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Line 1 Line 1
   *> \brief <b> DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DSTEVR + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevr.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevr.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevr.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
   *                          M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
   *                          LIWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ, RANGE
   *       INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
   *       DOUBLE PRECISION   ABSTOL, VL, VU
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            ISUPPZ( * ), IWORK( * )
   *       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSTEVR 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, DSTEVR calls DSTEMR to compute the
   *> eigenspectrum using Relatively Robust Representations.  DSTEMR
   *> 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
   *> orthogonalization 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 eigenvalue/eigenvector problem", by Inderjit Dhillon,
   *> Computer Science Division Technical Report No. UCB//CSD-97-971,
   *> UC Berkeley, May 1997.
   *>
   *>
   *> Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
   *> on machines which conform to the ieee-754 floating point standard.
   *> DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
   *> when partial spectrum requests are made.
   *>
   *> Normal execution of DSTEMR 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.
   *> \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.
   *>          For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
   *>          DSTEIN are called
   *> \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
   *>          A.
   *>          On exit, D may be multiplied by a constant factor chosen
   *>          to avoid over/underflow in computing the eigenvalues.
   *> \endverbatim
   *>
   *> \param[in,out] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension (max(1,N-1))
   *>          On entry, the (n-1) subdiagonal elements of the tridiagonal
   *>          matrix A in elements 1 to N-1 of E.
   *>          On exit, E may be multiplied by a constant factor chosen
   *>          to avoid over/underflow in computing the eigenvalues.
   *> \endverbatim
   *>
   *> \param[in] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION
   *> \endverbatim
   *>
   *> \param[in] VU
   *> \verbatim
   *>          VU is 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'.
   *> \endverbatim
   *>
   *> \param[in] IL
   *> \verbatim
   *>          IL is INTEGER
   *> \endverbatim
   *>
   *> \param[in] IU
   *> \verbatim
   *>          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; IL = 1 and IU = 0 if N = 0.
   *>          Not referenced if RANGE = 'A' or 'V'.
   *> \endverbatim
   *>
   *> \param[in] ABSTOL
   *> \verbatim
   *>          ABSTOL is DOUBLE PRECISION
   *>          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
   *>          DLAMCH( '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. 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.
   *> \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 DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          LDZ is INTEGER
   *>          The leading dimension of the array Z.  LDZ >= 1, and if
   *>          JOBZ = 'V', LDZ >= max(1,N).
   *> \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 eigenvector
   *>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
   *>          ISUPPZ( 2*i ).
   *>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,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(1,20*N).
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal sizes of the WORK and IWORK
   *>          arrays, returns these values as the first entries of the WORK
   *>          and IWORK arrays, and no error message related to LWORK or
   *>          LIWORK is issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
   *>          On exit, if INFO = 0, IWORK(1) returns the optimal (and
   *>          minimal) LIWORK.
   *> \endverbatim
   *>
   *> \param[in] LIWORK
   *> \verbatim
   *>          LIWORK is 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 sizes of the WORK and
   *>          IWORK arrays, returns these values as the first entries of
   *>          the WORK and IWORK arrays, and no error message related to
   *>          LWORK or LIWORK is issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *>          > 0:  Internal error
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHEReigen
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Inderjit Dhillon, IBM Almaden, USA \n
   *>     Osni Marques, LBNL/NERSC, USA \n
   *>     Ken Stanley, Computer Science Division, University of
   *>       California at Berkeley, USA \n
   *>
   *  =====================================================================
       SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,        SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
      $                   M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,       $                   M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
      $                   LIWORK, INFO )       $                   LIWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- 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..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBZ, RANGE        CHARACTER          JOBZ, RANGE
Line 17 Line 312
       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )        DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSTEVR 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, DSTEVR calls DSTEMR to compute the  
 *  eigenspectrum using Relatively Robust Representations.  DSTEMR  
 *  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  
 *  orthogonalization 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 eigenvalue/eigenvector problem", by Inderjit Dhillon,  
 *  Computer Science Division Technical Report No. UCB//CSD-97-971,  
 *  UC Berkeley, May 1997.  
 *  
 *  
 *  Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested  
 *  on machines which conform to the ieee-754 floating point standard.  
 *  DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and  
 *  when partial spectrum requests are made.  
 *  
 *  Normal execution of DSTEMR 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.  
 ********** For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and  
 ********** DSTEIN are called  
 *  
 *  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  
 *          A.  
 *          On exit, D may be multiplied by a constant factor chosen  
 *          to avoid over/underflow in computing the eigenvalues.  
 *  
 *  E       (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))  
 *          On entry, the (n-1) subdiagonal elements of the tridiagonal  
 *          matrix A in elements 1 to N-1 of E.  
 *          On exit, E may be multiplied by a constant factor chosen  
 *          to avoid over/underflow in computing the eigenvalues.  
 *  
 *  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; IL = 1 and IU = 0 if N = 0.  
 *          Not referenced if RANGE = 'A' or 'V'.  
 *  
 *  ABSTOL  (input) DOUBLE PRECISION  
 *          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  
 *          DLAMCH( '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. 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) DOUBLE PRECISION array, dimension (N)  
 *          The first M elements contain the selected eigenvalues in  
 *          ascending order.  
 *  
 *  Z       (output) DOUBLE PRECISION 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 ).  
 ********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1  
 *  
 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,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(1,20*N).  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal sizes of the WORK and IWORK  
 *          arrays, returns these values as the first entries of the WORK  
 *          and IWORK arrays, and no error message related to LWORK or  
 *          LIWORK is issued by XERBLA.  
 *  
 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))  
 *          On exit, if INFO = 0, IWORK(1) returns the optimal (and  
 *          minimal) 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 sizes of the WORK and  
 *          IWORK arrays, returns these values as the first entries of  
 *          the WORK and IWORK arrays, and no error message related to  
 *          LWORK or 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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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