       are needed to hold the eigenvectors.  This value is returned as
	       the first entry of the Z array, and no error message related to
	       NZC is issued by XERBLA.

       ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
	       The support of the eigenvectors in Z, i.e., the	indices	 indi‐
	       cating the nonzero elements in Z. The i-th computed eigenvector
	       is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i
	       ).  This	 is  relevant  in  the	case when the matrix is split.
	       ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

       TRYRAC  (input/output) LOGICAL
	       If TRYRAC.EQ..TRUE.,  indicates	that  the  code	 should	 check
	       whether	the tridiagonal matrix defines its eigenvalues to high
	       relative accuracy.  If so, the code uses relative-accuracy pre‐
	       serving	algorithms  that  might be (a bit) slower depending on
	       the matrix.  If the matrix does not define its  eigenvalues  to
	       high relative accuracy, the code can uses possibly faster algo‐
	       rithms.	If TRYRAC.EQ..FALSE., the  code	 is  not  required  to
	       guarantee  relatively  accurate	eigenvalues  and  can  use the
	       fastest possible techniques.  On exit, a .TRUE. TRYRAC will  be
	       set to .FALSE. if the matrix does not define its eigenvalues to
	       high relative accuracy.

       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 JOBZ =
	       'V', and LWORK >= max(1,12*N) if JOBZ = '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 the
	       eigenvectors  are desired, and LIWORK >= max(1,8*N) if only the
	       eigenvalues are to  be  computed.   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
	       On exit, INFO = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value
	       > 0:  if INFO = 1X, internal error in SLARRE,  if  INFO	=  2X,
	       internal	 error	in CLARRV.  Here, the digit X = ABS( IINFO ) <
	       10, where IINFO is the nonzero error code returned by SLARRE or
	       CLARRV, respectively.

FURTHER DETAILS
       Based on contributions by
	  Beresford Parlett, University of California, Berkeley, USA
	  Jim Demmel, University of California, Berkeley, USA
	  Inderjit Dhillon, University of Texas, Austin, USA
	  Osni Marques, LBNL/NERSC, USA
	  Christof Voemel, University of California, Berkeley, USA

 LAPACK computational routine (veNovember22008			     CSTEMR(1)
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