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cstemr.f(3)			    LAPACK			   cstemr.f(3)

       cstemr.f -

       subroutine cstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ,

Function/Subroutine Documentation
   subroutine cstemr (characterJOBZ, characterRANGE, integerN, real,
       dimension( * )D, real, dimension( * )E, realVL, realVU, integerIL,
       integerIU, integerM, real, dimension( * )W, complex, dimension( ldz, *
       )Z, integerLDZ, integerNZC, integer, dimension( * )ISUPPZ,
       logicalTRYRAC, real, dimension( * )WORK, integerLWORK, integer,
       dimension( * )IWORK, integerLIWORK, integerINFO)


	    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 corresponding
	    real eigenvectors are pairwise orthogonal.

	    The spectrum may be computed either completely or partially by specifying
	    either an interval (VL,VU] or a range of indices IL:IU for the desired

	    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 factorizations 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 compute
		   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 representations
	      to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
	      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
	    1.CSTEMR works only on machines which follow IEEE-754
	    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 diagonal
	    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 complex entries
	    in general.
	    Since LAPACK drivers overwrite the matrix data with the eigenvectors,
	    CSTEMR accepts complex workspace to facilitate interoperability
	    with CUNMTR or CUPMTR.


		     JOBZ is CHARACTER*1
		     = 'N':  Compute eigenvalues only;
		     = 'V':  Compute eigenvalues and eigenvectors.


		     = '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 is INTEGER
		     The order of the matrix.  N >= 0.


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


		     E is 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 is REAL


		     VU is 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 is INTEGER


		     IU is 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 is INTEGER
		     The total number of eigenvalues found.  0 <= M <= N.
		     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.


		     W is REAL array, dimension (N)
		     The first M elements contain the selected eigenvalues in
		     ascending order.


		     Z is 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
		     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 value of M
		     is not known in advance and can be computed with a workspace
		     query by setting NZC = -1, see below.


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


		     NZC is 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 is 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 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.


		     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 preserving
		     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 algorithms.
		     If TRYRAC.EQ..FALSE., the code is not required to guarantee
		     relatively accurate eigenvalues and can use the fastest possible
		     On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
		     does not define its eigenvalues to high relative accuracy.


		     WORK is REAL array, dimension (LWORK)
		     On exit, if INFO = 0, WORK(1) returns the optimal
		     (and minimal) LWORK.


		     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 is INTEGER array, dimension (LIWORK)
		     On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.


		     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 is 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.

	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

	   September 2012

	   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

       Definition at line 328 of file cstemr.f.

       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2			Tue Sep 25 2012			   cstemr.f(3)

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