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   *> \brief \b ZSTEMR
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZSTEMR + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zstemr.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zstemr.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zstemr.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
   *                          M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
   *                          IWORK, LIWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, RANGE
   *       LOGICAL            TRYRAC
   *       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
   *       DOUBLE PRECISION VL, VU
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            ISUPPZ( * ), IWORK( * )
   *       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
   *       COMPLEX*16         Z( LDZ, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZSTEMR 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
   *> 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 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.ZSTEMR 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 \phy_1}, ... , e^{i \phy_{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,
   *> ZSTEMR accepts complex workspace to facilitate interoperability
   *> with ZUNMTR or ZUPMTR.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBZ
   *> \verbatim
   *>          JOBZ is CHARACTER*1
   *>          = 'N':  Compute eigenvalues only;
   *>          = 'V':  Compute eigenvalues and eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] RANGE
   *> \verbatim
   *>          RANGE is 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.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          On entry, the N diagonal elements of the tridiagonal matrix
   *>          T. On exit, D is overwritten.
   *> \endverbatim
   *>
   *> \param[in,out] E
   *> \verbatim
   *>          E is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION
   *>
   *>          If RANGE='V', the lower bound of the interval to
   *>          be searched for eigenvalues. VL < VU.
   *>          Not referenced if RANGE = 'A' or 'I'.
   *> \endverbatim
   *>
   *> \param[in] VU
   *> \verbatim
   *>          VU is DOUBLE PRECISION
   *>
   *>          If RANGE='V', the upper bound of the interval to
   *>          be searched for eigenvalues. VL < VU.
   *>          Not referenced if RANGE = 'A' or 'I'.
   *> \endverbatim
   *>
   *> \param[in] IL
   *> \verbatim
   *>          IL is INTEGER
   *>
   *>          If RANGE='I', the index of the
   *>          smallest eigenvalue to be returned.
   *>          1 <= IL <= IU <= N, if N > 0.
   *>          Not referenced if RANGE = 'A' or 'V'.
   *> \endverbatim
   *>
   *> \param[in] IU
   *> \verbatim
   *>          IU is INTEGER
   *>
   *>          If RANGE='I', the index of the
   *>          largest eigenvalue to be returned.
   *>          1 <= IL <= IU <= N, if N > 0.
   *>          Not referenced if RANGE = 'A' or 'V'.
   *> \endverbatim
   *>
   *> \param[out] M
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension (N)
   *>          The first M elements contain the selected eigenvalues in
   *>          ascending order.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is COMPLEX*16 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.
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          LDZ is INTEGER
   *>          The leading dimension of the array Z.  LDZ >= 1, and if
   *>          JOBZ = 'V', then LDZ >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in] NZC
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] ISUPPZ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] TRYRAC
   *> \verbatim
   *>          TRYRAC is 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 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
   *>          techniques.
   *>          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
   *>          does not define its eigenvalues to high relative accuracy.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (LWORK)
   *>          On exit, if INFO = 0, WORK(1) returns the optimal
   *>          (and minimal) LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is 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.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (LIWORK)
   *>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
   *> \endverbatim
   *>
   *> \param[in] LIWORK
   *> \verbatim
   *>          LIWORK is 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.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          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 DLARRE,
   *>                if INFO = 2X, internal error in ZLARRV.
   *>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
   *>                the nonzero error code returned by DLARRE or
   *>                ZLARRV, respectively.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \date June 2016
   *
   *> \ingroup complex16OTHERcomputational
   *
   *> \par Contributors:
   *  ==================
   *>
   *> Beresford Parlett, University of California, Berkeley, USA \n
   *> Jim Demmel, University of California, Berkeley, USA \n
   *> Inderjit Dhillon, University of Texas, Austin, USA \n
   *> Osni Marques, LBNL/NERSC, USA \n
   *> Christof Voemel, University of California, Berkeley, USA \n
   *
   *  =====================================================================
       SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,        SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
      $                   M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,       $                   M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
      $                   IWORK, LIWORK, INFO )       $                   IWORK, LIWORK, INFO )
       IMPLICIT NONE  
 *  
 *  -- LAPACK computational routine (version 3.2.1)                    --  
 *  
 *  -- April 2009                                                      --  
 *  *
   *  -- LAPACK computational routine (version 3.7.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
   *     June 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBZ, RANGE        CHARACTER          JOBZ, RANGE
Line 22 Line 355
       COMPLEX*16         Z( LDZ, * )        COMPLEX*16         Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZSTEMR 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  
 *  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 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.ZSTEMR 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 \phy_1}, ... , e^{i \phy_{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,  
 *  ZSTEMR accepts complex workspace to facilitate interoperability  
 *  with ZUNMTR or ZUPMTR.  
 *  
 *  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) DOUBLE PRECISION array, dimension (N)  
 *          On entry, the N diagonal elements of the tridiagonal matrix  
 *          T. On exit, D is overwritten.  
 *  
 *  E       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION  
 *  VU      (input) DOUBLE PRECISION  
 *          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) DOUBLE PRECISION array, dimension (N)  
 *          The first M elements contain the selected eigenvalues in  
 *          ascending order.  
 *  
 *  Z       (output) COMPLEX*16 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     (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  
 *          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.  
 *  
 *  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 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  
 *          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) DOUBLE PRECISION 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 DLARRE,  
 *                if INFO = 2X, internal error in ZLARRV.  
 *                Here, the digit X = ABS( IINFO ) < 10, where IINFO is  
 *                the nonzero error code returned by DLARRE or  
 *                ZLARRV, 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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 293 Line 417
       WU = ZERO        WU = ZERO
       IIL = 0        IIL = 0
       IIU = 0        IIU = 0
         NSPLIT = 0
   
       IF( VALEIG ) THEN        IF( VALEIG ) THEN
 *        We do not reference VL, VU in the cases RANGE = 'I','A'  *        We do not reference VL, VU in the cases RANGE = 'I','A'
Line 410 Line 535
                IF (SN.NE.ZERO) THEN                 IF (SN.NE.ZERO) THEN
                   IF (CS.NE.ZERO) THEN                    IF (CS.NE.ZERO) THEN
                      ISUPPZ(2*M-1) = 1                       ISUPPZ(2*M-1) = 1
                      ISUPPZ(2*M-1) = 2                       ISUPPZ(2*M) = 2
                   ELSE                    ELSE
                      ISUPPZ(2*M-1) = 1                       ISUPPZ(2*M-1) = 1
                      ISUPPZ(2*M-1) = 1                       ISUPPZ(2*M) = 1
                   END IF                    END IF
                ELSE                 ELSE
                   ISUPPZ(2*M-1) = 2                    ISUPPZ(2*M-1) = 2
Line 434 Line 559
                IF (SN.NE.ZERO) THEN                 IF (SN.NE.ZERO) THEN
                   IF (CS.NE.ZERO) THEN                    IF (CS.NE.ZERO) THEN
                      ISUPPZ(2*M-1) = 1                       ISUPPZ(2*M-1) = 1
                      ISUPPZ(2*M-1) = 2                       ISUPPZ(2*M) = 2
                   ELSE                    ELSE
                      ISUPPZ(2*M-1) = 1                       ISUPPZ(2*M-1) = 1
                      ISUPPZ(2*M-1) = 1                       ISUPPZ(2*M) = 1
                   END IF                    END IF
                ELSE                 ELSE
                   ISUPPZ(2*M-1) = 2                    ISUPPZ(2*M-1) = 2
Line 445 Line 570
                END IF                 END IF
             ENDIF              ENDIF
          ENDIF           ENDIF
          RETURN        ELSE
       END IF  
   
 *     Continue with general N  *        Continue with general N
   
       INDGRS = 1           INDGRS = 1
       INDERR = 2*N + 1           INDERR = 2*N + 1
       INDGP = 3*N + 1           INDGP = 3*N + 1
       INDD = 4*N + 1           INDD = 4*N + 1
       INDE2 = 5*N + 1           INDE2 = 5*N + 1
       INDWRK = 6*N + 1           INDWRK = 6*N + 1
 *  *
       IINSPL = 1           IINSPL = 1
       IINDBL = N + 1           IINDBL = N + 1
       IINDW = 2*N + 1           IINDW = 2*N + 1
       IINDWK = 3*N + 1           IINDWK = 3*N + 1
 *  *
 *     Scale matrix to allowable range, if necessary.  *        Scale matrix to allowable range, if necessary.
 *     The allowable range is related to the PIVMIN parameter; see the  *        The allowable range is related to the PIVMIN parameter; see the
 *     comments in DLARRD.  The preference for scaling small values  *        comments in DLARRD.  The preference for scaling small values
 *     up is heuristic; we expect users' matrices not to be close to the  *        up is heuristic; we expect users' matrices not to be close to the
 *     RMAX threshold.  *        RMAX threshold.
 *  *
       SCALE = ONE           SCALE = ONE
       TNRM = DLANST( 'M', N, D, E )           TNRM = DLANST( 'M', N, D, E )
       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN           IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
          SCALE = RMIN / TNRM              SCALE = RMIN / TNRM
       ELSE IF( TNRM.GT.RMAX ) THEN           ELSE IF( TNRM.GT.RMAX ) THEN
          SCALE = RMAX / TNRM              SCALE = RMAX / TNRM
       END IF           END IF
       IF( SCALE.NE.ONE ) THEN           IF( SCALE.NE.ONE ) THEN
          CALL DSCAL( N, SCALE, D, 1 )              CALL DSCAL( N, SCALE, D, 1 )
          CALL DSCAL( N-1, SCALE, E, 1 )              CALL DSCAL( N-1, SCALE, E, 1 )
          TNRM = TNRM*SCALE              TNRM = TNRM*SCALE
          IF( VALEIG ) THEN              IF( VALEIG ) THEN
 *           If eigenvalues in interval have to be found,  *              If eigenvalues in interval have to be found,
 *           scale (WL, WU] accordingly  *              scale (WL, WU] accordingly
             WL = WL*SCALE                 WL = WL*SCALE
             WU = WU*SCALE                 WU = WU*SCALE
          ENDIF              ENDIF
       END IF           END IF
 *  *
 *     Compute the desired eigenvalues of the tridiagonal after splitting  *        Compute the desired eigenvalues of the tridiagonal after splitting
 *     into smaller subblocks if the corresponding off-diagonal elements  *        into smaller subblocks if the corresponding off-diagonal elements
 *     are small  *        are small
 *     THRESH is the splitting parameter for DLARRE  *        THRESH is the splitting parameter for DLARRE
 *     A negative THRESH forces the old splitting criterion based on the  *        A negative THRESH forces the old splitting criterion based on the
 *     size of the off-diagonal. A positive THRESH switches to splitting  *        size of the off-diagonal. A positive THRESH switches to splitting
 *     which preserves relative accuracy.  *        which preserves relative accuracy.
 *  *
       IF( TRYRAC ) THEN           IF( TRYRAC ) THEN
 *        Test whether the matrix warrants the more expensive relative approach.  *           Test whether the matrix warrants the more expensive relative approach.
          CALL DLARRR( N, D, E, IINFO )              CALL DLARRR( N, D, E, IINFO )
       ELSE           ELSE
 *        The user does not care about relative accurately eigenvalues  *           The user does not care about relative accurately eigenvalues
          IINFO = -1              IINFO = -1
       ENDIF           ENDIF
 *     Set the splitting criterion  *        Set the splitting criterion
       IF (IINFO.EQ.0) THEN           IF (IINFO.EQ.0) THEN
          THRESH = EPS              THRESH = EPS
       ELSE           ELSE
          THRESH = -EPS              THRESH = -EPS
 *        relative accuracy is desired but T does not guarantee it  *           relative accuracy is desired but T does not guarantee it
          TRYRAC = .FALSE.              TRYRAC = .FALSE.
       ENDIF           ENDIF
 *  *
       IF( TRYRAC ) THEN           IF( TRYRAC ) THEN
 *        Copy original diagonal, needed to guarantee relative accuracy  *           Copy original diagonal, needed to guarantee relative accuracy
          CALL DCOPY(N,D,1,WORK(INDD),1)              CALL DCOPY(N,D,1,WORK(INDD),1)
       ENDIF           ENDIF
 *     Store the squares of the offdiagonal values of T  *        Store the squares of the offdiagonal values of T
       DO 5 J = 1, N-1           DO 5 J = 1, N-1
          WORK( INDE2+J-1 ) = E(J)**2              WORK( INDE2+J-1 ) = E(J)**2
  5    CONTINUE   5    CONTINUE
   
 *     Set the tolerance parameters for bisection  *        Set the tolerance parameters for bisection
       IF( .NOT.WANTZ ) THEN           IF( .NOT.WANTZ ) THEN
 *        DLARRE computes the eigenvalues to full precision.  *           DLARRE computes the eigenvalues to full precision.
          RTOL1 = FOUR * EPS              RTOL1 = FOUR * EPS
          RTOL2 = FOUR * EPS              RTOL2 = FOUR * EPS
       ELSE           ELSE
 *        DLARRE computes the eigenvalues to less than full precision.  *           DLARRE computes the eigenvalues to less than full precision.
 *        ZLARRV will refine the eigenvalue approximations, and we only  *           ZLARRV will refine the eigenvalue approximations, and we only
 *        need less accurate initial bisection in DLARRE.  *           need less accurate initial bisection in DLARRE.
 *        Note: these settings do only affect the subset case and DLARRE  *           Note: these settings do only affect the subset case and DLARRE
          RTOL1 = SQRT(EPS)              RTOL1 = SQRT(EPS)
          RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )              RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
       ENDIF           ENDIF
       CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,           CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
      $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,       $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
      $             IWORK( IINSPL ), M, W, WORK( INDERR ),       $             IWORK( IINSPL ), M, W, WORK( INDERR ),
      $             WORK( INDGP ), IWORK( IINDBL ),       $             WORK( INDGP ), IWORK( IINDBL ),
      $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,       $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
      $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )       $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )
       IF( IINFO.NE.0 ) THEN           IF( IINFO.NE.0 ) THEN
          INFO = 10 + ABS( IINFO )              INFO = 10 + ABS( IINFO )
          RETURN              RETURN
       END IF           END IF
 *     Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired  *        Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
 *     part of the spectrum. All desired eigenvalues are contained in  *        part of the spectrum. All desired eigenvalues are contained in
 *     (WL,WU]  *        (WL,WU]
   
   
       IF( WANTZ ) THEN           IF( WANTZ ) THEN
 *  *
 *        Compute the desired eigenvectors corresponding to the computed  *           Compute the desired eigenvectors corresponding to the computed
 *        eigenvalues  *           eigenvalues
 *  *
          CALL ZLARRV( N, WL, WU, D, E,              CALL ZLARRV( N, WL, WU, D, E,
      $                PIVMIN, IWORK( IINSPL ), M,       $                PIVMIN, IWORK( IINSPL ), M,
      $                1, M, MINRGP, RTOL1, RTOL2,       $                1, M, MINRGP, RTOL1, RTOL2,
      $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),       $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
      $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,       $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
      $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )       $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
          IF( IINFO.NE.0 ) THEN              IF( IINFO.NE.0 ) THEN
             INFO = 20 + ABS( IINFO )                 INFO = 20 + ABS( IINFO )
             RETURN                 RETURN
          END IF              END IF
       ELSE           ELSE
 *        DLARRE computes eigenvalues of the (shifted) root representation  *           DLARRE computes eigenvalues of the (shifted) root representation
 *        ZLARRV returns the eigenvalues of the unshifted matrix.  *           ZLARRV returns the eigenvalues of the unshifted matrix.
 *        However, if the eigenvectors are not desired by the user, we need  *           However, if the eigenvectors are not desired by the user, we need
 *        to apply the corresponding shifts from DLARRE to obtain the  *           to apply the corresponding shifts from DLARRE to obtain the
 *        eigenvalues of the original matrix.  *           eigenvalues of the original matrix.
          DO 20 J = 1, M              DO 20 J = 1, M
             ITMP = IWORK( IINDBL+J-1 )                 ITMP = IWORK( IINDBL+J-1 )
             W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )                 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
  20      CONTINUE   20      CONTINUE
       END IF           END IF
 *  *
   
       IF ( TRYRAC ) THEN           IF ( TRYRAC ) THEN
 *        Refine computed eigenvalues so that they are relatively accurate  *           Refine computed eigenvalues so that they are relatively accurate
 *        with respect to the original matrix T.  *           with respect to the original matrix T.
          IBEGIN = 1              IBEGIN = 1
          WBEGIN = 1              WBEGIN = 1
          DO 39  JBLK = 1, IWORK( IINDBL+M-1 )              DO 39  JBLK = 1, IWORK( IINDBL+M-1 )
             IEND = IWORK( IINSPL+JBLK-1 )                 IEND = IWORK( IINSPL+JBLK-1 )
             IN = IEND - IBEGIN + 1                 IN = IEND - IBEGIN + 1
             WEND = WBEGIN - 1                 WEND = WBEGIN - 1
 *           check if any eigenvalues have to be refined in this block  *              check if any eigenvalues have to be refined in this block
  36         CONTINUE   36         CONTINUE
             IF( WEND.LT.M ) THEN                 IF( WEND.LT.M ) THEN
                IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN                    IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
                   WEND = WEND + 1                       WEND = WEND + 1
                   GO TO 36                       GO TO 36
                     END IF
                  END IF
                  IF( WEND.LT.WBEGIN ) THEN
                     IBEGIN = IEND + 1
                     GO TO 39
                END IF                 END IF
             END IF  
             IF( WEND.LT.WBEGIN ) THEN  
                IBEGIN = IEND + 1  
                GO TO 39  
             END IF  
   
             OFFSET = IWORK(IINDW+WBEGIN-1)-1                 OFFSET = IWORK(IINDW+WBEGIN-1)-1
             IFIRST = IWORK(IINDW+WBEGIN-1)                 IFIRST = IWORK(IINDW+WBEGIN-1)
             ILAST = IWORK(IINDW+WEND-1)                 ILAST = IWORK(IINDW+WEND-1)
             RTOL2 = FOUR * EPS                 RTOL2 = FOUR * EPS
             CALL DLARRJ( IN,                 CALL DLARRJ( IN,
      $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),       $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
      $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),       $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
      $                   WORK( INDERR+WBEGIN-1 ),       $                   WORK( INDERR+WBEGIN-1 ),
      $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,       $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
      $                   TNRM, IINFO )       $                   TNRM, IINFO )
             IBEGIN = IEND + 1                 IBEGIN = IEND + 1
             WBEGIN = WEND + 1                 WBEGIN = WEND + 1
  39      CONTINUE   39      CONTINUE
       ENDIF           ENDIF
 *  *
 *     If matrix was scaled, then rescale eigenvalues appropriately.  *        If matrix was scaled, then rescale eigenvalues appropriately.
 *  *
       IF( SCALE.NE.ONE ) THEN           IF( SCALE.NE.ONE ) THEN
          CALL DSCAL( M, ONE / SCALE, W, 1 )              CALL DSCAL( M, ONE / SCALE, W, 1 )
            END IF
       END IF        END IF
 *  *
 *     If eigenvalues are not in increasing order, then sort them,  *     If eigenvalues are not in increasing order, then sort them,
 *     possibly along with eigenvectors.  *     possibly along with eigenvectors.
 *  *
       IF( NSPLIT.GT.1 ) THEN        IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN
          IF( .NOT. WANTZ ) THEN           IF( .NOT. WANTZ ) THEN
             CALL DLASRT( 'I', M, W, IINFO )              CALL DLASRT( 'I', M, W, IINFO )
             IF( IINFO.NE.0 ) THEN              IF( IINFO.NE.0 ) THEN
Removed from v.1.7  
changed lines
  Added in v.1.18

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