OpenBLAS/lapack-netlib/SRC/sstevr.f

*> \brief <b> SSTEVR 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/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SSTEVR( 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
*       REAL               ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * ), IWORK( * )
*       REAL               D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> 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 SSTEMR to compute the
*> eigenspectrum using Relatively Robust Representations.  SSTEMR
*> 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 : SSTEVR calls SSTEMR 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 SSTEMR 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, SSTEBZ and
*>          SSTEIN 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 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.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is REAL 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 REAL
*>          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 REAL
*>          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; IL = 1 and IU = 0 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; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is 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. 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 REAL array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is 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.
*> \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 REAL 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 >= 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 >= 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 June 2016
*
*> \ingroup realOTHEReigen
*
*> \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
*>     Jason Riedy, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>
*  =====================================================================
      SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
     $                   M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
     $                   LIWORK, INFO )
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE
      INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
      REAL               ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * ), IWORK( * )
      REAL               D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
     $                   TRYRAC
      CHARACTER          ORDER
      INTEGER            I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
     $                   INDIWO, ISCALE, J, JJ, LIWMIN, LWMIN, NSPLIT
      REAL               BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
     $                   TMP1, TNRM, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      REAL               SLAMCH, SLANST
      EXTERNAL           LSAME, ILAENV, SLAMCH, SLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           SCOPY, SSCAL, SSTEBZ, SSTEMR, SSTEIN, SSTERF,
     $                   SSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*
*     Test the input parameters.
*
      IEEEOK = ILAENV( 10, 'SSTEVR', 'N', 1, 2, 3, 4 )
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
      LWMIN = MAX( 1, 20*N )
      LIWMIN = MAX(1, 10*N )
*
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -7
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -8
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -9
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -14
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -17
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -19
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SSTEVR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = D( 1 )
         ELSE
            IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
               M = 1
               W( 1 ) = D( 1 )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = SLAMCH( 'Safe minimum' )
      EPS = SLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      IF( VALEIG ) THEN
         VLL = VL
         VUU = VU
      END IF
*
      TNRM = SLANST( 'M', N, D, E )
      IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / TNRM
      ELSE IF( TNRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / TNRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL SSCAL( N, SIGMA, D, 1 )
         CALL SSCAL( N-1, SIGMA, E( 1 ), 1 )
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF

*     Initialize indices into workspaces.  Note: These indices are used only
*     if SSTERF or SSTEMR fail.

*     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
*     stores the block indices of each of the M<=N eigenvalues.
      INDIBL = 1
*     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
*     stores the starting and finishing indices of each block.
      INDISP = INDIBL + N
*     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
*     that corresponding to eigenvectors that fail to converge in
*     SSTEIN.  This information is discarded; if any fail, the driver
*     returns INFO > 0.
      INDIFL = INDISP + N
*     INDIWO is the offset of the remaining integer workspace.
      INDIWO = INDISP + N
*
*     If all eigenvalues are desired, then
*     call SSTERF or SSTEMR.  If this fails for some eigenvalue, then
*     try SSTEBZ.
*
*
      TEST = .FALSE.
      IF( INDEIG ) THEN
         IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
         CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
         IF( .NOT.WANTZ ) THEN
            CALL SCOPY( N, D, 1, W, 1 )
            CALL SSTERF( N, W, WORK, INFO )
         ELSE
            CALL SCOPY( N, D, 1, WORK( N+1 ), 1 )
            IF (ABSTOL .LE. TWO*N*EPS) THEN
               TRYRAC = .TRUE.
            ELSE
               TRYRAC = .FALSE.
            END IF
            CALL SSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
     $                   IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
     $                   WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
*
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 10
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF

      CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
     $             NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
     $                Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   10 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 30 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 20 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   20       CONTINUE
*
            IF( I.NE.0 ) THEN
               W( I ) = W( J )
               W( J ) = TMP1
               CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
            END IF
   30    CONTINUE
      END IF
*
*      Causes problems with tests 19 & 20:
*      IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
*
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of SSTEVR
*
      END